0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : tptp2X_and_run_prover9 %d %s 0.02/0.23 % Computer : n004.star.cs.uiowa.edu 0.02/0.23 % Model : x86_64 x86_64 0.02/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.02/0.23 % Memory : 32218.625MB 0.02/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.02/0.24 % CPULimit : 300 0.02/0.24 % DateTime : Sat Jul 14 05:57:54 CDT 2018 0.02/0.24 % CPUTime : 1.39/1.61 ============================== Prover9 =============================== 1.39/1.61 Prover9 (32) version 2009-11A, November 2009. 1.39/1.61 Process 20901 was started by sandbox2 on n004.star.cs.uiowa.edu, 1.39/1.61 Sat Jul 14 05:57:56 2018 1.39/1.61 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_20852_n004.star.cs.uiowa.edu". 1.39/1.61 ============================== end of head =========================== 1.39/1.61 1.39/1.61 ============================== INPUT ================================= 1.39/1.61 1.39/1.61 % Reading from file /tmp/Prover9_20852_n004.star.cs.uiowa.edu 1.39/1.61 1.39/1.61 set(prolog_style_variables). 1.39/1.61 set(auto2). 1.39/1.61 % set(auto2) -> set(auto). 1.39/1.61 % set(auto) -> set(auto_inference). 1.39/1.61 % set(auto) -> set(auto_setup). 1.39/1.61 % set(auto_setup) -> set(predicate_elim). 1.39/1.61 % set(auto_setup) -> assign(eq_defs, unfold). 1.39/1.61 % set(auto) -> set(auto_limits). 1.39/1.61 % set(auto_limits) -> assign(max_weight, "100.000"). 1.39/1.61 % set(auto_limits) -> assign(sos_limit, 20000). 1.39/1.61 % set(auto) -> set(auto_denials). 1.39/1.61 % set(auto) -> set(auto_process). 1.39/1.61 % set(auto2) -> assign(new_constants, 1). 1.39/1.61 % set(auto2) -> assign(fold_denial_max, 3). 1.39/1.61 % set(auto2) -> assign(max_weight, "200.000"). 1.39/1.61 % set(auto2) -> assign(max_hours, 1). 1.39/1.61 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.39/1.61 % set(auto2) -> assign(max_seconds, 0). 1.39/1.61 % set(auto2) -> assign(max_minutes, 5). 1.39/1.61 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.39/1.61 % set(auto2) -> set(sort_initial_sos). 1.39/1.61 % set(auto2) -> assign(sos_limit, -1). 1.39/1.61 % set(auto2) -> assign(lrs_ticks, 3000). 1.39/1.61 % set(auto2) -> assign(max_megs, 400). 1.39/1.61 % set(auto2) -> assign(stats, some). 1.39/1.61 % set(auto2) -> clear(echo_input). 1.39/1.61 % set(auto2) -> set(quiet). 1.39/1.61 % set(auto2) -> clear(print_initial_clauses). 1.39/1.61 % set(auto2) -> clear(print_given). 1.39/1.61 assign(lrs_ticks,-1). 1.39/1.61 assign(sos_limit,10000). 1.39/1.61 assign(order,kbo). 1.39/1.61 set(lex_order_vars). 1.39/1.61 clear(print_given). 1.39/1.61 1.39/1.61 % formulas(sos). % not echoed (1258 formulas) 1.39/1.61 1.39/1.61 ============================== end of input ========================== 1.39/1.61 1.39/1.61 % From the command line: assign(max_seconds, 300). 1.39/1.61 1.39/1.61 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.39/1.61 1.39/1.61 % Formulas that are not ordinary clauses: 1.39/1.61 1 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m)) # label(fact_Zero__not__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 2 (all V_b_H all V_b all V_a_H all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c) -> (c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_H),V_b_H),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c))))) # label(fact_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 3 (all V_k_2 all V_X_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_SEQ_OBseq(T_a,V_X_2) -> c_SEQ_OBseq(T_a,hAPP(c_COMBB(tc_Nat_Onat,T_a,tc_Nat_Onat,V_X_2),c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2)))))) # label(fact_Bseq__ignore__initial__segment) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 4 (all V_n all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)))))))) # label(fact_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 5 (all V_y_2 all V_x_2 all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),V_y_2)) & -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))))) # label(fact_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 6 (all V_q all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 7 (all V_q all V_b all V_r all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r))))))) # label(fact_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 8 (all V_y all V_x all T_a (class_Rings_Olinordered__ring(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_y))),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 9 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),hAPP(c_Nat_OSuc,V_n)))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)))))) # label(fact_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 10 (all V_c all V_b all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c)) # label(fact_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 11 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))))) # label(fact_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 12 (all T_1 (class_Groups_Ocancel__comm__monoid__add(T_1) -> class_Groups_Ocancel__ab__semigroup__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Ocancel__ab__semigroup__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 13 (all V_q all V_p all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Polynomial_Osmult(T_a,V_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),V_q)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Osmult(T_a,V_a,V_q)))) # label(fact_smult__add__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 14 (all V_m all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_m))))) # label(fact_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 15 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_z)))))) # label(fact_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 16 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Olinordered__semiring__strict(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Olinordered__semiring__strict) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 17 (all V_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))) # label(fact_le0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 18 (all V_y all V_x (hAPP(c_Nat_OSuc,V_x) = hAPP(c_Nat_OSuc,V_y) -> V_x = V_y)) # label(fact_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 19 (all V_c all V_b all V_a all T_a (class_Rings_Oordered__comm__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 20 (all V_p all V_b all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Polynomial_Osmult(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_p) = c_Polynomial_Osmult(T_a,V_a,c_Polynomial_Osmult(T_a,V_b,V_p)))) # label(fact_smult__smult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 21 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Divides_Odiv__class_Odiv(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_div__by__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 22 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x_2),V_y_2)) | V_y_2 = V_x_2 <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2),V_y_2)))) # label(fact_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 23 (all V_X_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_SEQ_OBseq(T_a,V_X_2) <-> (exists B_K ((all B_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(V_X_2,B_n))),B_K))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K))))))) # label(fact_Bseq__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 24 (all T_1 (class_Fields_Ofield(T_1) -> class_Divides_Osemiring__div(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Divides_Osemiring__div) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 25 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 26 (all T_1 (class_Rings_Ocomm__semiring__1(T_1) -> class_Rings_Ozero__neq__one(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Ozero__neq__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 27 (all V_b all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_div__nonneg__neg__le0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 28 (all V_z V_z = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) # label(fact_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 29 (all V_c all V_b all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c)) # label(fact_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 30 (all V_z (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z))))) # label(fact_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 31 (all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = c_Polynomial_Osmult(T_a,c_Groups_Ozero__class_Ozero(T_a),V_p))) # label(fact_smult__0__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 32 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> V_m = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n),V_n))) # label(fact_div__mult__self__is__m) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 33 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> (hAPP(c_Nat_OSuc,V_m) != V_n -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))))) # label(fact_Suc__lessI) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 34 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 35 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)))) # label(fact_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 36 (exists B_q (c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(tc_Complex_Ocomplex,B_q) = c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(tc_Complex_Ocomplex,v_pa____) & (all B_x hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),v_c____),B_x)) = hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,B_q),B_x)))) # label(fact__096EX_Aq_O_Apsize_Aq_A_061_Apsize_Ap_A_G_A_IALL_Ax_O_Apoly_Aq_Ax_A_061_Apoly_Ap_A_Ic_A_L_Ax_J_J_096) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 37 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_y_2)) -> (V_y_2 = V_x_2 <-> c_Transcendental_Oln(V_y_2) = c_Transcendental_Oln(V_x_2))))) # label(fact_ln__inj__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 38 (all V_X_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_SEQ_OBseq(T_a,V_X_2) -> (exists B_K ((all B_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(V_X_2,B_n))),B_K))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K))))))) # label(fact_BseqD) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 39 (all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),hAPP(c_Nat_OSuc,V_n))) -> (-hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) -> hAPP(c_Nat_OSuc,V_n) = V_m))) # label(fact_le__SucE) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 40 (all V_n_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_n_2)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_real__of__nat__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 41 (all V_n all V_m (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = V_m -> V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 42 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 43 (all V_x_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(T_a,V_x_2))) <-> c_Groups_Ozero__class_Ozero(T_a) != V_x_2))) # label(fact_zero__less__norm__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 44 (all V_y all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_y))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),c_RealVector_Onorm__class_Onorm(T_a,V_y)))))) # label(fact_norm__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 45 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Divides_Odiv__class_Odiv(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a))) # label(fact_div__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 46 (all V_n_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_x_2)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 47 (all V_q all V_r all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_r),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q)) = V_a -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_q),c_Groups_Oone__class_Oone(tc_Int_Oint))))))) # label(fact_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 48 (all T_1 (class_Groups_Ocancel__comm__monoid__add(T_1) -> class_Groups_Ocancel__comm__monoid__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Ocancel__comm__monoid__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 49 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))) # label(fact_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 50 (all V_n all V_p all T_a (class_Groups_Ozero(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p)),V_n)) -> (c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Polynomial_Ocoeff(T_a,V_p),V_n) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_p)),V_n)) | V_p = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)))))) # label(fact_eq__zero__or__degree__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 51 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_x)))) # label(fact_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 52 (all V_p all V_a all T_a (class_Rings_Oidom(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)) = c_Polynomial_Odegree(T_a,V_p)) & (c_Groups_Ozero__class_Ozero(T_a) = V_a -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p))))) # label(fact_degree__smult__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 53 (all V_w all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_w),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z),V_w)) # label(fact_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 54 (all V_p all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Polynomial_Osmult(T_a,c_Groups_Oone__class_Oone(T_a),V_p) = V_p)) # label(fact_smult__1__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 55 (all V_d_2 all V_b_2 all V_ca_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_ca_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b_2),V_a_2)),V_e_2)),V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2)),V_ca_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2)),V_d_2)))))) # label(fact_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 56 (all V_n all V_k all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_n)))) # label(fact_add__leD2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 57 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b)))) # label(fact_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 58 (all V_n_2 ((exists B_m V_n_2 = hAPP(c_Nat_OSuc,B_m)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)))) # label(fact_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 59 (all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) -> (V_n != V_m -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))))) # label(fact_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 60 (all V_y_2 all V_x_2 all V_b_2 all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b_2),V_x_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b_2),V_y_2))))))) # label(fact_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 61 (all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_n) # label(fact_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 62 (all V_y all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y)))))) # label(fact_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.61 63 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 64 (all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a)))))) # label(fact_zero__less__two) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 65 (all V_b all V_a all V_c all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b))))) # label(fact_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 66 (all V_b all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_a)) -> c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_div__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 67 (all V_p all V_b all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Osmult(T_a,V_b,V_p)) = c_Polynomial_Osmult(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_p))) # label(fact_smult__add__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 68 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))))))) # label(fact_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 69 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> V_a = c_Divides_Odiv__class_Odiv(T_a,V_a,c_Groups_Oone__class_Oone(T_a)))) # label(fact_div__by__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 70 (all V_n all T_a (class_Power_Opower(T_a) & class_Rings_Osemiring__0(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Nat_OSuc,V_n)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 71 (all V_n_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) # label(fact_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 72 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 73 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a),V_b))) # label(fact_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 74 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 75 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Oone__class_Oone(T_a) != c_Groups_Ozero__class_Ozero(T_a))) # label(fact_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 76 (all V_n all T_a (class_Groups_Ozero(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Polynomial_Ocoeff(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))),V_n))) # label(fact_coeff__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 77 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Groups_Olinordered__ab__group__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 78 (all V_n_2 all V_m_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 79 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_m)) -> c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n) = hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n))))) # label(fact_le__div__geq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 80 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 81 (all V_i all V_j all V_k (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_j)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k))) # label(fact_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 82 (all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m) # label(fact_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 83 (all T_a (class_Groups_Ozero(T_a) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Polynomial_Odegree(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))))) # label(fact_degree__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 84 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c))) # label(fact_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 85 (all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K ((all B_a all B_b hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),B_a),B_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,B_a)),c_RealVector_Onorm__class_Onorm(T_a,B_b))),B_K)))) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K)))))) # label(fact_mult_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 86 (all V_k all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_x,V_k)),V_x))))) # label(fact_int__div__less__self) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 87 (all V_n_2 all V_P_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (hBOOL(hAPP(hAPP(V_P_2,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_n_2,V_k_2)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2))) <-> (all B_i all B_j (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)) & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) -> hBOOL(hAPP(hAPP(V_P_2,B_i),B_j))))))) # label(fact_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 88 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_m_2)) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Int_Oint) <-> c_Groups_Oone__class_Oone(tc_Int_Oint) = V_m_2 & c_Groups_Oone__class_Oone(tc_Int_Oint) = V_n_2))) # label(fact_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 89 (all V_g_2 all V_f_2 ((all B_x hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x)) -> V_f_2 = V_g_2)) # label(fact_ext) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 90 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),V_d)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 91 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),c_Groups_Ozero__class_Ozero(T_a))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),c_Groups_Ozero__class_Ozero(T_a))) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 92 (all V_b all V_a all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a))))) # label(fact_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 93 (all T_1 (class_Rings_Ocomm__semiring__0(T_1) -> class_Rings_Osemiring__0(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Osemiring__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 94 (all V_b_H all V_b all V_a_H all V_c all V_a all T_a (class_Divides_Oring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c) -> (c_Divides_Odiv__class_Omod(T_a,V_b,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a_H),V_b_H),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c))))) # label(fact_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 95 (all V_ya all V_y all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_ya)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_ya)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_y)),V_ya))) # label(fact_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 96 (all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (c_Rings_Oinverse__class_Oinverse(T_a,V_b) = c_Rings_Oinverse__class_Oinverse(T_a,V_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> V_b = V_a))))) # label(fact_nonzero__inverse__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 97 (all V_a all T_a (class_Rings_Olinordered__ring(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a)),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 98 (all V_q all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Opcompose(T_a,V_p,V_q))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_p)),c_Polynomial_Odegree(T_a,V_q)))))) # label(fact_degree__pcompose__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 99 (all V_x all V_y (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_y)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> (exists B_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,B_n)),V_y)),V_x)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_n))),V_y)))))))) # label(fact_reals__Archimedean4) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 100 (all V_r2 all V_q2 all V_r1 all V_q1 all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q1,V_r1) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q2,V_r2) -> V_r2 = V_r1)))) # label(fact_pdivmod__rel__unique__mod) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 101 (all T_1 (class_Rings_Oidom(T_1) -> class_Rings_Oring__1__no__zero__divisors(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Oring__1__no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 102 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_c -> c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = c_Divides_Odiv__class_Odiv(T_a,V_a,V_b)))) # label(fact_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 103 (all T_a (class_Power_Opower(T_a) -> c_Power_Opower__class_Opower(T_a) = c_Power_Opower_Opower(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Otimes__class_Otimes(T_a)))) # label(fact_power__power__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 104 (all V_x all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(c_Polynomial_Opoly(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))),V_x) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_poly__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 105 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> V_n = V_m))) # label(fact_less__SucE) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 106 (all V_p all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Osmult(T_a,V_a,V_p))),c_Polynomial_Odegree(T_a,V_p))))) # label(fact_degree__smult__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 107 (all T_1 (class_Rings_Ocomm__semiring__0(T_1) -> class_Groups_Oab__semigroup__mult(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Oab__semigroup__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 108 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_add__0__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 109 (all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z)),V_z) != c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 110 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_m)),c_RealDef_Oreal(tc_Nat_Onat,V_n)) = c_RealDef_Oreal(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n))) # label(fact_real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 111 (all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_a)))))) # label(fact_positive__imp__inverse__positive) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 112 (all V_b all V_b_H all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_b_H),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b_H)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b))))))) # label(fact_zdiv__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 113 (all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_m)))) # label(fact_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 114 (all V_m all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_k)) -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),V_k))) # label(fact_Suc__times__mod__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 115 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> V_x != V_y))) # label(fact_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 116 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))))) # label(fact_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 117 (all V_P_2 all V_Q_2 (-hBOOL(V_Q_2) | hBOOL(hAPP(hAPP(c_fimplies,V_P_2),V_Q_2)))) # label(help_c__fimplies__2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 118 (all V_x_2 all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_x_2)),c_Groups_Oone__class_Oone(T_a))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),c_Groups_Ozero__class_Ozero(T_a))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a)),V_x_2))))) # label(fact_inverse__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 119 (all V_n_2 all V_m_2 (V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_add__is__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 120 (all V_n all V_m all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)))))) # label(fact_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 121 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),V_y_2)) -> (V_y_2 = V_x_2 <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2)))))) # label(fact_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 122 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Odiv(T_a,V_a,V_b)),c_Groups_Oone__class_Oone(T_a)) = c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_b)))) # label(fact_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 123 (all V_a all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> c_RComplete_Onatfloor(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),V_x),c_RealDef_Oreal(tc_Nat_Onat,V_a))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_RComplete_Onatfloor(V_x)),V_a))) # label(fact_natfloor__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 124 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_x))))) # label(fact_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 125 (all V_y_2 all V_x_2 all T_a (class_Groups_Ozero(T_a) -> (V_y_2 = V_x_2 <-> c_Polynomial_Ocoeff(T_a,V_y_2) = c_Polynomial_Ocoeff(T_a,V_x_2)))) # label(fact_coeff__inject) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 126 (all V_z all V_y all V_x hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))) # label(fact_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 127 (all V_n_2 all V_m_2 all V_k_2 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2) <-> V_m_2 = V_n_2)) # label(fact_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 128 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)) # label(fact_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 129 (all V_x_2 (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = V_x_2 <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x_2),V_x_2))))) # label(fact_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 130 (all V_a all V_r all V_q all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q,V_r) -> c_Polynomial_Opdivmod__rel(T_a,c_Polynomial_Osmult(T_a,V_a,V_x),V_y,c_Polynomial_Osmult(T_a,V_a,V_q),c_Polynomial_Osmult(T_a,V_a,V_r))))) # label(fact_pdivmod__rel__smult__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 131 (all V_v all V_u all V_y all V_a all V_x all T_a (class_Rings_Olinordered__semiring__1(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_u)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_v)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) = c_Groups_Oone__class_Oone(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y))),V_a))))))))) # label(fact_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 132 (all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_a))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a_2),V_b_2))))) # label(fact_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 133 (all V_n_2 all V_P_2 (-hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) -> (hBOOL(hAPP(V_P_2,V_n_2)) -> (exists B_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_k),V_n_2)) & hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_k),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) & (all B_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_i),B_k)) -> -hBOOL(hAPP(V_P_2,B_i))))))))) # label(fact_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 134 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)) = c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a)))) # label(fact_norm__minus__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 135 (all V_n all V_x (c_RealDef_Oreal(tc_Nat_Onat,c_RComplete_Onatfloor(V_x)) = V_x -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_RComplete_Onatfloor(V_x)),V_n) = c_RComplete_Onatfloor(hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),V_n)))) # label(fact_natfloor__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 136 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 137 (all V_b all V_a all V_c all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 138 (all V_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_a)) -> (exists B_r (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_r)) & hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B_r),V_n) = V_a))))) # label(fact_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 139 (all V_c all V_b all V_a all T_a (class_Groups_Oab__semigroup__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))) # label(fact_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 140 (all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (V_x_2 = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) <-> c_Transcendental_Oln(V_x_2) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))) # label(fact_ln__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 141 (all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Groups_Ozero__class_Ozero(T_a))) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_inverse__negative__imp__negative) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 142 (all V_pb_2 all V_a_2 all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Polynomial_Osmult(T_a,V_a_2,V_pb_2) = c_Polynomial_OAbs__poly(T_a,hAPP(c_COMBB(T_a,T_a,tc_Nat_Onat,hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2)),c_Polynomial_Ocoeff(T_a,V_pb_2))))) # label(fact_smult__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 143 (all V_n hAPP(c_Nat_OSuc,V_n) != V_n) # label(fact_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 144 (all V_m all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),V_j)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_m))))) # label(fact_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 145 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_n))) # label(fact_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 146 (all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z) = V_z) # label(fact_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 147 (all V_m hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_m))))) # label(fact_le__cube) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 148 (all V_q all V_n all V_p all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_p)),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_q)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),V_q))),V_n)))))) # label(fact_degree__add__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 149 (all V_n_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_n_2)) -> (-hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) -> c_Orderings_Oord__class_OLeast(tc_Nat_Onat,V_P_2) = hAPP(c_Nat_OSuc,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Nat_Onat,V_P_2),c_Nat_OSuc)))))) # label(fact_Least__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 150 (all V_z all V_y all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),V_z),V_x)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),V_z),V_y))))) # label(fact_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 151 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)))))) # label(fact_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 152 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n)),V_m))))) # label(fact_div__less__dividend) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 153 (all V_b all V_a all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Rings_Oinverse__class_Oinverse(T_a,V_a))),c_Rings_Oinverse__class_Oinverse(T_a,V_b)))))) # label(fact_inverse__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 154 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))) # label(fact_add__Suc__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 155 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 156 (all V_i_2 all V_j_2 all V_k_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2),V_j_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2)),V_j_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2)))))) # label(fact_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 157 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 158 (all V_ca_2 all V_a_2 all V_b_2 all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_ca_2),V_a_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b_2),V_a_2) <-> V_ca_2 = V_b_2))) # label(fact_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 159 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))))))) # label(fact_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 160 (all V_f_2 all T_b all T_a (c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant(T_a,T_b,V_f_2) <-> (all B_x all B_y hAPP(V_f_2,B_y) = hAPP(V_f_2,B_x)))) # label(fact_constant__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 161 (all V_y all V_x all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_z)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z),V_y)))))) # label(fact_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 162 (all V_c all V_a all V_b all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 163 (all V_Q all V_P all T_b all T_a V_P = hAPP(c_COMBK(T_a,T_b,V_P),V_Q)) # label(help_c__COMBK__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 164 (all V_b_2 all V_a_2 all V_ca_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_ca_2),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b_2),V_a_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_b_2))))))) # label(fact_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 165 (all T_1 (class_Groups_Ocancel__comm__monoid__add(T_1) -> class_Groups_Ocancel__semigroup__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Ocancel__semigroup__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 166 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mod__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 167 (all V_R_2 all V_Q_2 all V_P_2 all T_a all T_c all T_b hAPP(hAPP(V_P_2,V_R_2),V_Q_2) = hAPP(c_COMBC(T_b,T_c,T_a,V_P_2,V_Q_2),V_R_2)) # label(help_c__COMBC__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 168 (all V_r all V_q all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q,V_r) -> c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(T_a),V_x,V_y) = V_q))) # label(fact_div__poly__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 169 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Int_Oring__char__0(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Int_Oring__char__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 170 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_not__less0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 171 (all V_m all V_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_m)) -> V_m = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)))) # label(fact_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 172 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y_2),V_x_2)) | V_y_2 = V_x_2 <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))))) # label(fact_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 173 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> (V_b = V_c -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_a)))))) # label(fact_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 174 (all V_n all T_a (class_Power_Opower(T_a) & class_Rings_Osemiring__0(T_a) -> (V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n)) & (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n)))) # label(fact_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 175 (all V_a all T_a (class_Rings_Olinordered__ring(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))))) # label(fact_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 176 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_c)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_add__increasing) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 177 (all V_r_H all V_q_H all V_z all V_r all V_q all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q,V_r) -> (c_Polynomial_Opdivmod__rel(T_a,V_q,V_z,V_q_H,V_r_H) -> c_Polynomial_Opdivmod__rel(T_a,V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_y),V_z),V_q_H,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_y),V_r_H)),V_r)))))) # label(fact_pdivmod__rel__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 178 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 179 (all V_pb_2 all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Polynomial_Ocoeff(T_a,V_pb_2),c_Polynomial_Odegree(T_a,V_pb_2)))) <-> c_Polynomial_Opos__poly(T_a,V_pb_2)))) # label(fact_pos__poly__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 180 (all V_y all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Opdivmod__rel(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)),V_y,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))))) # label(fact_pdivmod__rel__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 181 (all V_x all V_r all T_a (class_RealVector_Oreal__normed__div__algebra(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_r),c_RealVector_Onorm__class_Onorm(T_a,V_x))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_r)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_x))),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,V_r))))))) # label(fact_Bseq__inverse__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 182 (all V_i all V_j -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_i))) # label(fact_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 183 (all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Transcendental_Oln(V_x)),V_x)))) # label(fact_ln__less__self) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 184 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n) = hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n)))) # label(fact_div__if) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 185 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Olinordered__semiring__1(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Olinordered__semiring__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 186 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) | V_n_2 = V_m_2 <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)))) # label(fact_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 187 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> V_a = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 188 (all V_m_2 all V_n_2 (V_m_2 = V_n_2 <-> c_RealDef_Oreal(tc_Nat_Onat,V_n_2) = c_RealDef_Oreal(tc_Nat_Onat,V_m_2))) # label(fact_real__of__nat__inject) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 189 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Divides_Odiv__class_Odiv(T_a,c_Divides_Odiv__class_Omod(T_a,V_a,V_b),V_b))) # label(fact_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 190 (all V_y all V_x all T_a (class_RealVector_Oreal__normed__div__algebra(T_a) -> c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),c_RealVector_Onorm__class_Onorm(T_a,V_y)))) # label(fact_norm__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 191 (all V_x all V_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_n)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))) -> V_n = c_RComplete_Onatfloor(V_x)))) # label(fact_natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 192 (all V_q all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))),V_q) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)))) # label(fact_mult__poly__0__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 193 (all T_1 (class_Rings_Ocomm__semiring__1(T_1) -> class_Rings_Ocomm__semiring__1(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Ocomm__semiring__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 194 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_j)))))) # label(fact_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 195 (all V_q all V_b all V_r all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c))))))) # label(fact_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 196 (all V_r_H all V_q_H all V_b_H all V_r all V_q all V_b (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b_H)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_b_H),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_q),V_q_H))))))))) # label(fact_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 197 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a_2,V_b_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_pos__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 198 (all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__le__lessD) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 199 (all V_l all V_k all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),V_j)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_l)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_l)))))) # label(fact_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 200 (all V_p all T_a (class_Groups_Ozero(T_a) -> (V_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) -> hAPP(c_Polynomial_Ocoeff(T_a,V_p),c_Polynomial_Odegree(T_a,V_p)) != c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_leading__coeff__neq__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 201 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (V_y = V_x -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))))) # label(fact_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 202 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))))) # label(fact_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 203 (all V_m all V_i hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_i))))) # label(fact_less__add__Suc2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 204 (all V_s all V_y all V_r all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_y)),V_s)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),V_s))))))) # label(fact_norm__mult__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 205 (all B_x hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_q____),B_x) = hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),v_c____),B_x))) # label(fact_q_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 206 (all V_n all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_a)) -> (exists B_r (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_r)) & V_a = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B_r),hAPP(c_Nat_OSuc,V_n)))))) # label(fact_realpow__pos__nth2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 207 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_b)),c_Rings_Oinverse__class_Oinverse(T_a,V_a))))))) # label(fact_le__imp__inverse__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 208 (all V_n V_n = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n)) # label(fact_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 209 (all V_b all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_RComplete_Onatfloor(V_a)),c_RComplete_Onatfloor(V_b))),c_RComplete_Onatfloor(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_a),V_b))))))) # label(fact_le__mult__natfloor) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 210 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 211 (all V_w all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z)) # label(fact_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 212 (all V_n all V_k all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),V_n)) -> -(hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) -> -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_n))))) # label(fact_add__leE) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 213 (all V_p all V_n all T_a (class_Groups_Ozero(T_a) -> ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),B_i)) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Polynomial_Ocoeff(T_a,V_p),B_i))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p)),V_n))))) # label(fact_degree__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 214 (all V_x_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = c_RealVector_Onorm__class_Onorm(T_a,V_x_2) <-> c_Groups_Ozero__class_Ozero(T_a) = V_x_2))) # label(fact_norm__eq__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 215 (all V_n all V_m all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)))))) # label(fact_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 216 (all V_k all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_k)) # label(fact_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 217 (all V_nat_H_2 all V_nat_2 (hAPP(c_Nat_OSuc,V_nat_2) = hAPP(c_Nat_OSuc,V_nat_H_2) <-> V_nat_2 = V_nat_H_2)) # label(fact_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 218 (all V_m all V_n all V_k c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)),V_m),V_n)) # label(fact_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 219 (all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 220 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m)) # label(fact_Suc__not__Zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 221 (all V_j all V_i -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_i))) # label(fact_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 222 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_a_2) = c_Groups_Ozero__class_Ozero(T_a) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 223 (all V_y all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K all B_x hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),B_x),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,B_x)),B_K)))))) # label(fact_mult__left_Obounded) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 224 (all V_n all V_p all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Polynomial_Ocoeff(T_a,V_p),V_n)) = hAPP(c_Polynomial_Ocoeff(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),V_n))) # label(fact_coeff__smult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 225 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))))) # label(fact_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 226 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 227 (all V_n all V_p all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(T_a)),V_p),V_n))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_p)),V_n))))) # label(fact_degree__power__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 228 (all V_z V_z = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 229 (all V_z3 all V_z2 all V_z1 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z1),V_z2)),V_z3) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z1),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z2),V_z3))) # label(fact_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 230 (all V_b all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_a)) -> V_a = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 231 (all V_ry all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry))))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 232 (all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))))) # label(fact_mult__poly__0__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 233 (all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> c_Rings_Oinverse__class_Oinverse(T_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_inverse__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 234 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Oordered__ring(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Oordered__ring) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 235 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (c_Rings_Oinverse__class_Oinverse(T_a,V_a) = c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = V_a))) # label(fact_inverse__zero__imp__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 236 (all V_w all V_z (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_z),V_w)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_w),V_z)) -> V_w = V_z))) # label(fact_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 237 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_z_2)))) # label(fact_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 238 (all V_k all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),V_j)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k))))) # label(fact_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 239 (all V_y_2 all V_x_2 all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_z_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2),V_y_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z_2),V_y_2)))))) # label(fact_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 240 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_z),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_z),V_x)))))) # label(fact_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 241 (all V_w all V_z (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_w),V_z)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_z),V_w)))) # label(fact_zle__linear) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 242 (all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_a))))))) # label(fact_one__le__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 243 (all T_1 (class_Fields_Ofield(T_1) -> class_Divides_Oring__div(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Divides_Oring__div) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 244 (all V_n all V_m c_RealDef_Oreal(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_m)),V_n)) # label(fact_power__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 245 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a = V_b -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_a)))))) # label(fact_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 246 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m)) # label(fact_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 247 (all V_n all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),V_n) = c_Rings_Oinverse__class_Oinverse(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))))) # label(fact_nonzero__power__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 248 (all V_b all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> (c_Rings_Oinverse__class_Oinverse(T_a,V_b) = c_Rings_Oinverse__class_Oinverse(T_a,V_a) -> V_a = V_b))) # label(fact_inverse__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 249 (all V_n_2 all V_m_2 all V_a_2 all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a_2)) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),V_m_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),V_n_2) <-> V_m_2 = V_n_2)))) # label(fact_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 250 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2))))) # label(fact_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 251 (all V_a_2 all V_pb_2 all T_a (class_Rings_Oidom(T_a) -> (c_Polynomial_Oorder(T_a,V_a_2,V_pb_2) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_pb_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) <-> hAPP(c_Polynomial_Opoly(T_a,V_pb_2),V_a_2) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_order__root) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 252 (all V_n_2 all V_m_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat) <-> c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_m_2 & V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 253 (all V_m V_m = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_div__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 254 (all V_y all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_x),V_y)))))) # label(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 255 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)) & V_n_2 != V_m_2)) # label(fact_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.62 256 (all V_b all V_a_H all V_a all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_H),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_a_H)),V_b))) # label(fact_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 257 (all V_d_2 all V_b_2 all V_ca_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_ca_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b_2),V_a_2)),V_e_2)),V_d_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2)),V_ca_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2)),V_d_2)))))) # label(fact_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 258 (all V_c all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c))) # label(fact_mod__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 259 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 260 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))))))) # label(fact_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 261 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)))))) # label(fact_one__less__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 262 (all V_n all V_b all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_n))) # label(fact_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 263 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))))) # label(fact_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 264 (all V_r all V_q all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q,V_r) -> V_r = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,V_y)))) # label(fact_mod__poly__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 265 (all V_b_H all V_b all V_a_H all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a,V_c) -> (c_Divides_Odiv__class_Omod(T_a,V_b,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_H),V_b_H),V_c))))) # label(fact_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 266 (all V_q all V_r all V_b all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r))))))) # label(fact_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 267 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_k)))))) # label(fact_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 268 (all V_z all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w),V_z)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_z)))) # label(fact_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 269 (all V_a all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 270 (all V_n_2 all V_m_2 ((exists B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_n_2)) & hAPP(c_Nat_OSuc,B_j) = V_m_2)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 271 (all V_n all V_m all V_l all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),V_l)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_l) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_n) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))))) # label(fact_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 272 (all V_c all V_b all V_a c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c)) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_a,V_b),V_c)) # label(fact_div__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 273 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))))) # label(fact_not__leE) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 274 (all V_m all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_j))))) # label(fact_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 275 (all V_P_2 all V_n_2 all V_m_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2))) -> ((V_m_2 = V_n_2 -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2))) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2))) -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)))))) # label(fact_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 276 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2) <-> V_n_2 = V_m_2))) # label(fact_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 277 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))))))) # label(fact_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 278 (all V_x all V_y all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) != V_y -> c_Polynomial_Opoly__gcd(T_a,V_y,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,V_y)) = c_Polynomial_Opoly__gcd(T_a,V_x,V_y)))) # label(fact_poly__gcd_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 279 (all V_X_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> ((exists B_N all B_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,hAPP(V_X_2,B_n))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_N))))) <-> c_SEQ_OBseq(T_a,V_X_2)))) # label(fact_Bseq__iff1a) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 280 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_a,T_b)),V_f_2),V_g_2)) <-> -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_g_2),V_f_2)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2),V_g_2))))) # label(fact_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 281 (all V_y_2 all V_x_2 all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_x_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_y_2)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x_2),V_y_2) = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ozero__class_Ozero(T_a) = V_y_2 & V_x_2 = c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 282 (all V_z V_z = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_z)) # label(fact_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 283 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__self) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 284 (all V_m all V_n (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_m)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,V_m))) -> V_n = V_m))) # label(fact_less__antisym) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 285 (all V_m_2 all V_Q_2 all V_n_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_n_2)) -> (hBOOL(hAPP(V_Q_2,V_m_2)) -> (-hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) -> ((all B_k (hBOOL(hAPP(V_P_2,hAPP(c_Nat_OSuc,B_k))) <-> hBOOL(hAPP(V_Q_2,B_k)))) -> c_Orderings_Oord__class_OLeast(tc_Nat_Onat,V_P_2) = hAPP(c_Nat_OSuc,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,V_Q_2))))))) # label(fact_Least__Suc2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 286 (all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))))) # label(fact_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 287 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))))) # label(fact_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 288 (all V_qa_2 all V_pb_2 all T_a (class_Int_Oring__char__0(T_a) & class_Rings_Oidom(T_a) -> (V_pb_2 = V_qa_2 <-> c_Polynomial_Opoly(T_a,V_pb_2) = c_Polynomial_Opoly(T_a,V_qa_2)))) # label(fact_poly__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 289 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> V_a = V_b))))) # label(fact_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 290 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m)))))) # label(fact_n__less__n__mult__m) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 291 (all V_n all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n))))))) # label(fact_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 292 (all V_z all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))))) # label(fact_termination__basic__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 293 (all V_b all V_a all V_c all T_a (class_Rings_Olinordered__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 294 (all T_a (class_Rings_Olinordered__semidom(T_a) -> -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a)),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_not__one__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 295 (all V_y all V_x all V_xa all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_y)))) # label(fact_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 296 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Olinordered__semiring__1__strict(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Olinordered__semiring__1__strict) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 297 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y_2),V_x_2)) <-> V_x_2 != V_y_2))) # label(fact_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 298 (all V_k all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b)),V_b)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_k)) # label(fact_zdiv__zmod__equality2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 299 (all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = V_n) # label(fact_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 300 (all V_c all V_b all V_a c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_c)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_b,V_c))),c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_c)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c))) # label(fact_zdiv__zadd1__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 301 (all V_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealDef_Oreal(tc_Nat_Onat,V_n)))) # label(fact_real__of__nat__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 302 (all V_a (c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) & V_a = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_DIVISION__BY__ZERO) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 303 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 304 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)))))) # label(fact_n__less__m__mult__n) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 305 (all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Polynomial_Osmult(T_a,V_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)))) # label(fact_smult__0__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 306 (all V_pb_2 all T_a (class_Groups_Ozero(T_a) -> (V_pb_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(T_a,V_pb_2)))) # label(fact_psize__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 307 (all V_m all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a))),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_m),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 308 (all V_b all V_a all T_a (class_Rings_Ono__zero__divisors(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) != c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 309 (all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> c_RComplete_Onatceiling(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),V_x),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_RComplete_Onatceiling(V_x)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_natceiling__add__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 310 (all V_q all V_p all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),c_Polynomial_Osmult(T_a,V_a,V_p)),V_q) = c_Polynomial_Osmult(T_a,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)))) # label(fact_mult__smult__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 311 (all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Transcendental_Oln(V_x))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x))))) # label(fact_ln__ge__zero__imp__ge__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 312 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Olinordered__semidom(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Olinordered__semidom) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 313 (all V_d_2 all V_m_2 (c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m_2,V_d_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) <-> (exists B_q V_m_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_d_2),B_q)))) # label(fact_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 314 (all V_b_2 all V_n_2 all V_a_2 all T_a (class_Groups_Ozero(T_a) -> (c_Polynomial_Omonom(T_a,V_a_2,V_n_2) = c_Polynomial_Omonom(T_a,V_b_2,V_n_2) <-> V_b_2 = V_a_2))) # label(fact_monom__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 315 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__less__SucD) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 316 (all V_X_2 all T_a (class_Orderings_Oorder(T_a) -> ((all B_m all B_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m),B_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(V_X_2,B_n)),hAPP(V_X_2,B_m))))) <-> c_SEQ_Odecseq(T_a,V_X_2)))) # label(fact_decseq__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 317 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_a)) -> (V_b = V_c -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),V_a)))))) # label(fact_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 318 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Olinordered__ring(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Olinordered__ring) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 319 (all V_a (V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Groups_Oone__class_Oone(tc_Int_Oint) = c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_a))) # label(fact_zdiv__self) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 320 (all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Transcendental_Oln(V_x_2)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x_2),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))))) # label(fact_ln__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 321 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n))),c_Groups_Oone__class_Oone(T_a))))))) # label(fact_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 322 (all V_x_2 all T_a (class_Groups_Oone(T_a) -> (c_Groups_Oone__class_Oone(T_a) = V_x_2 <-> V_x_2 = c_Groups_Oone__class_Oone(T_a)))) # label(fact_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 323 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Orderings_Oorder(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 324 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x),V_y)))) # label(fact_termination__basic__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 325 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (V_a != V_b -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 326 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,c_Groups_Oone__class_Oone(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 327 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 328 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)))))) # label(fact_inverse__less__imp__less__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 329 (all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_a))))))) # label(fact_one__less__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 330 (all V_m_2 all V_n_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2),V_m_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Nat_OSuc,V_m_2))))) # label(fact_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 331 (all V_w hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_w),V_w))) # label(fact_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 332 (all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Polynomial_Odegree(T_a,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(T_a))))) # label(fact_degree__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 333 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 334 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 335 (all V_y all V_z all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x),V_z)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))))) # label(fact_termination__basic__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 336 (all V_b all V_a all T_a (class_Rings_Ono__zero__divisors(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) -> c_Groups_Ozero__class_Ozero(T_a) = V_a | V_b = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 337 (all V_b_2 all V_ca_2 all V_a_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_ca_2),c_Groups_Ozero__class_Ozero(T_a))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_ca_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_ca_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_ca_2)))))) # label(fact_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 338 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Oordered__semiring(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Oordered__semiring) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 339 (all V_k_2 all V_n_2 all V_P_2 (hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n_2,V_k_2))) <-> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 -> hBOOL(hAPP(V_P_2,V_n_2))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_k_2 -> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_k_2)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),B_i)),B_j) = V_n_2 -> hBOOL(hAPP(V_P_2,B_j)))))))) # label(fact_split__mod) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 340 (all V_b all V_a all T_a (class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_b)) = c_Rings_Oinverse__class_Oinverse(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_inverse__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 341 (all V_n all V_z (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_RComplete_Onatfloor(V_z)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_z),c_RealDef_Oreal(tc_Nat_Onat,V_n))))) # label(fact_ge__natfloor__plus__one__imp__gt) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 342 (all V_p all V_q all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_q)),c_Polynomial_Odegree(T_a,V_p))) -> c_Polynomial_Odegree(T_a,V_p) = c_Polynomial_Odegree(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),V_q))))) # label(fact_degree__add__eq__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 343 (all V_n all V_m (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_mod__geq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 344 (all V_n V_n != hAPP(c_Nat_OSuc,V_n)) # label(fact_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 345 (all V_b c_Divides_Odiv__class_Odiv(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zdiv__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 346 (all V_n all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x),c_RealDef_Oreal(tc_Nat_Onat,V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_RComplete_Onatfloor(V_x)),V_n))))) # label(fact_less__natfloor) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 347 (all V_X_2 all T_b (class_RealVector_Oreal__normed__vector(T_b) -> ((exists B_K (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K)) & (all B_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_b,hAPP(V_X_2,B_n))),B_K))))) <-> (exists B_N all B_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_b,hAPP(V_X_2,B_n))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_N)))))))) # label(fact_lemma__NBseq__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 348 (all V_k all V_j all V_u all V_i hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_u)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u)),V_k))) # label(fact_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 349 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2))))) # label(fact_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 350 (all V_v all V_u all V_y all V_a all V_x all T_a (class_Rings_Olinordered__semiring__1__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_u)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_v)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) = c_Groups_Oone__class_Oone(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y))),V_a))))))))) # label(fact_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 351 (all V_m_2 all V_n_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)))) # label(fact_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 352 (all V_z all V_y all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 353 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,V_n)))) # label(fact_lessI) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 354 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n))))) # label(fact_real__of__nat__Suc__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 355 (all V_c all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Polynomial_Osynthetic__div(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)),V_c) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)))) # label(fact_synthetic__div__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 356 (all V_d_2 all V_b_2 all V_ca_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a_2),V_b_2)),V_e_2)),V_ca_2)),V_d_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2)),V_ca_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2)),V_d_2)))))) # label(fact_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 357 (all V_ry all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 358 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x),V_y)) | V_x = V_y | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_y),V_x)))) # label(fact_zless__linear) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 359 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b)))) # label(fact_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 360 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a_2),c_Groups_Ozero__class_Ozero(T_a))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 361 (all V_k all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i),V_j)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_j),V_k)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i),V_k))))) # label(fact_zle__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 362 (all V_y all V_x all V_a all T_a (class_Fields_Ofield(T_a) -> c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),c_Polynomial_Osmult(T_a,V_a,V_x),V_y) = c_Polynomial_Osmult(T_a,V_a,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,V_y)))) # label(fact_mod__smult__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 363 (all V_z_2 all V_x_2 all V_y_2 all V_w_2 all T_a (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a) -> (V_z_2 = V_y_2 | V_w_2 = V_x_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w_2),V_y_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x_2),V_z_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w_2),V_z_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x_2),V_y_2))))) # label(fact_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 364 (all V_y all V_x all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> ((all B_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),B_z)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),B_z),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),B_z),V_x)),V_y))))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))))) # label(fact_field__le__mult__one__interval) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 365 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c))) # label(fact_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 366 (all V_b all V_c all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)))))) # label(fact_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 367 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Olinordered__ring__strict(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Olinordered__ring__strict) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 368 (all V_z all V_y all V_x hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_y),V_z)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y)),V_z)) # label(fact_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 369 (all V_n all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n)),V_x)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),V_x),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),V_n))))) # label(fact_LIMSEQ__inverse__realpow__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 370 (all V_m_2 all V_n_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_m_2)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))))) # label(fact_nat__le__real__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 371 (all V_x_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_RComplete_Onatfloor(V_x_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x_2)))) # label(fact_le__natfloor__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 372 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k))))) # label(fact_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 373 (all V_k all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_m)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_k)),V_m)))) # label(fact_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 374 (all T_1 (class_Groups_Ocomm__monoid__add(T_1) -> class_Groups_Ocomm__monoid__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Ocomm__monoid__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 375 (all V_x_2 all T_a (class_Groups_Ozero(T_a) -> (V_x_2 = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ozero__class_Ozero(T_a) = V_x_2))) # label(fact_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 376 (all V_qa_2 all V_pb_2 all T_a (class_Groups_Ocomm__monoid__add(T_a) -> c_Polynomial_OAbs__poly(T_a,hAPP(hAPP(c_COMBS(tc_Nat_Onat,T_a,T_a),hAPP(c_COMBB(T_a,tc_fun(T_a,T_a),tc_Nat_Onat,c_Groups_Oplus__class_Oplus(T_a)),c_Polynomial_Ocoeff(T_a,V_pb_2))),c_Polynomial_Ocoeff(T_a,V_qa_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_pb_2),V_qa_2))) # label(fact_plus__poly__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 377 (all T_2 all T_1 (class_Orderings_Opreorder(T_1) -> class_Orderings_Opreorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Opreorder) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 378 (all T_1 (class_Rings_Ocomm__ring__1(T_1) -> class_Rings_Oring__1(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Oring__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 379 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> V_a = c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_b)))) # label(fact_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 380 (all V_n all V_b all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Polynomial_Osmult(T_a,V_a,c_Polynomial_Omonom(T_a,V_b,V_n)) = c_Polynomial_Omonom(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_n))) # label(fact_smult__monom) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 381 (all V_Q_2 all V_P_2 (-hBOOL(hAPP(hAPP(c_fimplies,V_P_2),V_Q_2)) | hBOOL(V_Q_2) | -hBOOL(V_P_2))) # label(help_c__fimplies__3) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 382 (all V_b_H all V_b all V_a all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b_H)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_b_H)))) # label(fact_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 383 (all V_pb_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(tc_Complex_Ocomplex,V_pb_2)),c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(tc_Complex_Ocomplex,v_pa____))) -> (-c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant(tc_Complex_Ocomplex,tc_Complex_Ocomplex,c_Polynomial_Opoly(tc_Complex_Ocomplex,V_pb_2)) -> (exists B_z c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,V_pb_2),B_z))))) # label(fact_less_Ohyps) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 384 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_y)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_Transcendental_Oln(V_x)),c_Transcendental_Oln(V_y)) = c_Transcendental_Oln(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_y))))) # label(fact_ln__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 385 (all V_n_2 all V_m_2 (V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat) & V_m_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat) <-> c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 386 (all V_w hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_w),V_w))) # label(fact_zle__refl) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 387 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_y_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Transcendental_Oln(V_x_2)),c_Transcendental_Oln(V_y_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2),V_y_2)))))) # label(fact_ln__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 388 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))) <-> V_n_2 = V_m_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 389 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_N)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))))) # label(fact_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 390 (all V_b all V_a c_Divides_Odiv__class_Odiv(tc_Int_Oint,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zmod__zdiv__trivial) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 391 (all V_c all V_a all V_b all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))))) # label(fact_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 392 (all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_a) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zmod__self) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 393 (all V_n all V_m (V_m = V_n -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)))) # label(fact_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 394 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_fequal,V_x_2),V_y_2)) | V_x_2 != V_y_2)) # label(help_c__fequal__2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 395 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 396 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))) # label(fact_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 397 (all V_l_2 all V_k_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2),V_l_2)) <-> (exists B_n V_l_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),B_n)))) # label(fact_le__Suc__ex__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 398 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),V_y_2)) <-> V_y_2 = V_x_2)))) # label(fact_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 399 (all V_m_2 all V_n_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n_2)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),c_RealDef_Oreal(tc_Nat_Onat,V_m_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)))) # label(fact_nat__less__real__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 400 (all V_x_2 all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x_2)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),c_Groups_Oone__class_Oone(T_a))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_x_2)))))) # label(fact_one__le__inverse__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 401 (all V_y_2 all V_x_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (V_x_2 != c_Groups_Ozero__class_Ozero(T_a) | c_Groups_Ozero__class_Ozero(T_a) != V_y_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y_2),V_y_2))))))) # label(fact_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 402 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_j)))))) # label(fact_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 403 (all V_p all V_b all V_a all T_a (class_Rings_Ocomm__ring(T_a) -> c_Polynomial_Osmult(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_p) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(T_a)),c_Polynomial_Osmult(T_a,V_a,V_p)),c_Polynomial_Osmult(T_a,V_b,V_p)))) # label(fact_smult__diff__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 404 (all T_1 (class_Rings_Ocomm__semiring__0(T_1) -> class_Rings_Omult__zero(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Omult__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 405 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_z)))))) # label(fact_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 406 (all V_z3 all V_z2 all V_z1 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)),V_z3) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z2),V_z3))) # label(fact_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 407 (all V_n all V_m all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 408 (all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_negative__imp__inverse__negative) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 409 (all V_n all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_i)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_n))))) # label(fact_nat__one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 410 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))) # label(fact_not__real__of__nat__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 411 (all V_n_2 all V_m_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 412 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 413 (all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_mult__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 414 (all V_ca_2 all V_pb_2 all T_a (class_Rings_Ocomm__semiring__0(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = c_Polynomial_Osynthetic__div(T_a,V_pb_2,V_ca_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Polynomial_Odegree(T_a,V_pb_2)))) # label(fact_synthetic__div__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 415 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_a,V_c)),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_b,V_c))),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,V_c)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_b,V_c)),V_c)) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_a),V_b),V_c)) # label(fact_div__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 416 (all V_q all V_b all V_r all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_r),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_q,V_c))),V_r)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c)))))) # label(fact_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 417 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> (exists B_k hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k)) = V_n_2))) # label(fact_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 418 (all V_p all T_a (class_Groups_Ozero(T_a) -> (V_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) -> hAPP(c_Nat_OSuc,c_Polynomial_Odegree(T_a,V_p)) = c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(T_a,V_p)) & (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_p -> c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(T_a,V_p) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_psize__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 419 (all V_k_2 all V_n_2 all V_P_2 (hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (all B_i all B_j (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)) & hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 -> hBOOL(hAPP(V_P_2,B_j))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 -> hBOOL(hAPP(V_P_2,B_j))))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> hBOOL(hAPP(V_P_2,V_n_2))))) # label(fact_split__zmod) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 420 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c))) # label(fact_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 421 (all V_n all V_b all V_m all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Polynomial_Omonom(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),c_Polynomial_Omonom(T_a,V_a,V_m)),c_Polynomial_Omonom(T_a,V_b,V_n)))) # label(fact_mult__monom) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 422 (all V_n_2 all V_m_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2)),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 423 (all V_q all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_p)),c_Polynomial_Odegree(T_a,V_q)))))) # label(fact_degree__mult__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 424 (all V_m all V_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_m)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 425 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)))) # label(fact_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 426 (all V_p all T_a (class_Groups_Ocomm__monoid__add(T_a) -> V_p = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))))) # label(fact_add__poly__code_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.63 427 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_b,V_c))),c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c)) = c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c)) # label(fact_zdiv__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 428 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_a)))))) # label(fact_inverse__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 429 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_x)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y)))) # label(fact_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 430 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_a_2)))))) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 431 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Orderings_Opreorder(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Orderings_Opreorder) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 432 (all V_x_2 all V_y_2 all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y_2),V_x_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2)) <-> V_y_2 = V_x_2)))) # label(fact_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 433 (all V_n c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) # label(fact_real__of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 434 (all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(T_a),V_x,V_y),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,V_y)))) # label(fact_pdivmod__rel) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 435 (all V_x_2 all T_a (class_Fields_Ofield__inverse__zero(T_a) -> (c_Groups_Oone__class_Oone(T_a) = c_Rings_Oinverse__class_Oinverse(T_a,V_x_2) <-> V_x_2 = c_Groups_Oone__class_Oone(T_a)))) # label(fact_inverse__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 436 (all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,c_RComplete_Onatfloor(V_x))),V_x)))) # label(fact_real__natfloor__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 437 (all V_i all V_j all V_k (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_j)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)),V_i) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_k))) # label(fact_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 438 (all V_b_2 all V_a_2 all V_ca_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_ca_2),V_a_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_ca_2),V_b_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a_2),V_b_2))))) # label(fact_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 439 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))))))) # label(fact_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 440 (all V_k all V_j all V_i hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_k)),V_j)) # label(fact_diff__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 441 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))))) # label(fact_less__SucI) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 442 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2),V_m_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2),V_n_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_k_2))))))) # label(fact_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 443 (all T_1 (class_Rings_Ocomm__ring(T_1) -> class_Rings_Ocomm__ring(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Ocomm__ring) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 444 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Orderings_Oord(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Orderings_Oord) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 445 (all V_w all V_z2 all V_z1 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),V_z1),V_z2)),V_w) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z1),V_w)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z2),V_w))) # label(fact_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 446 (all V_k_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_i_2)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i_2),V_k_2)) | V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) <-> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_i_2,V_k_2))) # label(fact_zdiv__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 447 (all V_y_2 all V_x_2 all V_b_2 all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_b_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b_2),V_x_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b_2),V_y_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x_2),V_y_2)))))) # label(fact_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 448 (all V_n_2 all V_k_2 all V_m_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 449 (all V_b all V_b_H all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_b_H),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b_H))))))) # label(fact_zdiv__mono2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 450 (all V_y all V_x all T_a (class_Rings_Olinordered__ring(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_y)))))) # label(fact_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 451 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),c_Divides_Odiv__class_Odiv(T_a,V_a,V_b))),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)))) # label(fact_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 452 (all V_r all V_q all V_r_H all V_q_H all V_b (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q_H)),V_r_H)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_q_H),V_q))))))) # label(fact_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 453 (all V_k all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_k,V_n)),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_k,V_m)))))) # label(fact_div__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 454 (all V_b_2 all V_a_2 all V_ca_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a_2),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_ca_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_ca_2),c_Groups_Ozero__class_Ozero(T_a))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_b_2)))))) # label(fact_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 455 (all V_a all V_b all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)))))) # label(fact_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 456 (all V_y_2 all V_x_2 (V_y_2 = V_x_2 | -hBOOL(hAPP(hAPP(c_fequal,V_x_2),V_y_2)))) # label(help_c__fequal__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 457 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),V_x))))))) # label(fact_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 458 (all V_l all V_k all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),V_j)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_l)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_l)))))) # label(fact_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 459 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a_2,V_b_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_a_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_neg__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 460 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) | V_x = V_y))) # label(fact_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 461 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 462 (all V_m all V_i hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_m))))) # label(fact_less__add__Suc1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 463 (all V_c all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),c_Divides_Odiv__class_Odiv(T_a,V_a,V_b))),c_Divides_Odiv__class_Omod(T_a,V_a,V_b))),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) # label(fact_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 464 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))))) # label(fact_Suc__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 465 (all V_n all V_x all T_a (class_RealVector_Oreal__normed__algebra__1(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),V_n))))) # label(fact_norm__power__ineq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 466 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a != V_b -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)))))) # label(fact_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 467 (all V_x all V_q all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) = hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x))) # label(fact_poly__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 468 (all V_x_2 all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2),V_g_2)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))))) # label(fact_le__funD) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 469 (all V_P_2 all V_k_2 all T_a (class_Orderings_Owellorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_k_2),c_Orderings_Oord__class_OLeast(T_a,V_P_2))) -> -hBOOL(hAPP(V_P_2,V_k_2))))) # label(fact_not__less__Least) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 470 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 471 (all V_n_2 c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2)) = hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2)) # label(fact_less__eq__Suc__le__raw) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 472 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 473 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) # label(fact_mult__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 474 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))))))) # label(fact_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 475 (all V_X_2 all T_a (class_Orderings_Oorder(T_a) -> (c_SEQ_Oincseq(T_a,V_X_2) -> c_SEQ_Omonoseq(T_a,V_X_2)))) # label(fact_incseq__imp__monoseq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 476 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) | V_n = V_m -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)))) # label(fact_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 477 (all V_b all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_div__nonpos__pos__le0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 478 (all V_x_2 all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_x_2)),c_Groups_Oone__class_Oone(T_a))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_x_2)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_inverse__less__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 479 (all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Opoly__gcd(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)))) # label(fact_poly__gcd__0__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 480 (all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 481 (all V_k all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_i),V_j)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_j),V_k)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_i),V_k))))) # label(fact_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 482 (all V_m_2 all V_x_2 (hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_m_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 | V_x_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_nat__power__eq__Suc__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 483 (all V_b_2 all V_a_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a_2,V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_b_2),V_a_2))))) # label(fact_nonneg1__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 484 (all V_ya all V_y all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_y)),V_ya) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_ya)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_ya)))) # label(fact_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 485 (all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),c_Groups_Oone__class_Oone(T_a))))) # label(fact_zero__le__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 486 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2)) <-> V_x_2 != V_y_2 & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),V_y_2))))) # label(fact_order__less__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 487 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),V_d)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 488 (all V_n all V_m c_RealDef_Oreal(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_m)),V_n)) # label(fact_real__of__nat__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 489 (all V_y all V_m all V_x c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_m)),V_y),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y),V_m)) # label(fact_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 490 (all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_x & c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_y -> hAPP(c_Polynomial_Ocoeff(T_a,c_Polynomial_Opoly__gcd(T_a,V_x,V_y)),c_Polynomial_Odegree(T_a,c_Polynomial_Opoly__gcd(T_a,V_x,V_y))) = c_Groups_Ozero__class_Ozero(T_a)) & (-(c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_y & V_x = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Polynomial_Ocoeff(T_a,c_Polynomial_Opoly__gcd(T_a,V_x,V_y)),c_Polynomial_Odegree(T_a,c_Polynomial_Opoly__gcd(T_a,V_x,V_y)))))) # label(fact_poly__gcd__monic) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 491 (all T_a (class_Rings_Olinordered__idom(T_a) -> -c_Polynomial_Opos__poly(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))))) # label(fact_not__pos__poly__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 492 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_a)))))) # label(fact_inverse__le__imp__le__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 493 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__comm__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 494 (all V_a_2 all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ozero__class_Ozero(T_a) = c_Rings_Oinverse__class_Oinverse(T_a,V_a_2)))) # label(fact_inverse__nonzero__iff__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 495 (all T_a (class_Rings_Olinordered__semidom(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_not__one__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 496 (all V_b all V_a all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)))) # label(fact_zmod__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 497 (all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))) # label(fact_norm__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 498 (all V_x all V_y all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) != V_y -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,V_y))),c_Polynomial_Odegree(T_a,V_y))) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,V_y)))) # label(fact_degree__mod__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 499 (all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Transcendental_Oln(V_x_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x_2))))) # label(fact_ln__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 500 (all V_a all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x),c_RealDef_Oreal(tc_Nat_Onat,V_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_RComplete_Onatceiling(V_x)),V_a)))) # label(fact_natceiling__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 501 (all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K)) & (all B_x hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),B_x))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,B_x)),B_K)))))))) # label(fact_mult__right_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 502 (all V_ry all V_rx all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),V_ry))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 503 (all V_l all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),V_l)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_l)))))) # label(fact_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 504 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n_2)),c_RealDef_Oreal(tc_Nat_Onat,V_m_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)))) # label(fact_real__of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 505 (all V_a_2 all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a_2),c_Groups_Ozero__class_Ozero(T_a))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a_2)),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_inverse__negative__iff__negative) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 506 (all V_x all V_n all V_p all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(T_a)),V_p),V_n)),V_x) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),V_n))) # label(fact_poly__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 507 (all V_y_2 all V_x_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_y_2 & V_x_2 = c_Groups_Ozero__class_Ozero(T_a) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y_2),V_y_2))),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 508 (all V_n all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> c_Rings_Oinverse__class_Oinverse(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),V_n))) # label(fact_power__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 509 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Groups_Oordered__ab__group__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Oordered__ab__group__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 510 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) # label(fact_mult__Suc__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 511 (all V_m all V_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_m)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 512 (all V_m all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 513 (all V_m all V_n all V_k c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)),V_m)),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)) # label(fact_mod__mult__self4) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 514 (all V_x hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_RComplete_Onatceiling(V_x)))) # label(fact_zero__le__natceiling) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 515 (all V_P_2 all T_a (class_Orderings_Owellorder(T_a) -> ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) -> hBOOL(hAPP(V_P_2,c_Orderings_Oord__class_OLeast(T_a,V_P_2)))))) # label(fact_LeastI__ex) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 516 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,c_Divides_Odiv__class_Omod(T_a,V_a,V_b),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 517 (all V_r2 all V_q2 all V_r1 all V_q1 all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q1,V_r1) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q2,V_r2) -> V_q2 = V_q1 & V_r1 = V_r2)))) # label(fact_pdivmod__rel__unique) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 518 (all V_q all V_p all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_p),V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 519 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> V_m = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m),V_n))) # label(fact_div__mult__self1__is__m) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 520 (all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2)))) # label(fact_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 521 (all V_y all V_z all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))))) # label(fact_termination__basic__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 522 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Groups_Oordered__ab__semigroup__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 523 (all V_y all V_q_1 all V_b_1 all V_a_1 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_1),V_q_1)),V_y) = V_a_1 -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_y),V_b_1))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_y),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_1),V_y))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b_1 -> V_y = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a_1,V_b_1))))) # label(fact_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 524 (all V_b_2 all V_a_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_a))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b_2)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a_2),c_Groups_Ozero__class_Ozero(T_a))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a_2)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b_2),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 525 (all V_b all V_a all V_c all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)))))) # label(fact_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 526 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_a = V_b -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_c)))))) # label(fact_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 527 (all V_c all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),c_Divides_Odiv__class_Odiv(T_a,V_a,V_b)) = c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)),V_b)))) # label(fact_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 528 (all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),V_ly) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 529 (all V_b_2 all V_ca_2 all V_a_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_ca_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b_2),V_ca_2)))))) # label(fact_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 530 (all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_x)),c_Polynomial_Odegree(T_a,V_y))) -> c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,V_y) = V_x))) # label(fact_mod__poly__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 531 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_a)) -> (V_b != V_a -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)))))) # label(fact_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 532 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) # label(fact_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 533 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c))) # label(fact_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 534 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> (V_y != V_x -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))))) # label(fact_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 535 (all V_y all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Opoly__gcd(T_a,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(T_a)),V_y) = c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(T_a)))) # label(fact_poly__gcd__1__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 536 (all V_r_1 all V_y all V_b_1 all V_a_1 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_1),V_y)),V_r_1) = V_a_1 -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_r_1),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_1),V_r_1))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_1)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_1),V_b_1))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b_1 -> V_y = c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a_1,V_b_1))))) # label(fact_divmod__int__rel__div__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 537 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) # label(fact_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 538 (all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) -> V_x = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | V_x = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nat__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 539 (all V_n all V_m all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k)) -> c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n))) # label(fact_nat__mult__div__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 540 (all V_n_2 all V_m_2 all V_k_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))))) # label(fact_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 541 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a_2,V_b_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2))))) # label(fact_pos__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 542 (all V_b_2 all V_a_2 all V_ca_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_ca_2),V_a_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_ca_2),V_b_2)))))) # label(fact_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 543 (all V_b_2 all V_a_2 all V_ca_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_ca_2),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b_2),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_b_2))))))) # label(fact_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 544 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))))) # label(fact_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 545 (all V_b all V_m all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_m)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 546 (all V_y all V_x (V_x != V_y -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_y),V_x))))) # label(fact_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 547 (all V_n -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n)),V_n))) # label(fact_Suc__n__not__le__n) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 548 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_b)),c_Rings_Oinverse__class_Oinverse(T_a,V_a))))))) # label(fact_less__imp__inverse__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 549 (all V_a_2 all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_a))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a_2),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 550 (all V_n_2 all V_a_2 all V_times_2 all V_one_2 all T_a hAPP(hAPP(V_times_2,V_a_2),hAPP(hAPP(c_Power_Opower_Opower(T_a,V_one_2,V_times_2),V_a_2),V_n_2)) = hAPP(hAPP(c_Power_Opower_Opower(T_a,V_one_2,V_times_2),V_a_2),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_power_Opower_Opower__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 551 (all V_X_2 all T_a (class_Orderings_Oorder(T_a) -> (c_SEQ_Omonoseq(T_a,V_X_2) <-> (all B_m all B_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m),B_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(V_X_2,B_n)),hAPP(V_X_2,B_m))))) | (all B_m all B_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m),B_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(V_X_2,B_m)),hAPP(V_X_2,B_n)))))))) # label(fact_monoseq__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 552 (all V_nat_H_1 c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_nat_H_1)) # label(fact_nat_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 553 (all V_n c_RComplete_Onatceiling(c_RealDef_Oreal(tc_Nat_Onat,V_n)) = V_n) # label(fact_natceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 554 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 555 (all T_1 (class_Rings_Ocomm__semiring__0(T_1) -> class_Rings_Ocomm__semiring(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Ocomm__semiring) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 556 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b)))))) # label(fact_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 557 (all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a))),c_Rings_Oinverse__class_Oinverse(T_a,V_b)))))) # label(fact_division__ring__inverse__diff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 558 (all V_b_2 all V_a_2 all V_ca_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_ca_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a_2),V_b_2)))))) # label(fact_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 559 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 560 (all T_1 (class_Groups_Ocomm__monoid__add(T_1) -> class_Groups_Oab__semigroup__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Oab__semigroup__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 561 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_b))) # label(fact_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 562 (all V_x all V_y all T_a (class_Fields_Ofield(T_a) -> (V_y = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) -> c_Polynomial_Opoly__gcd(T_a,V_x,V_y) = c_Polynomial_Osmult(T_a,c_Rings_Oinverse__class_Oinverse(T_a,hAPP(c_Polynomial_Ocoeff(T_a,V_x),c_Polynomial_Odegree(T_a,V_x))),V_x)) & (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) != V_y -> c_Polynomial_Opoly__gcd(T_a,V_y,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,V_y)) = c_Polynomial_Opoly__gcd(T_a,V_x,V_y)))) # label(fact_poly__gcd__code) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 563 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_k)))) # label(fact_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 564 (all V_x all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x)),c_Groups_Oone__class_Oone(T_a)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),c_Groups_Oone__class_Oone(T_a))),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),c_Groups_Oone__class_Oone(T_a))))) # label(fact_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 565 (all V_a all V_m all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a))),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 566 -(all B_c -(all B_w hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),B_c))),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),B_w)))))) # label(fact__096_B_Bthesis_O_A_I_B_Bc_O_AALL_Aw_O_Acmod_A_Ipoly_Ap_Ac_J_A_060_061_Acmod_A_Ipoly_Ap_Aw_J_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 567 (all V_c all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c))) # label(fact_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 568 (all V_z2 all V_z1 all V_w hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z1)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z2))) # label(fact_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 569 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))) # label(fact_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 570 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))))))) # label(fact_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 571 (all V_i_2 all V_k_2 all V_j_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2)),V_i_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2))))) # label(fact_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 572 (all V_n all V_x all T_a (class_RealVector_Oreal__normed__div__algebra(T_a) -> c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),V_n))) # label(fact_norm__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 573 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Olinordered__semiring(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Olinordered__semiring) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 574 (all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a)) # label(fact_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 575 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_x)))) # label(fact_order__refl) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 576 (all V_n all V_m c_RealDef_Oreal(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_m)),c_RealDef_Oreal(tc_Nat_Onat,V_n))) # label(fact_real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 577 (all V_X_2 all T_a (class_Orderings_Oorder(T_a) -> (c_SEQ_Odecseq(T_a,V_X_2) -> c_SEQ_Omonoseq(T_a,V_X_2)))) # label(fact_decseq__imp__monoseq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 578 (all V_n all V_q all V_p all T_a (class_Groups_Ocomm__monoid__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Polynomial_Ocoeff(T_a,V_p),V_n)),hAPP(c_Polynomial_Ocoeff(T_a,V_q),V_n)) = hAPP(c_Polynomial_Ocoeff(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_n))) # label(fact_coeff__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 579 (all V_q all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Polynomial_Opcompose(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)),V_q) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)))) # label(fact_pcompose__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 580 (all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_RComplete_Onatfloor(V_x))) # label(fact_natfloor__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 581 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_z),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_x)))))) # label(fact_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 582 (all V_c all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_c),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_a),V_c)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_b),V_c)))))) # label(fact_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 583 (all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Transcendental_Oln(V_x))))) # label(fact_ln__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 584 (all V_i all V_j all V_k (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),V_j))) # label(fact_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 585 (all V_n hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_power__Suc__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 586 (all V_qa_2 all V_pb_2 all T_a (class_Groups_Ozero(T_a) -> ((all B_n hAPP(c_Polynomial_Ocoeff(T_a,V_pb_2),B_n) = hAPP(c_Polynomial_Ocoeff(T_a,V_qa_2),B_n)) <-> V_pb_2 = V_qa_2))) # label(fact_expand__poly__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 587 (all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))))) # label(fact_le__imp__less__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 588 (all T_2 all T_1 (class_Orderings_Oorder(T_1) -> class_Orderings_Oorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 589 (all V_c all V_b all V_a all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Opoly__gcd(T_a,c_Polynomial_Opoly__gcd(T_a,V_a,V_b),V_c) = c_Polynomial_Opoly__gcd(T_a,V_a,c_Polynomial_Opoly__gcd(T_a,V_b,V_c)))) # label(fact_poly__gcd_Oassoc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 590 (all V_x_2 all T_a (class_Groups_Ozero(T_a) -> V_x_2 = c_Polynomial_OAbs__poly(T_a,c_Polynomial_Ocoeff(T_a,V_x_2)))) # label(fact_coeff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 591 (all V_c all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),c_Divides_Odiv__class_Odiv(T_a,V_a,V_b)) = c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_b)))) # label(fact_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 592 (all V_b_2 all V_a_2 all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> (c_Rings_Oinverse__class_Oinverse(T_a,V_b_2) = c_Rings_Oinverse__class_Oinverse(T_a,V_a_2) <-> V_b_2 = V_a_2))) # label(fact_inverse__eq__iff__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 593 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),V_d)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 594 (all V_n_2 (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = c_RealDef_Oreal(tc_Nat_Onat,V_n_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_real__of__nat__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 595 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Odiv(T_a,V_a,V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b))),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) # label(fact_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 596 (all V_q all T_a (class_Groups_Ocomm__monoid__add(T_a) -> V_q = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))),V_q))) # label(fact_add__poly__code_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 597 (all V_u all V_y all V_x1 all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x1)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x1),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_u))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,V_x)),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,V_x1)),V_u))))))) # label(fact_real__mult__inverse__cancel) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 598 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k))) # label(fact_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 599 (all V_Q_2 all V_P_2 (hBOOL(hAPP(hAPP(c_fimplies,V_P_2),V_Q_2)) | hBOOL(V_P_2))) # label(help_c__fimplies__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 600 (all V_w all V_z2 all V_z1 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_w)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_w)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)),V_w)) # label(fact_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 601 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))) # label(fact_add__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 602 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a))))) # label(fact_right__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 603 (all V_r all V_q all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_r) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_r)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_q),V_r)))) # label(fact_mult__poly__add__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 604 (all V_k_2 all V_P_2 all T_a (class_Orderings_Owellorder(T_a) -> (hBOOL(hAPP(V_P_2,V_k_2)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Orderings_Oord__class_OLeast(T_a,V_P_2)),V_k_2))))) # label(fact_Least__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 605 (all T_1 (class_Rings_Oidom(T_1) -> class_Rings_Oidom(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Oidom) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 606 (all T_a (class_Rings_Odivision__ring(T_a) -> c_Rings_Oinverse__class_Oinverse(T_a,c_Groups_Oone__class_Oone(T_a)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_inverse__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 607 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n))) # label(fact_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 608 (all T_1 (class_Groups_Ozero(T_1) -> class_Groups_Ozero(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Ozero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 609 (all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Transcendental_Oln(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x))),V_x)))) # label(fact_ln__add__one__self__le__self) # label(axiom) # label(non_clause). [assumption]. 1.39/1.64 610 (all V_b_2 all V_a_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b_2),c_Groups_Ozero__class_Ozero(T_a))) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a_2),c_Groups_Ozero__class_Ozero(T_a))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b_2)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_b_2)))))) # label(fact_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 611 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) | V_w_2 = V_z_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))))) # label(fact_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 612 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)))))) # label(fact_inverse__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 613 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_k)) # label(fact_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 614 (all V_a_2 all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_a_2)))))) # label(fact_inverse__positive__iff__positive) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 615 (all V_y_2 all V_x_2 (V_y_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) & V_x_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_y_2),V_y_2)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) # label(fact_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 616 (all T_1 (class_Rings_Ocomm__semiring__1(T_1) -> class_Groups_Omonoid__mult(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Omonoid__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 617 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)))))) # label(fact_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 618 (all V_n_2 all V_m_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n_2)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_one__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 619 (all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y)) -> V_y = V_x)))) # label(fact_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 620 (all T_1 (class_Rings_Ocomm__ring__1(T_1) -> class_Rings_Ocomm__ring__1(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Ocomm__ring__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 621 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_N)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N))))))) # label(fact_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 622 (all V_w all V_z (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_z),V_w)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_w),V_z)) -> V_w = V_z))) # label(fact_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 623 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> V_m = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_mod__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 624 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)),hAPP(c_Nat_OSuc,V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_k)) # label(fact_Suc__diff__diff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 625 (all V_y all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_RComplete_Onatfloor(V_x)),c_RComplete_Onatfloor(V_y))))) # label(fact_natfloor__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 626 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (V_c != c_Groups_Ozero__class_Ozero(T_a) -> c_Divides_Odiv__class_Odiv(T_a,V_a,V_b) = c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))) # label(fact_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 627 (all V_pb_2 all V_a_2 all T_a (class_Rings_Oidom(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_pb_2 | V_a_2 = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = c_Polynomial_Osmult(T_a,V_a_2,V_pb_2)))) # label(fact_smult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 628 (all V_x hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_RComplete_Onatfloor(V_x)))) # label(fact_zero__le__natfloor) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 629 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_z)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_z)))))) # label(fact_order__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 630 (all V_z all V_w hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_w)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),V_w),V_z))),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_z))))) # label(fact_complex__mod__triangle__sub) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 631 (all V_y_2 all V_x_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (V_y_2 = c_Groups_Ozero__class_Ozero(T_a) & c_Groups_Ozero__class_Ozero(T_a) = V_x_2 <-> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y_2),V_y_2))))) # label(fact_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 632 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> V_a = c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a),V_b)))) # label(fact_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 633 (all V_m all V_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)))) # label(fact_le__add1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 634 (all V_m all V_n all V_k (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)),V_k))))) # label(fact_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 635 (all V_c all V_a all V_b all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))))) # label(fact_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 636 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_c))) # label(fact_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 637 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 638 (all V_y all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_y)))))) # label(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 639 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_z)))))) # label(fact_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 640 (all V_X_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> ((exists B_N all B_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(V_X_2,B_n))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_N))))) <-> c_SEQ_OBseq(T_a,V_X_2)))) # label(fact_Bseq__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 641 (all V_x all V_p all V_n all T_a (class_Rings_Ocomm__ring__1(T_a) -> hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),c_Polynomial_Omonom(T_a,c_Groups_Oone__class_Oone(T_a),V_n)),V_p)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)))) # label(fact_poly__replicate__append) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 642 (all V_b_2 all V_a_2 all V_ca_2 (V_ca_2 != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) -> (V_b_2 = V_a_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_ca_2),V_b_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_ca_2),V_a_2)))) # label(fact_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 643 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))))) # label(fact_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 644 (all V_K_2 all V_X_2 all V_N_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> ((all B_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_N_2),B_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(V_X_2,B_n))),V_K_2)))) -> c_SEQ_OBseq(T_a,V_X_2)))) # label(fact_BseqI2_H) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 645 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n) = hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n))))) # label(fact_div__geq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 646 (all V_b all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mult_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 647 (all V_x all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Osmult(T_a,c_Rings_Oinverse__class_Oinverse(T_a,hAPP(c_Polynomial_Ocoeff(T_a,V_x),c_Polynomial_Odegree(T_a,V_x))),V_x) = c_Polynomial_Opoly__gcd(T_a,V_x,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))))) # label(fact_poly__gcd_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 648 (all V_ry all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 649 (all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2)),V_z_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 650 (all V_pb_2 all T_a (class_Rings_Oidom(T_a) & class_Int_Oring__char__0(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_pb_2 <-> c_Polynomial_Opoly(T_a,V_pb_2) = c_Polynomial_Opoly(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)))))) # label(fact_poly__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 651 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)))))) # label(fact_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 652 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_OSuc,V_n)))) # label(fact_zero__less__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 653 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n)),V_n)) # label(fact_div__mod__equality_H) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 654 (all V_n all V_m all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_i)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))))) # label(fact_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 655 (all V_n all V_r all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_r)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_r),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_r),hAPP(c_Nat_OSuc,V_n))),V_r)))))) # label(fact_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 656 (all V_m hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_m)))) # label(fact_le__square) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 657 (all V_n_2 all V_k_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 658 (all V_n all V_m all V_k ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_k -> c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n))))) # label(fact_nat__mult__div__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 659 (all V_n all V_m ((V_n = hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) != V_n -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))))) # label(fact_mod__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 660 (all V_d all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_d)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 661 (all V_y all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),c_RealVector_Onorm__class_Onorm(T_a,V_y)))))) # label(fact_norm__mult__ineq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 662 (all V_k_2 all V_P_2 all T_a (class_Orderings_Owellorder(T_a) -> (hBOOL(hAPP(V_P_2,V_k_2)) -> hBOOL(hAPP(V_P_2,c_Orderings_Oord__class_OLeast(T_a,V_P_2)))))) # label(fact_LeastI) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 663 (all V_nat_H hAPP(c_Nat_OSuc,V_nat_H) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 664 (all V_m_2 all V_a_2 all T_a (class_Groups_Ozero(T_a) -> c_Polynomial_Omonom(T_a,V_a_2,V_m_2) = c_Polynomial_OAbs__poly(T_a,c_COMBC(tc_Nat_Onat,T_a,T_a,c_COMBC(tc_Nat_Onat,T_a,tc_fun(T_a,T_a),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_a,tc_fun(T_a,T_a)),tc_Nat_Onat,c_If(T_a)),hAPP(c_fequal,V_m_2)),V_a_2),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_monom__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 665 (all V_b all V_n all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> c_Polynomial_Omonom(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),c_Polynomial_Omonom(T_a,V_a,V_n)),c_Polynomial_Omonom(T_a,V_b,V_n)))) # label(fact_add__monom) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 666 (all T_1 (class_Groups_Ocomm__monoid__add(T_1) -> class_Groups_Omonoid__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Omonoid__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 667 (all V_m_2 all V_n_2 (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2))) <-> V_n_2 = V_m_2))) # label(fact_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 668 (all T_1 (class_Groups_Oab__group__add(T_1) -> class_Groups_Oab__group__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Oab__group__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 669 (all T_1 (class_Rings_Ocomm__semiring__1(T_1) -> class_Groups_Ocomm__monoid__mult(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Ocomm__monoid__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 670 (all V_k_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2)),V_j_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2))))) # label(fact_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 671 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 672 (all V_n all V_m ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> V_m = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)))) # label(fact_mod__if) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 673 (all V_n_2 all V_m_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m_2)) # label(fact_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 674 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> V_b = V_a)))))) # label(fact_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 675 (all V_n all T_a (class_Rings_Ocomm__semiring__1(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Polynomial_Ocoeff(T_a,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(T_a))),V_n)) & (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Polynomial_Ocoeff(T_a,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(T_a))),V_n)))) # label(fact_coeff__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 676 (all V_y_2 all V_x_2 all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_z_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2),V_y_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x_2),V_z_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_y_2),V_z_2)))))) # label(fact_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 677 (all V_a all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> c_RComplete_Onatceiling(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),V_x),c_RealDef_Oreal(tc_Nat_Onat,V_a))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_RComplete_Onatceiling(V_x)),V_a))) # label(fact_natceiling__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 678 (all V_d_2 all V_ca_2 all V_b_2 all V_a_2 all T_a (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a) -> (V_d_2 != V_ca_2 & V_b_2 != V_a_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_d_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_ca_2)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_ca_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_d_2))))) # label(fact_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 679 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 680 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b)))))) # label(fact_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 681 (all V_n_2 all V_m_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) <-> -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)))) # label(fact_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 682 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (V_x != V_y -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))))) # label(fact_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 683 (all V_k all V_a all V_b hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_k)) # label(fact_zdiv__zmod__equality) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 684 (all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> V_a = c_Rings_Oinverse__class_Oinverse(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_a)))) # label(fact_inverse__inverse__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 685 (all V_y all V_x all V_a all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Osmult(T_a,V_a,c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(T_a),V_x,V_y)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(T_a),c_Polynomial_Osmult(T_a,V_a,V_x),V_y))) # label(fact_div__smult__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 686 (all V_z all V_z_H all V_w all V_w_H (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_H),V_w)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_z_H),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_H),V_z_H)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z)))))) # label(fact_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 687 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 688 (all V_x (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != V_x -> c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,V_x)),V_x))) # label(fact_real__mult__inverse__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 689 (all V_n_2 all V_m_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)) -> (V_n_2 = V_m_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2)))))) # label(fact_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 690 (all V_z all V_y all V_x hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_z))) # label(fact_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 691 (all V_n all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n))))) # label(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 692 (all V_z all V_y all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))))) # label(fact_termination__basic__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 693 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 694 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n))) # label(fact_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 695 (all V_a all T_a (class_Rings_Omult__zero(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 696 (all V_n all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n)),c_Transcendental_Oln(V_x)) = c_Transcendental_Oln(hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),V_n)))) # label(fact_ln__realpow) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 697 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_b)),c_Rings_Oinverse__class_Oinverse(T_a,V_a))))))) # label(fact_le__imp__inverse__le__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 698 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_n))) # label(fact_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 699 (all V_x hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,c_RComplete_Onatfloor(V_x))),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))))) # label(fact_real__natfloor__add__one__gt) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 700 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Olinordered__comm__semiring__strict(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Olinordered__comm__semiring__strict) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 701 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 702 (all T_1 (class_Rings_Ocomm__semiring__1(T_1) -> class_Groups_Oone(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Oone) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 703 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2))))) # label(fact_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 704 (all V_k all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k)) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_k),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_k),V_m),V_k)))) # label(fact_int__power__div__base) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 705 (all V_qa_2 all V_pb_2 all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(T_a)),V_pb_2),V_qa_2) = c_Polynomial_OAbs__poly(T_a,hAPP(hAPP(c_COMBS(tc_Nat_Onat,T_a,T_a),hAPP(c_COMBB(T_a,tc_fun(T_a,T_a),tc_Nat_Onat,c_Groups_Ominus__class_Ominus(T_a)),c_Polynomial_Ocoeff(T_a,V_pb_2))),c_Polynomial_Ocoeff(T_a,V_qa_2))))) # label(fact_minus__poly__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 706 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 707 (all V_y all V_n all V_x (c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_y,V_n) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_y),V_x)) -> (exists B_q hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),B_q)) = V_x)))) # label(fact_nat__mod__eq__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 708 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 709 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2),V_g_2)) <-> (all B_x hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))))))) # label(fact_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 710 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 711 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a != V_b -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 712 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_x)))))) # label(fact_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 713 (all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 714 (all V_a all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),V_a) = c_Groups_Oone__class_Oone(T_a)))) # label(fact_field__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 715 (all V_b all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> V_a = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 716 (all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K ((all B_a all B_b hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),B_a),B_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,B_a)),c_RealVector_Onorm__class_Onorm(T_a,B_b))),B_K)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K)))))) # label(fact_mult_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 717 (all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))))) # label(fact_less__add__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 718 (all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 719 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_k))) # label(fact_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 720 (all V_x all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Opoly__gcd(T_a,V_x,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(T_a))) = c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(T_a)))) # label(fact_poly__gcd__1__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 721 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),V_a) = c_Groups_Oone__class_Oone(T_a)))) # label(fact_left__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 722 (all V_n all V_a all T_a (class_Groups_Ozero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> V_n = c_Polynomial_Odegree(T_a,c_Polynomial_Omonom(T_a,V_a,V_n))))) # label(fact_degree__monom__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 723 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> V_y = V_x | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)))) # label(fact_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 724 (all V_y all V_x all V_a all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,c_Polynomial_Osmult(T_a,V_a,V_y)) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(T_a),V_x,V_y)))) # label(fact_mod__smult__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 725 (all V_r all V_q all V_y all V_x all V_a all T_a (class_Fields_Ofield(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q,V_r) -> c_Polynomial_Opdivmod__rel(T_a,V_x,c_Polynomial_Osmult(T_a,V_a,V_y),c_Polynomial_Osmult(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_a),V_q),V_r))))) # label(fact_pdivmod__rel__smult__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 726 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (V_b = V_c -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_c)))))) # label(fact_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 727 (all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 728 (all V_a all V_p all T_a (class_Rings_Oidom(T_a) -> (V_p != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Oorder(T_a,V_a,V_p)),c_Polynomial_Odegree(T_a,V_p)))))) # label(fact_order__degree) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 729 -(all B_q (c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(tc_Complex_Ocomplex,v_pa____) = c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(tc_Complex_Ocomplex,B_q) -> -(all B_x hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),v_c____),B_x)) = hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,B_q),B_x)))) # label(fact__096_B_Bthesis_O_A_I_B_Bq_O_A_091_124_Apsize_Aq_A_061_Apsize_Ap_059_AALL_Ax_O_Apoly_Aq_Ax_A_061_Apoly_Ap_A_Ic_A_L_Ax_J_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 730 (all V_q all V_y all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),V_q))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 731 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealDef_Oreal(tc_Nat_Onat,V_n_2))))) # label(fact_real__of__nat__gt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 732 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Groups_Oordered__comm__monoid__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Oordered__comm__monoid__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 733 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 734 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 735 (all B_w hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_q____),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)))),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),B_w))))) # label(fact_cq0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 736 (all V_a all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_x)),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x),c_RComplete_Onatfloor(V_a))))) # label(fact_le__natfloor) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 737 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)),V_k)) # label(fact_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 738 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_b)),c_Rings_Oinverse__class_Oinverse(T_a,V_a))))))) # label(fact_less__imp__inverse__less__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 739 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 740 (all V_n_2 all V_m_2 (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2))))) # label(fact_not__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 741 (all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m)),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)))) # label(fact_Suc__leD) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 742 (all V_x hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x),c_RealDef_Oreal(tc_Nat_Onat,c_RComplete_Onatceiling(V_x))))) # label(fact_real__natceiling__ge) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 743 (all V_b all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_div__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 744 (all V_pb_2 all T_a (class_Groups_Ozero(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_pb_2 <-> hAPP(c_Polynomial_Ocoeff(T_a,V_pb_2),c_Polynomial_Odegree(T_a,V_pb_2)) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_leading__coeff__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 745 (all V_k all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),V_j)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_j))))) # label(fact_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 746 (all V_b c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 747 (all V_q all V_a all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> c_Polynomial_Osmult(T_a,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),c_Polynomial_Osmult(T_a,V_a,V_q)))) # label(fact_mult__smult__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 748 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))))))) # label(fact_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 749 (all V_n all V_a all T_a (class_Groups_Ozero(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,c_Polynomial_Omonom(T_a,V_a,V_n))),V_n)))) # label(fact_degree__monom__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 750 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 751 (all V_x_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_x_2 <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_x_2)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))))) # label(fact_norm__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 752 (all V_s all V_y all V_r all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_y)),V_s)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_y))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),V_r),V_s))))))) # label(fact_norm__add__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 753 (all V_b all V_c all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 754 (all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) = hAPP(c_Nat_OSuc,V_n)) # label(fact_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 755 (all V_b_2 all V_a_2 all T_a (class_Rings_Oring__no__zero__divisors(T_a) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_a) | c_Groups_Ozero__class_Ozero(T_a) = V_b_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 756 (all T_a (class_Groups_Ozero(T_a) -> c_Polynomial_OAbs__poly(T_a,c_COMBK(T_a,tc_Nat_Onat,c_Groups_Ozero__class_Ozero(T_a))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)))) # label(fact_zero__poly__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 757 (all V_a_2 all V_x_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_RComplete_Onatceiling(V_x_2)),V_a_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2),c_RealDef_Oreal(tc_Nat_Onat,V_a_2)))))) # label(fact_natceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 758 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x_2),V_y_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2),V_y_2)) & V_x_2 != V_y_2)) # label(fact_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 759 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (V_b = V_c -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_c)))))) # label(fact_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 760 (all V_r all V_q all V_r_H all V_q_H all V_b (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q_H)),V_r_H)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r_H)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_q),V_q_H))))))) # label(fact_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 761 (all V_z all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(T_a),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(T_a),V_x,V_y),V_z) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(T_a),V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_y),V_z)))) # label(fact_poly__div__mult__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 762 (all V_n all V_a all T_a (class_Power_Opower(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 763 (all V_n all V_m all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_m)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_m),V_n))))))) # label(fact_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 764 (all B_w hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),v_c____))),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),B_w))))) # label(fact_c) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 765 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_j),V_k))))) # label(fact_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 766 (all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(T_a,V_x))))) # label(fact_norm__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 767 (all V_c all V_b all V_a all T_a (class_Groups_Oab__semigroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 768 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),V_y_2)) | V_y_2 = V_x_2))) # label(fact_order__le__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 769 (all V_y all V_x all V_c all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_c),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_c),c_RealVector_Onorm__class_Onorm(T_a,V_y)))) -> V_x = c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_norm__ratiotest__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 770 (all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_rx)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 771 (all V_b all V_c all V_a all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))))) # label(fact_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 772 (all V_a_2 all V_times_2 all V_one_2 all T_a V_one_2 = hAPP(hAPP(c_Power_Opower_Opower(T_a,V_one_2,V_times_2),V_a_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_power_Opower_Opower__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 773 (all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> (all B_y exists B_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B_y),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,B_n)),V_x)))))) # label(fact_reals__Archimedean3) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 774 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 775 (all V_r2 all V_q2 all V_r1 all V_q1 all V_y all V_x all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q1,V_r1) -> (c_Polynomial_Opdivmod__rel(T_a,V_x,V_y,V_q2,V_r2) -> V_q2 = V_q1)))) # label(fact_pdivmod__rel__unique__div) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 776 (all V_y_2 all V_x_2 all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_z_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x_2),V_z_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_y_2),V_z_2)))))) # label(fact_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 777 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_a_2)))))) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 778 (all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_norm__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 779 (all V_n_2 all V_m_2 all V_k_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2) <-> V_m_2 = V_n_2)) # label(fact_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 780 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (V_y_2 = V_x_2 <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),V_y_2))))) # label(fact_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 781 (all V_q all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),V_x))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 782 (all V_k_2 all V_j_2 all V_i_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2),V_j_2)) -> (V_k_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_i_2) = V_j_2))) # label(fact_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 783 (all B_z hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_q____),B_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,c_Polynomial_Osmult(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_q____),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex))),v_q____)),B_z)),hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_q____),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)))) # label(fact_qr) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 784 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 785 (all V_n_2 all V_a_2 all T_a (class_Power_Opower(T_a) & class_Rings_Ono__zero__divisors(T_a) & class_Rings_Ozero__neq__one(T_a) & class_Rings_Omult__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),V_n_2) <-> c_Groups_Ozero__class_Ozero(T_a) = V_a_2 & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2))) # label(fact_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 786 (all V_p all V_r exists B_m ((all B_z (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B_z)),V_r)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,V_p),B_z))),B_m)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_m)))) # label(fact_poly__bound__exists) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 787 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Groups_Oordered__cancel__ab__semigroup__add(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 788 c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),v_c____) -> (exists B_z c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_pa____),B_z)) # label(fact_calculation) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 789 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 790 (all V_n V_n = c_RComplete_Onatfloor(c_RealDef_Oreal(tc_Nat_Onat,V_n))) # label(fact_natfloor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 791 (all V_a all V_p exists B_q (c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(tc_Complex_Ocomplex,B_q) = c_Fundamental__Theorem__Algebra__Mirabelle_Opsize(tc_Complex_Ocomplex,V_p) & (all B_x hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,V_p),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),V_a),B_x)) = hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,B_q),B_x)))) # label(fact_poly__offset) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 792 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = V_a)) # label(fact_add__0__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 793 (all V_n_2 all V_m_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)) <-> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 794 (all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = c_Groups_Oone__class_Oone(T_a) -> V_b = c_Rings_Oinverse__class_Oinverse(T_a,V_a)))) # label(fact_inverse__unique) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 795 (all V_w all V_z (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_w),V_z)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_z),V_w)))) # label(fact_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 796 (all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) & class_RealVector_Oreal__normed__div__algebra(T_a) -> c_RealVector_Onorm__class_Onorm(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_a)) = c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a)))) # label(fact_norm__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 797 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 798 (all V_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))) # label(fact_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 799 (all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Groups_Oone__class_Oone(T_a))))) # label(fact_zero__less__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 800 (all V_b_2 all V_a_2 all V_ca_2 (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != V_ca_2 -> (V_a_2 = V_b_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_a_2),V_ca_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_b_2),V_ca_2)))) # label(fact_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 801 (all V_x_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_RComplete_Onatceiling(V_x_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))) # label(fact_natceiling__le__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 802 (all T_a (class_Fields_Ofield__inverse__zero(T_a) -> c_Rings_Oinverse__class_Oinverse(T_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_field__inverse__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 803 (all V_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_a)) -> (exists B_x (V_a = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B_x),V_n) & (all B_y (hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B_y),V_n) = V_a & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_y)) -> B_y = B_x)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_x))))))) # label(fact_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 804 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_i)),V_k))))) # label(fact_less__trans__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 805 (all V_R_2 all V_Q_2 all V_P_2 all T_a all T_c all T_b hAPP(hAPP(V_P_2,V_R_2),hAPP(V_Q_2,V_R_2)) = hAPP(hAPP(hAPP(c_COMBS(T_b,T_c,T_a),V_P_2),V_Q_2),V_R_2)) # label(help_c__COMBS__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 806 (all T_1 (class_Rings_Ocomm__semiring__0(T_1) -> class_Rings_Osemiring(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Osemiring) # label(axiom) # label(non_clause). [assumption]. 1.39/1.65 807 (all V_a all V_b all V_c all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 808 (all V_n_2 all V_m_2 (hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_n_2 & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_one__is__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 809 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Transcendental_Oln(V_x_2)),c_Transcendental_Oln(V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x_2),V_y_2)))))) # label(fact_ln__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 810 (all V_u all V_y all V_x1 all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x1)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x1),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_u))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_y),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,V_x))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_u),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,V_x1)))))))) # label(fact_real__mult__inverse__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 811 (all V_d all V_c all V_b all V_a all V_r all T_a (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_r -> (V_a = V_b & V_c != V_d -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_c)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_d)))))) # label(fact_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 812 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_m))))) # label(fact_diff__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 813 (all V_a all T_a (class_Power_Opower(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_power__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 814 (all V_n_2 all V_m_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)) | hAPP(c_Nat_OSuc,V_n_2) = V_m_2 <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 815 (all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 816 (all V_q all V_p all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),V_q))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 817 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a_2),c_Groups_Ozero__class_Ozero(T_a))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 818 (all V_k all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),V_j)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_k))))) # label(fact_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 819 (all V_d all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),V_d))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 820 (all V_n all V_m all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),V_n))) # label(fact_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 821 (all V_q all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 822 (all V_r_H all V_q_H all V_b_H (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_q_H),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))))) # label(fact_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 823 (all V_y all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))) # label(fact_mult__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 824 (all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Transcendental_Oln(V_x_2)))))) # label(fact_ln__ge__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 825 (all V_k all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i),V_j)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_i)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_j))))) # label(fact_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 826 (all V_q all V_p all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_p)),c_Polynomial_Odegree(T_a,V_q))) -> c_Polynomial_Odegree(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),V_q)) = c_Polynomial_Odegree(T_a,V_q)))) # label(fact_degree__add__eq__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 827 (all V_b_H all V_b all V_a all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b_H)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_b_H)))) # label(fact_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 828 (all V_b all V_a all V_c all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))))) # label(fact_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 829 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2)) # label(fact_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 830 (all V_r_2 all V_qa_2 all V_x_2 all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x_2,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)),V_qa_2,V_r_2) <-> V_r_2 = V_x_2 & c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_qa_2))) # label(fact_pdivmod__rel__by__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 831 (all V_q all V_n all V_p all T_a (class_Groups_Ocomm__monoid__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_p)),V_n)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,V_q)),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Polynomial_Odegree(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),V_q))),V_n)))))) # label(fact_degree__add__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 832 (all V_r_2 all V_qa_2 all V_y_2 all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)),V_y_2,V_qa_2,V_r_2) <-> c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_r_2 & c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_qa_2))) # label(fact_pdivmod__rel__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 833 (all V_x all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n)),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))) -> c_RComplete_Onatceiling(V_x) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))))) # label(fact_natceiling__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 834 (all V_m all V_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_m)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n) = V_m)) # label(fact_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 835 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 836 (all V_a_2 all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_a_2)))))) # label(fact_inverse__nonnegative__iff__nonnegative) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 837 (all V_c all V_a all V_b all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a) -> V_b = V_c))) # label(fact_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 838 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))) # label(fact_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 839 (all V_n all V_m (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = V_m -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m | c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_n)) # label(fact_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 840 (all V_n_2 all V_a_2 all T_a (class_Groups_Ozero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_a_2 <-> c_Polynomial_Omonom(T_a,V_a_2,V_n_2) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a))))) # label(fact_monom__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 841 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)))) # label(fact_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 842 (all V_i all V_j all V_k (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_j)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)),V_i) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_k))) # label(fact_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 843 (all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Transcendental_Oln(V_x))))) # label(fact_ln__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 844 (all T_1 (class_Rings_Oidom(T_1) -> class_Rings_Oring__no__zero__divisors(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Oring__no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 845 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_N)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N))))))) # label(fact_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 846 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),V_d)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))))))) # label(fact_add__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 847 (all V_c all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c))) # label(fact_mod__diff__left__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 848 (all V_x_2 all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_x_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x_2),c_Groups_Oone__class_Oone(T_a)))))) # label(fact_one__less__inverse__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 849 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_x)))))) # label(fact_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 850 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_x))))) # label(fact_leI) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 851 (all V_ry all V_rx all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 852 (all V_n all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)))) # label(fact_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 853 (all V_n all V_m (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> V_m = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)))) # label(fact_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 854 (all V_d_2 all V_b_2 all V_ca_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a_2),V_b_2)),V_e_2)),V_ca_2)),V_d_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2)),V_ca_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_e_2)),V_d_2)))))) # label(fact_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 855 (all V_X_2 all V_K_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_K_2)) -> ((all B_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(V_X_2,B_n))),V_K_2))) -> c_SEQ_OBseq(T_a,V_X_2))))) # label(fact_BseqI) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 856 (all V_y all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_x,V_y)))))) # label(fact_Divides_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 857 (all V_X_2 all T_a (class_Orderings_Oorder(T_a) -> ((all B_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(V_X_2,B_n)),hAPP(V_X_2,hAPP(c_Nat_OSuc,B_n))))) | (all B_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(V_X_2,hAPP(c_Nat_OSuc,B_n))),hAPP(V_X_2,B_n)))) <-> c_SEQ_Omonoseq(T_a,V_X_2)))) # label(fact_monoseq__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 858 (all V_k all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_k)),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_n,V_k))))) # label(fact_div__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 859 (all V_q all V_p all T_a (class_Rings_Oidom(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) != V_p -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) != V_q -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_p)),c_Polynomial_Odegree(T_a,V_q)) = c_Polynomial_Odegree(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)))))) # label(fact_degree__mult__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 860 (all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_a))) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)))))) # label(fact_inverse__positive__imp__positive) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 861 (all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))) # label(fact_mult__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 862 (all V_r_H all V_q_H all V_b_H all V_r all V_q all V_b (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_b_H),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_q_H),V_q))))))))) # label(fact_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 863 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))) # label(fact_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 864 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_x)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))))) # label(fact_leD) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 865 (all V_q all V_p all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Polynomial_Opos__poly(T_a,V_p) -> (c_Polynomial_Opos__poly(T_a,V_q) -> c_Polynomial_Opos__poly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)))))) # label(fact_pos__poly__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 866 (all V_r (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_r)) -> (exists B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_r),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_n)))) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,B_n)),V_r)))))) # label(fact_reals__Archimedean6a) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 867 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Oone__class_Oone(T_a) = c_Divides_Odiv__class_Odiv(T_a,V_a,V_a)))) # label(fact_div__self) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 868 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Odiv(T_a,V_a,V_b)),V_b)))) # label(fact_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 869 (all V_i all V_j all V_k (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)))) # label(fact_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 870 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))))))) # label(fact_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 871 (all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)))) # label(fact_gr0I) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 872 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_N)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))))) # label(fact_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 873 (all V_y all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K)) & (all B_x hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),B_x),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,B_x)),B_K)))))))) # label(fact_mult__left_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 874 (all V_c all V_b all V_a all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) -> V_b = V_c))) # label(fact_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 875 (all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_RComplete_Onatfloor(V_x)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_RComplete_Onatfloor(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),V_x),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))))) # label(fact_natfloor__add__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 876 (all V_y_2 all V_x_2 all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = c_Polynomial_Opoly__gcd(T_a,V_x_2,V_y_2) <-> c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_x_2 & c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_y_2))) # label(fact_poly__gcd__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 877 (all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),c_Groups_Oone__class_Oone(tc_Int_Oint)) = V_z) # label(fact_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 878 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y_2),V_x_2)) <-> -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),V_y_2))))) # label(fact_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 879 (all V_R_2 all V_Q_2 all V_P_2 all T_c all T_a all T_b hAPP(V_P_2,hAPP(V_Q_2,V_R_2)) = hAPP(hAPP(c_COMBB(T_b,T_a,T_c,V_P_2),V_Q_2),V_R_2)) # label(help_c__COMBB__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 880 (all V_n all V_k all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)))) # label(fact_add__leD1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 881 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))))) # label(fact_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 882 (all V_m all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n))) # label(fact_mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 883 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 884 (all V_c all V_b all V_e all V_a all T_a (class_Rings_Osemiring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_e)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_e)),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_e)),V_c))) # label(fact_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 885 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Olinordered__idom(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Olinordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 886 (all V_c all V_a all V_b all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 887 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))))))) # label(fact_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 888 (all V_q all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Ocoeff(T_a,V_p),c_Polynomial_Odegree(T_a,V_p))),hAPP(c_Polynomial_Ocoeff(T_a,V_q),c_Polynomial_Odegree(T_a,V_q))) = hAPP(c_Polynomial_Ocoeff(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_p)),c_Polynomial_Odegree(T_a,V_q))))) # label(fact_coeff__mult__degree__sum) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 889 (all V_b all V_a_H all V_a all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_a_H)),V_b) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_H),V_b)))) # label(fact_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 890 (all T_a (class_RealVector_Oreal__normed__algebra__1(T_a) -> c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oone__class_Oone(T_a)))) # label(fact_norm__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 891 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 892 (all V_X_2 all T_b (class_RealVector_Oreal__normed__vector(T_b) -> ((exists B_K ((all B_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_b,hAPP(V_X_2,B_n))),B_K))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K)))) <-> (exists B_N all B_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_b,hAPP(V_X_2,B_n))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_N)))))))) # label(fact_lemma__NBseq__def2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 893 (all V_n all T_a (class_Groups_Ozero(T_a) -> c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = c_Polynomial_Omonom(T_a,c_Groups_Ozero__class_Ozero(T_a),V_n))) # label(fact_monom__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 894 (all V_n all V_m hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_m))) # label(fact_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 895 (all V_k_2 all V_n_2 all V_P_2 ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (all B_i ((exists B_j (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)) & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j))) -> hBOOL(hAPP(V_P_2,B_i))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (all B_i ((exists B_j (V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j)))) -> hBOOL(hAPP(V_P_2,B_i))))) <-> hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_n_2,V_k_2))))) # label(fact_split__zdiv) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 896 (all V_b all V_c all V_a all T_a (class_Rings_Olinordered__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 897 (all V_a all T_a (class_RealVector_Oreal__normed__div__algebra(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a)) = c_RealVector_Onorm__class_Onorm(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_a))))) # label(fact_nonzero__norm__inverse) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 898 (all V_a_2 all V_b_2 all T_a (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a) -> (V_b_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b_2),V_a_2) <-> c_Groups_Ozero__class_Ozero(T_a) = V_a_2))) # label(fact_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 899 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n)),V_n))) # label(fact_Divides_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 900 (all V_ca_2 all V_b_2 all V_a_2 all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (V_b_2 = V_ca_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_ca_2)))) # label(fact_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 901 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Oone__class_Oone(T_a) != c_Groups_Ozero__class_Ozero(T_a))) # label(fact_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 902 (all V_x all V_q all V_p all T_a (class_Rings_Ocomm__ring(T_a) -> hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)))) # label(fact_poly__diff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 903 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_x)),V_x))))))) # label(fact_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 904 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))))) # label(fact_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 905 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))) # label(fact_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 906 (all V_c all V_b all V_a all T_a (class_Groups_Ocancel__ab__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) -> V_c = V_b))) # label(fact_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 907 (all V_y all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_RComplete_Onatceiling(V_x)),c_RComplete_Onatceiling(V_y))))) # label(fact_natceiling__mono) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 908 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_w_2),V_z_2)))) # label(fact_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 909 (all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K ((all B_x hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),B_x))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,B_x)),B_K)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K)))))) # label(fact_mult__right_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 910 (all V_i_2 all V_P_2 all V_pb_2 ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_pb_2)) -> (hBOOL(hAPP(V_P_2,B_i)) -> hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_i),V_pb_2)))))) -> (hBOOL(hAPP(V_P_2,V_i_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),V_pb_2)) -> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))))) # label(fact_mod__induct__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 911 (all V_n all V_a all T_a (class_Rings_Oring__1__no__zero__divisors(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n) != c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 912 (all V_p all V_n all T_a (class_Groups_Ozero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Polynomial_Odegree(T_a,V_p))) -> (exists B_i (c_Groups_Ozero__class_Ozero(T_a) != hAPP(c_Polynomial_Ocoeff(T_a,V_p),B_i) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),B_i))))))) # label(fact_less__degree__imp) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 913 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 914 (all V_w all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_w)) # label(fact_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 915 (all T_1 (class_Rings_Oidom(T_1) -> class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 916 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) <-> V_n_2 != V_m_2)) # label(fact_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 917 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_m)) -> V_m != V_n)) # label(fact_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 918 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Odiv(T_a,V_a,V_b)),c_Groups_Oone__class_Oone(T_a)) = c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a),V_b)))) # label(fact_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 919 (all V_m all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),V_j)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_j))))) # label(fact_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 920 (all V_b_2 all V_a_2 all V_ca_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_ca_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ca_2),V_b_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a_2),V_b_2)))))) # label(fact_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 921 (all V_b all V_a_H all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_a),V_a_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a_H,V_b)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b)))))) # label(fact_zdiv__mono1__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 922 (all V_t all V_s (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_s),V_t)) -> V_s != V_t)) # label(fact_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 923 (all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Nat_OSuc,V_n)) # label(fact_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 924 (all V_b all V_a all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Opoly__gcd(T_a,V_a,V_b) = c_Polynomial_Opoly__gcd(T_a,V_b,V_a))) # label(fact_poly__gcd_Ocommute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 925 (all V_n all V_b (c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != V_b -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> (exists B_z hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),V_b),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),B_z),V_n))))),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))))))) # label(fact_reduce__poly__simple) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 926 (all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_m)) -> V_m = V_n))) # label(fact_le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 927 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 928 (all V_n_2 all V_P_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(V_P_2,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_n_2,V_k_2)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2))) <-> (all B_i all B_j (V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j)) -> hBOOL(hAPP(hAPP(V_P_2,B_i),B_j))))))) # label(fact_split__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 929 (all V_n_2 all V_m_2 all V_k_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) <-> V_n_2 = V_m_2 | V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 930 (all V_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),V_n))) # label(fact_le__refl) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 931 (all V_n_2 all V_m_2 all V_k_2 (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_n_2 = V_m_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))) # label(fact_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 932 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)))) # label(fact_Suc__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 933 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 934 (all V_w_2 all V_z_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_z_2),V_w_2)) & V_w_2 != V_z_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),V_w_2)))) # label(fact_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 935 (all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> (exists B_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_n)))),V_x))))) # label(fact_reals__Archimedean) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 936 (all V_n_2 all V_m_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2),V_n_2)) <-> (exists B_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k) = V_n_2))) # label(fact_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 937 (all V_z all V_y all V_x hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_y),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_y),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_z))) # label(fact_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 938 (all V_b all V_a all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))))) # label(fact_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 939 (all V_x all V_n all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) = hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Omonom(T_a,V_a,V_n)),V_x))) # label(fact_poly__monom) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 940 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c))) # label(fact_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 941 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c))) # label(fact_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 942 (all V_X_2 all T_a (class_Orderings_Oorder(T_a) -> (c_SEQ_Oincseq(T_a,V_X_2) <-> (all B_m all B_n (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m),B_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(V_X_2,B_m)),hAPP(V_X_2,B_n)))))))) # label(fact_incseq__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 943 (all V_a_2 all V_x_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_a_2)),V_x_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_a_2),c_RComplete_Onatfloor(V_x_2)))))) # label(fact_le__natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 944 (all V_z3 all V_z2 all V_z1 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_z3)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_z2)),V_z3)) # label(fact_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 945 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (V_y != V_x -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))))) # label(fact_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 946 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 947 (all V_c all V_a all V_b all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Opoly__gcd(T_a,V_b,c_Polynomial_Opoly__gcd(T_a,V_a,V_c)) = c_Polynomial_Opoly__gcd(T_a,V_a,c_Polynomial_Opoly__gcd(T_a,V_b,V_c)))) # label(fact_poly__gcd_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 948 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 949 (all V_n all V_p all T_a (class_Groups_Ozero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_p)),V_n)) -> hAPP(c_Polynomial_Ocoeff(T_a,V_p),V_n) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_coeff__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 950 (all V_a_2 all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_a_2)),c_Groups_Ozero__class_Ozero(T_a))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a_2),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_inverse__nonpositive__iff__nonpositive) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 951 (all V_b all V_a all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))))) # label(fact_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 952 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n))))))) # label(fact_power__gt1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 953 (all V_q all V_r all V_b all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))))) # label(fact_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 954 (all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Transcendental_Oln(V_x))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x))))) # label(fact_ln__gt__zero__imp__gt__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 955 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Oordered__cancel__semiring(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Oordered__cancel__semiring) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 956 (all V_x_2 all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2),V_g_2)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))))) # label(fact_le__funE) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 957 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_div__neg__pos__less0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 958 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m)),V_n)))) # label(fact_Suc__leI) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 959 (all V_k all V_j all V_i (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),V_j)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j),V_k)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i),V_k))))) # label(fact_le__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 960 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Rings_Oordered__comm__semiring(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Oordered__comm__semiring) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 961 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> c_Groups_Ozero__class_Ozero(T_a) != c_Rings_Oinverse__class_Oinverse(T_a,V_a)))) # label(fact_nonzero__imp__inverse__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 962 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Polynomial_Opoly(T_a,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(T_a))),V_x))) # label(fact_poly__1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.66 963 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_y)))))) # label(fact_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 964 (all T_1 (class_Rings_Oidom(T_1) -> class_Rings_Ono__zero__divisors(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Ono__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 965 (all V_m_2 all V_n_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2),V_m_2)) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_n_2)),c_RealDef_Oreal(tc_Nat_Onat,V_m_2))))) # label(fact_real__of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 966 (all T_2 all T_1 (class_Orderings_Oord(T_1) -> class_Orderings_Oord(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oord) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 967 (all V_b all V_a_H all V_a (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_a),V_a_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a_H,V_b)))))) # label(fact_zdiv__mono1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 968 (all V_n_2 all V_k_2 all V_m_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2) <-> V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_m_2 = V_n_2)) # label(fact_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 969 (all V_x (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) -> c_RComplete_Onatceiling(V_x) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_natceiling__neg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 970 (all V_q all V_p all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Polynomial_Opos__poly(T_a,V_p) -> (c_Polynomial_Opos__poly(T_a,V_q) -> c_Polynomial_Opos__poly(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),V_p),V_q)))))) # label(fact_pos__poly__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 971 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)))) # label(fact_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 972 (all V_n all V_m hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) # label(fact_add__Suc) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 973 (all V_b_2 all V_ca_2 all V_a_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_ca_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b_2),V_ca_2))) <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a_2),V_b_2))))) # label(fact_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 974 (all V_r_H all V_q_H all V_b_H (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_q_H)))))) # label(fact_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 975 (all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) = V_a) # label(fact_zmod__zdiv__equality) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 976 (all V_pb_2 all T_a (class_Groups_Ozero(T_a) -> c_Orderings_Oord__class_OLeast(tc_Nat_Onat,hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat,c_HOL_OAll(tc_Nat_Onat)),c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat,c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat,c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat,c_fimplies)),c_Orderings_Oord__class_Oless(tc_Nat_Onat))),c_COMBC(tc_Nat_Onat,T_a,tc_HOL_Obool,hAPP(c_COMBB(T_a,tc_fun(T_a,tc_HOL_Obool),tc_Nat_Onat,c_fequal),c_Polynomial_Ocoeff(T_a,V_pb_2)),c_Groups_Ozero__class_Ozero(T_a))))) = c_Polynomial_Odegree(T_a,V_pb_2))) # label(fact_degree__def) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 977 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_add__is__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 978 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a_2,V_b_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2))))) # label(fact_neg__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 979 (all T_1 (class_Rings_Olinordered__idom(T_1) -> class_Orderings_Olinorder(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Orderings_Olinorder) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 980 (all V_x all V_q all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(c_Polynomial_Opoly(T_a,V_p),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) = hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Opcompose(T_a,V_p,V_q)),V_x))) # label(fact_poly__pcompose) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 981 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_n)))) # label(fact_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 982 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y))))) # label(fact_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 983 (all V_q all V_r all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_r),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q)) = V_a -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_a)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)),V_q)))))) # label(fact_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 984 (all V_y all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K ((all B_x hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),B_x),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,B_x)),B_K)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B_K)))))) # label(fact_mult__left_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 985 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> V_y != V_x))) # label(fact_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 986 (all V_x all V_q all V_p all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)),hAPP(c_Polynomial_Opoly(T_a,V_q),V_x)) = hAPP(c_Polynomial_Opoly(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_x))) # label(fact_poly__mult) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 987 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_n)))) # label(fact_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 988 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (V_c = c_Groups_Ozero__class_Ozero(T_a) -> c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = c_Groups_Ozero__class_Ozero(T_a)) & (c_Groups_Ozero__class_Ozero(T_a) != V_c -> c_Divides_Odiv__class_Odiv(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = c_Divides_Odiv__class_Odiv(T_a,V_a,V_b)))) # label(fact_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 989 (all V_d_2 all V_ca_2 all V_b_2 all V_a_2 all T_a (class_Groups_Oab__group__add(T_a) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_ca_2),V_d_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a_2),V_b_2) -> (V_d_2 = V_ca_2 <-> V_a_2 = V_b_2)))) # label(fact_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 990 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 991 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))))))) # label(fact_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 992 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_x)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 993 (all V_x_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) -> c_SEQ_Omonoseq(tc_RealDef_Oreal,hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x_2))))) # label(fact_monoseq__realpow) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 994 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Odiv(T_a,V_a,V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) = V_a)) # label(fact_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 995 (all V_i_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2),V_i_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_i_2,V_k_2)))))) # label(fact_pos__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 996 (all V_b all V_a all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),c_Groups_Ozero__class_Ozero(T_a))) & hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))))) # label(fact_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 997 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a = V_b -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_c),V_a)))))) # label(fact_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 998 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__lessD) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 999 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_b = V_a -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_b),V_c)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_c)))))) # label(fact_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1000 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1001 (all V_k_2 all V_X_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_SEQ_OBseq(T_a,hAPP(c_COMBB(tc_Nat_Onat,T_a,tc_Nat_Onat,V_X_2),c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2))) -> c_SEQ_OBseq(T_a,V_X_2)))) # label(fact_Bseq__offset) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1002 (all T_1 (class_Rings_Ocomm__semiring__0(T_1) -> class_Rings_Ocomm__semiring__0(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Rings_Ocomm__semiring__0) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1003 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> c_Rings_Oinverse__class_Oinverse(T_a,c_Rings_Oinverse__class_Oinverse(T_a,V_a)) = V_a))) # label(fact_nonzero__inverse__inverse__eq) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1004 (all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))),c_Rings_Oinverse__class_Oinverse(T_a,V_b)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_a)),c_Rings_Oinverse__class_Oinverse(T_a,V_b)))))) # label(fact_division__ring__inverse__add) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1005 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))))) # label(fact_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1006 (all V_n all V_q all V_p all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(c_Polynomial_Ocoeff(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(T_a)),V_p),V_q)),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Polynomial_Ocoeff(T_a,V_p),V_n)),hAPP(c_Polynomial_Ocoeff(T_a,V_q),V_n)))) # label(fact_coeff__diff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1007 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_2),V_a_2) = c_Groups_Ozero__class_Ozero(T_a) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1008 (all V_n all T_a (class_Groups_Omonoid__mult(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),V_n))) # label(fact_power__one) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1009 (all V_n all V_m hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n)),V_m))) # label(fact_div__le__dividend) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1010 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x),V_y)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y),V_x)) -> V_y = V_x)))) # label(fact_order__antisym) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1011 (all V_b all V_c all V_a all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c))) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_a),V_b))))) # label(fact_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1012 (all V_a all T_a (class_Rings_Omult__zero(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1013 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_m_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1014 (all V_n all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1015 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1016 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)))) # label(fact_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.39/1.67 1017 (all V_b all V_n all V_a all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(T_a)),c_Polynomial_Omonom(T_a,V_a,V_n)),c_Polynomial_Omonom(T_a,V_b,V_n)) = c_Polynomial_Omonom(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_n))) # label(fact_diff__monom) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1018 (all V_x_2 all V_y_2 all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_y_2),V_x_2)) -> (V_x_2 = V_y_2 <-> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a,V_x_2),V_y_2)))))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1019 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> V_y != V_x))) # label(fact_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1020 (all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),V_x),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Transcendental_Oln(V_x)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))))) # label(fact_ln__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1021 (all V_x all T_a (class_Fields_Ofield(T_a) -> c_Polynomial_Opdivmod__rel(T_a,V_x,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)),V_x))) # label(fact_pdivmod__rel__by__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1022 (all V_n all V_m (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),V_n)) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m),hAPP(c_Nat_OSuc,V_n))))) # label(fact_le__SucI) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1023 (all V_n_2 all V_x_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_n_2))))) # label(fact_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1024 (all V_n all V_m hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),hAPP(c_Nat_OSuc,V_m)))) # label(fact_diff__less__Suc) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1025 (all V_a all V_n all V_m all T_a (class_Groups_Ozero(T_a) -> (V_n != V_m -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Polynomial_Ocoeff(T_a,c_Polynomial_Omonom(T_a,V_a,V_m)),V_n)) & (V_m = V_n -> hAPP(c_Polynomial_Ocoeff(T_a,c_Polynomial_Omonom(T_a,V_a,V_m)),V_n) = V_a))) # label(fact_coeff__monom) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1026 (all V_m all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a))),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1027 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))))) # label(fact_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1028 (all V_n_2 all V_k_2 all V_m_2 (V_m_2 = V_n_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_k_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_k_2))) # label(fact_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1029 (all V_d_2 all V_m_2 (c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,V_d_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> (exists B_q V_m_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_d_2),B_q)))) # label(fact_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1030 (all V_r_2 all V_qa_2 all V_y_2 all V_x_2 all T_a (class_Fields_Ofield(T_a) -> (c_Polynomial_Opdivmod__rel(T_a,V_x_2,V_y_2,V_qa_2,V_r_2) <-> (V_y_2 = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) -> c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_qa_2) & (V_y_2 != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) -> c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a)) = V_r_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(T_a,V_r_2)),c_Polynomial_Odegree(T_a,V_y_2)))) & hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(T_a)),V_qa_2),V_y_2)),V_r_2) = V_x_2))) # label(fact_pdivmod__rel__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1031 (all V_x all V_p all V_a all T_a (class_Rings_Ocomm__semiring__0(T_a) -> hAPP(c_Polynomial_Opoly(T_a,c_Polynomial_Osmult(T_a,V_a,V_p)),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Polynomial_Opoly(T_a,V_p),V_x)))) # label(fact_poly__smult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1032 (all V_y all V_x all V_xa all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_y)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_y)))) # label(fact_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1033 (all T_1 (class_Rings_Ocomm__semiring__1(T_1) -> class_Power_Opower(tc_Polynomial_Opoly(T_1)))) # label(arity_Polynomial__Opoly__Power_Opower) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1034 (all V_x_2 (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),V_x_2)) -> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) -> c_SEQ_OBseq(tc_RealDef_Oreal,hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x_2))))) # label(fact_Bseq__realpow) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1035 (all V_m all V_n hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)))) # label(fact_le__add2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1036 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,V_m,V_n))) # label(fact_div__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1037 (all V_n all V_p all T_a (class_Groups_Ozero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != hAPP(c_Polynomial_Ocoeff(T_a,V_p),V_n) -> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n),c_Polynomial_Odegree(T_a,V_p)))))) # label(fact_le__degree) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1038 (all V_b all V_a all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)) = c_Divides_Odiv__class_Odiv(tc_Int_Oint,c_Divides_Odiv__class_Odiv(tc_Int_Oint,V_a,V_b),V_c))) # label(fact_zdiv__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1039 (all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Rings_Oinverse__class_Oinverse(T_a,V_b)),c_Rings_Oinverse__class_Oinverse(T_a,V_a)) = c_Rings_Oinverse__class_Oinverse(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))))) # label(fact_nonzero__inverse__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1040 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_n)) # label(fact_mod__Suc__eq__Suc__mod) # label(axiom) # label(non_clause). [assumption]. 1.52/1.67 1041 -((c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = c_Polynomial_Osmult(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_q____),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex))),v_q____) -> (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != v_q____ -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_OSuc,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat,c_HOL_OAll(tc_Nat_Onat)),c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat,c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat,c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat,c_fimplies)),c_Orderings_Oord__class_Oless(tc_Nat_Onat))),c_COMBC(tc_Nat_Onat,tc_Complex_Ocomplex,tc_HOL_Obool,hAPP(c_COMBB(tc_Complex_Ocomplex,tc_fun(tc_Complex_Ocomplex,tc_HOL_Obool),tc_Nat_Onat,c_fequal),c_Polynomial_Ocoeff(tc_Complex_Ocomplex,v_q____)),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)))))))) & (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != c_Polynomial_Osmult(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_q____),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex))),v_q____) -> (v_q____ != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) -> hAPP(c_Nat_OSuc,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat,c_HOL_OAll(tc_Nat_Onat)),c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat,c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat,c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat,c_fimplies)),c_Orderings_Oord__class_Oless(tc_Nat_Onat))),c_COMBC(tc_Nat_Onat,tc_Complex_Ocomplex,tc_HOL_Obool,hAPP(c_COMBB(tc_Complex_Ocomplex,tc_fun(tc_Complex_Ocomplex,tc_HOL_Obool),tc_Nat_Onat,c_fequal),c_Polynomial_Ocoeff(tc_Complex_Ocomplex,c_Polynomial_Osmult(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_q____),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex))),v_q____))),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)))))) = hAPP(c_Nat_OSuc,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat,c_HOL_OAll(tc_Nat_Onat)),c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat,c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat,c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat,c_fimplies)),c_Orderings_Oord__class_Oless(tc_Nat_Onat))),c_COMBC(tc_Nat_Onat,tc_Complex_Ocomplex,tc_HOL_Obool,hAPP(c_COMBB(tc_Complex_Ocomplex,tc_fun(tc_Complex_Ocomplex,tc_HOL_Obool),tc_Nat_Onat,c_fequal),c_Polynomial_Ocoeff(tc_Complex_Ocomplex,v_q____)),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex))))))) & (c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = v_q____ -> hAPP(c_Nat_OSuc,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat,c_HOL_OAll(tc_Nat_Onat)),c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat,c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat,c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat,c_fimplies)),c_Orderings_Oord__class_Oless(tc_Nat_Onat))),c_COMBC(tc_Nat_Onat,tc_Complex_Ocomplex,tc_HOL_Obool,hAPP(c_COMBB(tc_Complex_Ocomplex,tc_fun(tc_Complex_Ocomplex,tc_HOL_Obool),tc_Nat_Onat,c_fequal),c_Polynomial_Ocoeff(tc_Complex_Ocomplex,c_Polynomial_Osmult(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(c_Polynomial_Opoly(tc_Complex_Ocomplex,v_q____),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex))),v_q____))),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)))))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 1.52/1.68 1.52/1.68 ============================== end of process non-clausal formulas === 1.52/1.68 1.52/1.68 ============================== PROCESS INITIAL CLAUSES =============== 1.52/1.68 1.52/1.68 ============================== PREDICATE ELIMINATION ================= 1.52/1.68 1042 -class_Fields_Ofield(A) | class_Divides_Osemiring__div(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Divides_Osemiring__div) # label(axiom). [clausify(24)]. 1.52/1.68 1043 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,E,C) != c_Divides_Odiv__class_Omod(A,F,C) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),F),C) # label(fact_mod__mult__cong) # label(axiom). [clausify(2)]. 1.52/1.68 1044 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Odiv(A,B,c_Groups_Ozero__class_Ozero(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_div__by__0) # label(axiom). [clausify(21)]. 1.52/1.68 Derived: -class_Fields_Ofield(A) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),E,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),F,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),E),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),F),C). [resolve(1042,b,1043,a)]. 1.52/1.68 Derived: -class_Fields_Ofield(A) | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)). [resolve(1042,b,1044,a)]. 1.52/1.68 1045 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,c_Groups_Ozero__class_Ozero(A),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__0) # label(axiom). [clausify(34)]. 1.52/1.68 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1045,a,1042,b)]. 1.52/1.68 1046 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Odiv(A,c_Groups_Ozero__class_Ozero(A),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_div__0) # label(axiom). [clausify(45)]. 1.52/1.68 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1046,a,1042,b)]. 1.52/1.68 1047 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Odiv(A,B,c_Groups_Oone__class_Oone(A)) = B # label(fact_div__by__1) # label(axiom). [clausify(69)]. 1.52/1.68 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) = B | -class_Fields_Ofield(A). [resolve(1047,a,1042,b)]. 1.52/1.69 1048 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__mult__self1__is__0) # label(axiom). [clausify(73)]. 1.52/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1048,a,1042,b)]. 1.52/1.69 1049 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D),C) # label(fact_zmod__simps_I4_J) # label(axiom). [clausify(84)]. 1.52/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),D),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1049,a,1042,b)]. 1.52/1.69 1050 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)) = c_Divides_Odiv__class_Odiv(A,C,D) # label(fact_div__mult__mult2) # label(axiom). [clausify(102)]. 1.52/1.69 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,D) | -class_Fields_Ofield(A). [resolve(1050,a,1042,b)]. 1.52/1.69 1051 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),B),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Odiv(A,C,B)),c_Groups_Oone__class_Oone(A)) # label(fact_div__add__self2) # label(axiom). [clausify(122)]. 1.52/1.69 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),B),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) | -class_Fields_Ofield(A). [resolve(1051,a,1042,b)]. 1.52/1.69 1052 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Omod(A,B,C)),c_Divides_Odiv__class_Omod(A,D,C)),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D),C) # label(fact_mod__add__eq) # label(axiom). [clausify(155)]. 1.52/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1052,a,1042,b)]. 1.52/1.69 1053 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C),C) = c_Divides_Odiv__class_Omod(A,B,C) # label(fact_mod__add__self2) # label(axiom). [clausify(178)]. 1.52/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) | -class_Fields_Ofield(A). [resolve(1053,a,1042,b)]. 1.52/1.69 1054 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Odiv(A,c_Divides_Odiv__class_Omod(A,B,C),C) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__div__trivial) # label(axiom). [clausify(189)]. 1.52/1.69 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C),C) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1054,a,1042,b)]. 1.54/1.69 1055 class_Divides_Osemiring__div(tc_Nat_Onat) # label(arity_Nat__Onat__Divides_Osemiring__div) # label(axiom). [assumption]. 1.54/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,D,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,E,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),E),B). [resolve(1055,a,1043,a)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Odiv(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1055,a,1044,a)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1055,a,1045,a)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Odiv(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1055,a,1046,a)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Odiv(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1055,a,1047,a)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1055,a,1048,a)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C),B). [resolve(1055,a,1049,a)]. 1.54/1.69 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A)) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,B,C). [resolve(1055,a,1050,a)]. 1.54/1.69 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),A),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,B,A)),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(1055,a,1051,a)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C),B). [resolve(1055,a,1052,a)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B). [resolve(1055,a,1053,a)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Odiv(tc_Nat_Onat,c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B),B) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1055,a,1054,a)]. 1.54/1.69 1056 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,E,C) != c_Divides_Odiv__class_Omod(A,F,C) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),E),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),F),C) # label(fact_mod__add__cong) # label(axiom). [clausify(265)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),E,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),F,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),E),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),F),C) | -class_Fields_Ofield(A). [resolve(1056,a,1042,b)]. 1.54/1.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,D,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,E,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),D),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),E),B). [resolve(1056,a,1055,a)]. 1.54/1.71 1057 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,B) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__self) # label(axiom). [clausify(283)]. 1.54/1.71 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1057,a,1042,b)]. 1.54/1.71 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1057,a,1055,a)]. 1.54/1.71 1058 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,c_Groups_Oone__class_Oone(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__by__1) # label(axiom). [clausify(326)]. 1.54/1.71 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1058,a,1042,b)]. 1.54/1.71 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1058,a,1055,a)]. 1.54/1.71 1059 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Omod(A,B,C)),c_Divides_Odiv__class_Omod(A,D,C)),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D),C) # label(fact_mod__mult__eq) # label(axiom). [clausify(365)]. 1.54/1.71 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1059,a,1042,b)]. 1.54/1.71 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C),B). [resolve(1059,a,1055,a)]. 1.54/1.71 1060 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B),B) = C # label(fact_div__mult__self2__is__id) # label(axiom). [clausify(379)]. 1.54/1.71 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B),B) = C | -class_Fields_Ofield(A). [resolve(1060,a,1042,b)]. 1.54/1.71 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A),A) = B. [resolve(1060,a,1055,a)]. 1.54/1.71 1061 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D),C) # label(fact_mod__add__left__eq) # label(axiom). [clausify(420)]. 1.54/1.71 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),D),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1061,a,1042,b)]. 1.54/1.71 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C),B). [resolve(1061,a,1055,a)]. 1.54/1.71 1062 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Divides_Odiv__class_Odiv(A,C,B))),c_Divides_Odiv__class_Omod(A,C,B)) = C # label(fact_mod__div__equality2) # label(axiom). [clausify(451)]. 1.57/1.73 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B))),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B)) = C | -class_Fields_Ofield(A). [resolve(1062,a,1042,b)]. 1.57/1.73 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,B,A))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,A)) = B. [resolve(1062,a,1055,a)]. 1.57/1.73 1063 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Divides_Odiv__class_Odiv(A,C,B))),c_Divides_Odiv__class_Omod(A,C,B))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D) # label(fact_div__mod__equality2) # label(axiom). [clausify(463)]. 1.57/1.73 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B))),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),D) | -class_Fields_Ofield(A). [resolve(1063,a,1042,b)]. 1.57/1.73 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,B,A))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C). [resolve(1063,a,1055,a)]. 1.57/1.73 1064 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,c_Divides_Odiv__class_Omod(A,B,C),C) = c_Divides_Odiv__class_Omod(A,B,C) # label(fact_mod__mod__trivial) # label(axiom). [clausify(516)]. 1.57/1.73 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) | -class_Fields_Ofield(A). [resolve(1064,a,1042,b)]. 1.57/1.73 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B). [resolve(1064,a,1055,a)]. 1.57/1.73 1065 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),c_Divides_Odiv__class_Odiv(A,C,B)) # label(fact_div__mult__self1) # label(axiom). [clausify(527)]. 1.57/1.73 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B)) | -class_Fields_Ofield(A). [resolve(1065,a,1042,b)]. 1.57/1.73 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,B,A)). [resolve(1065,a,1055,a)]. 1.57/1.73 1066 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D),C) # label(fact_mod__mult__left__eq) # label(axiom). [clausify(533)]. 1.57/1.73 1067 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,c_Groups_Ozero__class_Ozero(A)) = B # label(fact_mod__by__0) # label(axiom). [clausify(554)]. 1.57/1.73 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) = B | -class_Fields_Ofield(A). [resolve(1067,a,1042,b)]. 1.58/1.74 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(1067,a,1055,a)]. 1.58/1.74 1068 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),C) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__mult__self2__is__0) # label(axiom). [clausify(561)]. 1.58/1.74 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),C) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1068,a,1042,b)]. 1.58/1.74 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),B) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1068,a,1055,a)]. 1.58/1.74 1069 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),c_Divides_Odiv__class_Odiv(A,C,B)) # label(fact_div__mult__self2) # label(axiom). [clausify(591)]. 1.58/1.74 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B)) | -class_Fields_Ofield(A). [resolve(1069,a,1042,b)]. 1.58/1.74 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,B,A)). [resolve(1069,a,1055,a)]. 1.58/1.74 1070 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Odiv(A,B,C)),C)),c_Divides_Odiv__class_Omod(A,B,C))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D) # label(fact_div__mod__equality) # label(axiom). [clausify(595)]. 1.58/1.74 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,C)),C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D) | -class_Fields_Ofield(A). [resolve(1070,a,1042,b)]. 1.58/1.74 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,A,B)),B)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C). [resolve(1070,a,1055,a)]. 1.58/1.74 1071 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = c_Divides_Odiv__class_Odiv(A,C,D) # label(fact_div__mult__mult1) # label(axiom). [clausify(626)]. 1.58/1.74 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,D) | -class_Fields_Ofield(A). [resolve(1071,a,1042,b)]. 1.58/1.74 1072 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),B) = C # label(fact_div__mult__self1__is__id) # label(axiom). [clausify(632)]. 1.58/1.76 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),B) = C | -class_Fields_Ofield(A). [resolve(1072,a,1042,b)]. 1.58/1.76 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),A) = B. [resolve(1072,a,1055,a)]. 1.58/1.76 1073 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Omod(A,B,D)),C) # label(fact_mod__mult__mult2) # label(axiom). [clausify(636)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,D)),C) | -class_Fields_Ofield(A). [resolve(1073,a,1042,b)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,C)),B). [resolve(1073,a,1055,a)]. 1.58/1.76 1074 class_Divides_Osemiring__div(tc_Int_Oint) # label(arity_Int__Oint__Divides_Osemiring__div) # label(axiom). [assumption]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,D,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,E,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),E),B). [resolve(1074,a,1043,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Odiv(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1074,a,1044,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1074,a,1045,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Odiv(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1074,a,1046,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Odiv(tc_Int_Oint,A,c_Groups_Oone__class_Oone(tc_Int_Oint)) = A. [resolve(1074,a,1047,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1074,a,1048,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C),B). [resolve(1074,a,1049,a)]. 1.58/1.76 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)) = c_Divides_Odiv__class_Odiv(tc_Int_Oint,B,C). [resolve(1074,a,1050,a)]. 1.58/1.76 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),A),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,B,A)),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(1074,a,1051,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C),B). [resolve(1074,a,1052,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B). [resolve(1074,a,1053,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Odiv(tc_Int_Oint,c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B),B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1074,a,1054,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,D,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,E,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),D),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),E),B). [resolve(1074,a,1056,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1074,a,1057,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1074,a,1058,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C),B). [resolve(1074,a,1059,a)]. 1.58/1.76 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A),A) = B. [resolve(1074,a,1060,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C),B). [resolve(1074,a,1061,a)]. 1.58/1.76 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Divides_Odiv__class_Odiv(tc_Int_Oint,B,A))),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C). [resolve(1074,a,1063,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B). [resolve(1074,a,1064,a)]. 1.58/1.76 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),c_Divides_Odiv__class_Odiv(tc_Int_Oint,B,A)). [resolve(1074,a,1065,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(1074,a,1067,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1074,a,1068,a)]. 1.58/1.76 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),c_Divides_Odiv__class_Odiv(tc_Int_Oint,B,A)). [resolve(1074,a,1069,a)]. 1.58/1.76 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,A,B)),B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C). [resolve(1074,a,1070,a)]. 1.58/1.76 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)) = c_Divides_Odiv__class_Odiv(tc_Int_Oint,B,C). [resolve(1074,a,1071,a)]. 1.58/1.76 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),A) = B. [resolve(1074,a,1072,a)]. 1.58/1.76 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,C)),B). [resolve(1074,a,1073,a)]. 1.58/1.78 1075 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),C) = c_Divides_Odiv__class_Omod(A,B,C) # label(fact_mod__mult__self2) # label(axiom). [clausify(789)]. 1.58/1.78 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) | -class_Fields_Ofield(A). [resolve(1075,a,1042,b)]. 1.58/1.78 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B). [resolve(1075,a,1055,a)]. 1.58/1.78 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B). [resolve(1075,a,1074,a)]. 1.58/1.78 1076 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Oone__class_Oone(A) = c_Divides_Odiv__class_Odiv(A,B,B) # label(fact_div__self) # label(axiom). [clausify(867)]. 1.58/1.78 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,B) | -class_Fields_Ofield(A). [resolve(1076,a,1042,b)]. 1.58/1.78 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,A,A). [resolve(1076,a,1055,a)]. 1.58/1.78 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Oone__class_Oone(tc_Int_Oint) = c_Divides_Odiv__class_Odiv(tc_Int_Oint,A,A). [resolve(1076,a,1074,a)]. 1.58/1.78 1077 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Omod(A,B,C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Odiv(A,B,C)),C)) = B # label(fact_semiring__div__class_Omod__div__equality_H) # label(axiom). [clausify(868)]. 1.58/1.78 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,C)),C)) = B | -class_Fields_Ofield(A). [resolve(1077,a,1042,b)]. 1.58/1.78 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,A,B)),B)) = A. [resolve(1077,a,1055,a)]. 1.58/1.78 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,A,B)),B)) = A. [resolve(1077,a,1074,a)]. 1.58/1.78 1078 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Divides_Odiv__class_Omod(A,C,D)),D) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C),D) # label(fact_mod__add__right__eq) # label(axiom). [clausify(891)]. 1.58/1.78 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D)),D) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C),D) | -class_Fields_Ofield(A). [resolve(1078,a,1042,b)]. 1.58/1.78 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B),C). [resolve(1078,a,1055,a)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B),C). [resolve(1078,a,1074,a)]. 1.66/1.81 1079 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Odiv(A,C,B)),c_Groups_Oone__class_Oone(A)) # label(fact_div__add__self1) # label(axiom). [clausify(918)]. 1.66/1.81 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) | -class_Fields_Ofield(A). [resolve(1079,a,1042,b)]. 1.66/1.81 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,B,A)),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(1079,a,1055,a)]. 1.66/1.81 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,B,A)),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(1079,a,1074,a)]. 1.66/1.81 1080 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D),C) # label(fact_zmod__simps_I1_J) # label(axiom). [clausify(940)]. 1.66/1.81 1081 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Divides_Odiv__class_Omod(A,C,D)),D) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C),D) # label(fact_zmod__simps_I2_J) # label(axiom). [clausify(941)]. 1.66/1.81 1082 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Divides_Odiv__class_Omod(A,C,D)),D) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D) # label(fact_mod__mult__right__eq) # label(axiom). [clausify(948)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D)),D) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),D) | -class_Fields_Ofield(A). [resolve(1082,a,1042,b)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),C). [resolve(1082,a,1055,a)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C). [resolve(1082,a,1074,a)]. 1.66/1.81 1083 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Divides_Odiv__class_Omod(A,C,D)) # label(fact_mod__mult__mult1) # label(axiom). [clausify(971)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D)) | -class_Fields_Ofield(A). [resolve(1083,a,1042,b)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,C)). [resolve(1083,a,1055,a)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C)). [resolve(1083,a,1074,a)]. 1.66/1.81 1084 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = c_Groups_Ozero__class_Ozero(A) # label(fact_div__mult__mult1__if) # label(axiom). [clausify(988)]. 1.66/1.81 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1084,a,1042,b)]. 1.66/1.81 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1084,a,1055,a)]. 1.66/1.81 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Divides_Odiv__class_Odiv(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1084,a,1074,a)]. 1.66/1.81 1085 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Odiv(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = c_Divides_Odiv__class_Odiv(A,C,D) # label(fact_div__mult__mult1__if) # label(axiom). [clausify(988)]. 1.66/1.81 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Divides_Odiv__class_Odiv(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)) = c_Divides_Odiv__class_Odiv(tc_Nat_Onat,B,C). [resolve(1085,a,1055,a)]. 1.66/1.81 1086 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C),B) = c_Divides_Odiv__class_Omod(A,C,B) # label(fact_mod__add__self1) # label(axiom). [clausify(990)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B) | -class_Fields_Ofield(A). [resolve(1086,a,1042,b)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B),A) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,A). [resolve(1086,a,1055,a)]. 1.66/1.81 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B),A) = c_Divides_Odiv__class_Omod(tc_Int_Oint,B,A). [resolve(1086,a,1074,a)]. 1.66/1.81 1087 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Odiv(A,B,C)),C)),c_Divides_Odiv__class_Omod(A,B,C)) = B # label(fact_mod__div__equality) # label(axiom). [clausify(994)]. 1.66/1.81 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,C)),C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)) = B | -class_Fields_Ofield(A). [resolve(1087,a,1042,b)]. 1.66/1.81 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Odiv(tc_Nat_Onat,A,B)),B)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)) = A. [resolve(1087,a,1055,a)]. 1.72/1.88 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint,A,B)),B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)) = A. [resolve(1087,a,1074,a)]. 1.72/1.88 1088 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),D) = c_Divides_Odiv__class_Omod(A,B,D) # label(fact_mod__mult__self1) # label(axiom). [clausify(1015)]. 1.72/1.88 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),D) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,D) | -class_Fields_Ofield(A). [resolve(1088,a,1042,b)]. 1.72/1.88 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)),C) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,C). [resolve(1088,a,1055,a)]. 1.72/1.88 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),C) = c_Divides_Odiv__class_Omod(tc_Int_Oint,A,C). [resolve(1088,a,1074,a)]. 1.72/1.88 1089 class_RealVector_Oreal__normed__vector(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Oreal__normed__vector) # label(axiom). [assumption]. 1.72/1.88 1090 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | c_SEQ_OBseq(A,hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat,B),c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C))) # label(fact_Bseq__ignore__initial__segment) # label(axiom). [clausify(3)]. 1.72/1.88 1091 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,C))),f1(B,A))) # label(fact_Bseq__def) # label(axiom). [clausify(23)]. 1.72/1.88 1092 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f1(B,A))) # label(fact_Bseq__def) # label(axiom). [clausify(23)]. 1.72/1.88 1093 -class_RealVector_Oreal__normed__vector(A) | c_SEQ_OBseq(A,B) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f2(B,A,C)))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) # label(fact_Bseq__def) # label(axiom). [clausify(23)]. 1.72/1.88 1094 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,C))),f3(B,A))) # label(fact_BseqD) # label(axiom). [clausify(38)]. 1.72/1.88 1095 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f3(B,A))) # label(fact_BseqD) # label(axiom). [clausify(38)]. 1.72/1.88 1096 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(A,B))) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_zero__less__norm__iff) # label(axiom). [clausify(43)]. 1.72/1.88 1097 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(A,B))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_zero__less__norm__iff) # label(axiom). [clausify(43)]. 1.72/1.88 1098 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),c_RealVector_Onorm__class_Onorm(A,C)))) # label(fact_norm__triangle__ineq) # label(axiom). [clausify(44)]. 1.72/1.88 1099 -class_RealVector_Oreal__normed__vector(A) | c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)) = c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),B)) # label(fact_norm__minus__commute) # label(axiom). [clausify(134)]. 1.72/1.88 1100 -class_RealVector_Oreal__normed__vector(A) | c_RealVector_Onorm__class_Onorm(A,B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_norm__eq__zero) # label(axiom). [clausify(214)]. 1.72/1.88 1101 -class_RealVector_Oreal__normed__vector(A) | c_RealVector_Onorm__class_Onorm(A,B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_norm__eq__zero) # label(axiom). [clausify(214)]. 1.72/1.88 1102 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,hAPP(B,f18(B,A,C)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)))) | c_SEQ_OBseq(A,B) # label(fact_Bseq__iff1a) # label(axiom). [clausify(279)]. 1.72/1.88 1103 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,hAPP(B,C))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f19(B,A))))) | -c_SEQ_OBseq(A,B) # label(fact_Bseq__iff1a) # label(axiom). [clausify(279)]. 1.72/1.88 1104 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,f26(C,A,B)))),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,D))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f27(C,A))))) # label(fact_lemma__NBseq__def) # label(axiom). [clausify(347)]. 1.72/1.88 1105 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f28(B,A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f29(B,A,C)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)))) # label(fact_lemma__NBseq__def) # label(axiom). [clausify(347)]. 1.72/1.88 1106 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,C))),f28(B,A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f29(B,A,D)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,D)))) # label(fact_lemma__NBseq__def) # label(axiom). [clausify(347)]. 1.72/1.88 Derived: -c_SEQ_OBseq(tc_RealDef_Oreal,A) | c_SEQ_OBseq(tc_RealDef_Oreal,hAPP(c_COMBB(tc_Nat_Onat,tc_RealDef_Oreal,tc_Nat_Onat,A),c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B))). [resolve(1089,a,1090,a)]. 1.72/1.88 Derived: -c_SEQ_OBseq(tc_RealDef_Oreal,A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,B))),f1(A,tc_RealDef_Oreal))). [resolve(1089,a,1091,a)]. 1.72/1.88 Derived: -c_SEQ_OBseq(tc_RealDef_Oreal,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f1(A,tc_RealDef_Oreal))). [resolve(1089,a,1092,a)]. 1.72/1.88 Derived: c_SEQ_OBseq(tc_RealDef_Oreal,A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,f2(A,tc_RealDef_Oreal,B)))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)). [resolve(1089,a,1093,a)]. 1.72/1.88 Derived: -c_SEQ_OBseq(tc_RealDef_Oreal,A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,B))),f3(A,tc_RealDef_Oreal))). [resolve(1089,a,1094,a)]. 1.72/1.88 Derived: -c_SEQ_OBseq(tc_RealDef_Oreal,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f3(A,tc_RealDef_Oreal))). [resolve(1089,a,1095,a)]. 1.72/1.90 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A))) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1089,a,1096,a)]. 1.72/1.90 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A))) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1089,a,1097,a)]. 1.72/1.90 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B)))). [resolve(1089,a,1098,a)]. 1.72/1.90 Derived: c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),B)) = c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),B),A)). [resolve(1089,a,1099,a)]. 1.72/1.90 Derived: c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1089,a,1100,a)]. 1.72/1.90 Derived: c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1089,a,1101,a)]. 1.72/1.90 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,f18(A,tc_RealDef_Oreal,B)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)))) | c_SEQ_OBseq(tc_RealDef_Oreal,A). [resolve(1089,a,1102,a)]. 1.72/1.90 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,B))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f19(A,tc_RealDef_Oreal))))) | -c_SEQ_OBseq(tc_RealDef_Oreal,A). [resolve(1089,a,1103,a)]. 1.72/1.90 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(B,f26(B,tc_RealDef_Oreal,A)))),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(B,C))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f27(B,tc_RealDef_Oreal))))). [resolve(1089,a,1104,a)]. 1.72/1.90 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f28(A,tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,f29(A,tc_RealDef_Oreal,B)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)))). [resolve(1089,a,1105,a)]. 1.72/1.90 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,B))),f28(A,tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,f29(A,tc_RealDef_Oreal,C)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)))). [resolve(1089,a,1106,a)]. 1.72/1.90 1107 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) # label(fact_norm__not__less__zero) # label(axiom). [clausify(497)]. 1.72/1.90 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1107,a,1089,a)]. 1.72/1.90 1108 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f44(B,A,C)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)))) | c_SEQ_OBseq(A,B) # label(fact_Bseq__iff) # label(axiom). [clausify(640)]. 1.72/1.92 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,f44(A,tc_RealDef_Oreal,B)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)))) | c_SEQ_OBseq(tc_RealDef_Oreal,A). [resolve(1108,a,1089,a)]. 1.72/1.92 1109 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,C))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f45(B,A))))) | -c_SEQ_OBseq(A,B) # label(fact_Bseq__iff) # label(axiom). [clausify(640)]. 1.72/1.92 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,B))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f45(A,tc_RealDef_Oreal))))) | -c_SEQ_OBseq(tc_RealDef_Oreal,A). [resolve(1109,a,1089,a)]. 1.72/1.92 1110 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),f46(C,D,B,A))) | c_SEQ_OBseq(A,D) # label(fact_BseqI2_H) # label(axiom). [clausify(644)]. 1.72/1.92 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),f46(B,C,A,tc_RealDef_Oreal))) | c_SEQ_OBseq(tc_RealDef_Oreal,C). [resolve(1110,a,1089,a)]. 1.72/1.92 1111 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f46(C,B,D,A)))),C)) | c_SEQ_OBseq(A,B) # label(fact_BseqI2_H) # label(axiom). [clausify(644)]. 1.72/1.92 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,f46(B,A,C,tc_RealDef_Oreal)))),B)) | c_SEQ_OBseq(tc_RealDef_Oreal,A). [resolve(1111,a,1089,a)]. 1.72/1.92 1112 -class_RealVector_Oreal__normed__vector(A) | c_Groups_Ozero__class_Ozero(A) != B | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) # label(fact_norm__le__zero__iff) # label(axiom). [clausify(751)]. 1.72/1.92 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1112,a,1089,a)]. 1.72/1.92 1113 -class_RealVector_Oreal__normed__vector(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) # label(fact_norm__le__zero__iff) # label(axiom). [clausify(751)]. 1.72/1.92 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1113,a,1089,a)]. 1.72/1.92 1114 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,D)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),E))) # label(fact_norm__add__less) # label(axiom). [clausify(752)]. 1.72/1.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),D))). [resolve(1114,a,1089,a)]. 1.72/1.92 1115 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(A,B))) # label(fact_norm__ge__zero) # label(axiom). [clausify(766)]. 1.72/1.92 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A))). [resolve(1115,a,1089,a)]. 1.72/1.92 1116 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),c_RealVector_Onorm__class_Onorm(A,D)))) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_norm__ratiotest__lemma) # label(axiom). [clausify(769)]. 1.72/1.92 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,C)))) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B. [resolve(1116,a,1089,a)]. 1.72/1.92 1117 -class_RealVector_Oreal__normed__vector(A) | c_RealVector_Onorm__class_Onorm(A,c_Groups_Ozero__class_Ozero(A)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) # label(fact_norm__zero) # label(axiom). [clausify(778)]. 1.72/1.92 Derived: c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1117,a,1089,a)]. 1.72/1.92 1118 class_RealVector_Oreal__normed__vector(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Oreal__normed__vector) # label(axiom). [assumption]. 1.72/1.92 Derived: -c_SEQ_OBseq(tc_Complex_Ocomplex,A) | c_SEQ_OBseq(tc_Complex_Ocomplex,hAPP(c_COMBB(tc_Nat_Onat,tc_Complex_Ocomplex,tc_Nat_Onat,A),c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B))). [resolve(1118,a,1090,a)]. 1.72/1.92 Derived: -c_SEQ_OBseq(tc_Complex_Ocomplex,A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,B))),f1(A,tc_Complex_Ocomplex))). [resolve(1118,a,1091,a)]. 1.72/1.92 Derived: -c_SEQ_OBseq(tc_Complex_Ocomplex,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f1(A,tc_Complex_Ocomplex))). [resolve(1118,a,1092,a)]. 1.72/1.92 Derived: c_SEQ_OBseq(tc_Complex_Ocomplex,A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,f2(A,tc_Complex_Ocomplex,B)))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)). [resolve(1118,a,1093,a)]. 1.72/1.92 Derived: -c_SEQ_OBseq(tc_Complex_Ocomplex,A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,B))),f3(A,tc_Complex_Ocomplex))). [resolve(1118,a,1094,a)]. 1.72/1.92 Derived: -c_SEQ_OBseq(tc_Complex_Ocomplex,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f3(A,tc_Complex_Ocomplex))). [resolve(1118,a,1095,a)]. 1.72/1.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A))) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A. [resolve(1118,a,1096,a)]. 1.72/1.92 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A))) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A. [resolve(1118,a,1097,a)]. 1.72/1.92 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B)))). [resolve(1118,a,1098,a)]. 1.72/1.93 Derived: c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),A),B)) = c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),B),A)). [resolve(1118,a,1099,a)]. 1.72/1.93 Derived: c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A. [resolve(1118,a,1100,a)]. 1.72/1.93 Derived: c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A. [resolve(1118,a,1101,a)]. 1.72/1.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,f18(A,tc_Complex_Ocomplex,B)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)))) | c_SEQ_OBseq(tc_Complex_Ocomplex,A). [resolve(1118,a,1102,a)]. 1.72/1.93 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,B))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f19(A,tc_Complex_Ocomplex))))) | -c_SEQ_OBseq(tc_Complex_Ocomplex,A). [resolve(1118,a,1103,a)]. 1.72/1.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(B,f26(B,tc_Complex_Ocomplex,A)))),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(B,C))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f27(B,tc_Complex_Ocomplex))))). [resolve(1118,a,1104,a)]. 1.72/1.93 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f28(A,tc_Complex_Ocomplex))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,f29(A,tc_Complex_Ocomplex,B)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)))). [resolve(1118,a,1105,a)]. 1.72/1.93 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,B))),f28(A,tc_Complex_Ocomplex))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,f29(A,tc_Complex_Ocomplex,C)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)))). [resolve(1118,a,1106,a)]. 1.72/1.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1118,a,1107,a)]. 1.72/1.93 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,f44(A,tc_Complex_Ocomplex,B)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)))) | c_SEQ_OBseq(tc_Complex_Ocomplex,A). [resolve(1118,a,1108,a)]. 1.72/1.93 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,B))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f45(A,tc_Complex_Ocomplex))))) | -c_SEQ_OBseq(tc_Complex_Ocomplex,A). [resolve(1118,a,1109,a)]. 1.72/1.93 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),f46(B,C,A,tc_Complex_Ocomplex))) | c_SEQ_OBseq(tc_Complex_Ocomplex,C). [resolve(1118,a,1110,a)]. 1.72/1.93 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,f46(B,A,C,tc_Complex_Ocomplex)))),B)) | c_SEQ_OBseq(tc_Complex_Ocomplex,A). [resolve(1118,a,1111,a)]. 1.72/1.93 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1118,a,1112,a)]. 1.72/1.94 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1118,a,1113,a)]. 1.72/1.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),C))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),D))). [resolve(1118,a,1114,a)]. 1.72/1.94 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A))). [resolve(1118,a,1115,a)]. 1.72/1.94 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,C)))) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B. [resolve(1118,a,1116,a)]. 1.72/1.94 Derived: c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1118,a,1117,a)]. 1.72/1.94 1119 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,f54(C,B,A)))),B)) | c_SEQ_OBseq(A,C) # label(fact_BseqI) # label(axiom). [clausify(855)]. 1.72/1.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(B,f54(B,A,tc_RealDef_Oreal)))),A)) | c_SEQ_OBseq(tc_RealDef_Oreal,B). [resolve(1119,a,1089,a)]. 1.72/1.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(B,f54(B,A,tc_Complex_Ocomplex)))),A)) | c_SEQ_OBseq(tc_Complex_Ocomplex,B). [resolve(1119,a,1118,a)]. 1.72/1.94 1120 -class_RealVector_Oreal__normed__vector(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f59(B,A,C)))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,hAPP(B,D))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f60(B,A))))) # label(fact_lemma__NBseq__def2) # label(axiom). [clausify(892)]. 1.72/1.94 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,f59(A,tc_RealDef_Oreal,B)))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,C))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f60(A,tc_RealDef_Oreal))))). [resolve(1120,a,1089,a)]. 1.72/1.94 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,f59(A,tc_Complex_Ocomplex,B)))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,C))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,f60(A,tc_Complex_Ocomplex))))). [resolve(1120,a,1118,a)]. 1.87/2.02 1121 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,C))),f61(B,A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,hAPP(B,f62(B,A,D)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,D)))) # label(fact_lemma__NBseq__def2) # label(axiom). [clausify(892)]. 1.87/2.02 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,B))),f61(A,tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,f62(A,tc_RealDef_Oreal,C)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)))). [resolve(1121,a,1089,a)]. 1.87/2.02 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,B))),f61(A,tc_Complex_Ocomplex))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,f62(A,tc_Complex_Ocomplex,C)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)))). [resolve(1121,a,1118,a)]. 1.87/2.02 1122 -class_RealVector_Oreal__normed__vector(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f61(B,A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,hAPP(B,f62(B,A,C)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)))) # label(fact_lemma__NBseq__def2) # label(axiom). [clausify(892)]. 1.87/2.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f61(A,tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(A,f62(A,tc_RealDef_Oreal,B)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)))). [resolve(1122,a,1089,a)]. 1.87/2.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f61(A,tc_Complex_Ocomplex))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(A,f62(A,tc_Complex_Ocomplex,B)))),c_RealDef_Oreal(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)))). [resolve(1122,a,1118,a)]. 1.87/2.02 1123 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat,B),c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C))) | c_SEQ_OBseq(A,B) # label(fact_Bseq__offset) # label(axiom). [clausify(1001)]. 1.87/2.02 Derived: -c_SEQ_OBseq(tc_RealDef_Oreal,hAPP(c_COMBB(tc_Nat_Onat,tc_RealDef_Oreal,tc_Nat_Onat,A),c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B))) | c_SEQ_OBseq(tc_RealDef_Oreal,A). [resolve(1123,a,1089,a)]. 1.87/2.02 Derived: -c_SEQ_OBseq(tc_Complex_Ocomplex,hAPP(c_COMBB(tc_Nat_Onat,tc_Complex_Ocomplex,tc_Nat_Onat,A),c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B))) | c_SEQ_OBseq(tc_Complex_Ocomplex,A). [resolve(1123,a,1118,a)]. 1.87/2.02 1124 -class_Rings_Olinordered__idom(A) | class_Rings_Olinordered__semidom(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Olinordered__semidom) # label(axiom). [clausify(312)]. 1.87/2.02 1125 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D)),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),D))) # label(fact_power__strict__mono) # label(axiom). [clausify(4)]. 1.87/2.02 1126 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C))),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),hAPP(c_Nat_OSuc,C)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),D)) # label(fact_power__le__imp__le__base) # label(axiom). [clausify(9)]. 1.87/2.02 1127 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_zero__le__power) # label(axiom). [clausify(49)]. 1.87/2.02 1128 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) # label(fact_power__strict__increasing__iff) # label(axiom). [clausify(60)]. 1.87/2.02 1129 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) # label(fact_power__strict__increasing__iff) # label(axiom). [clausify(60)]. 1.87/2.02 1130 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),c_Groups_Oone__class_Oone(A)))) # label(fact_zero__less__two) # label(axiom). [clausify(64)]. 1.87/2.02 1131 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_one__less__power) # label(axiom). [clausify(68)]. 1.87/2.02 1132 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) # label(fact_power__less__imp__less__exp) # label(axiom). [clausify(120)]. 1.87/2.02 1133 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) # label(fact_power__le__imp__le__exp) # label(axiom). [clausify(215)]. 1.87/2.02 1134 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) | C = D # label(fact_power__inject__exp) # label(axiom). [clausify(249)]. 1.87/2.02 1135 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) | C != D # label(fact_power__inject__exp) # label(axiom). [clausify(249)]. 1.87/2.02 1136 -class_Rings_Olinordered__semidom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)) != hAPP(hAPP(c_Power_Opower__class_Opower(A),D),hAPP(c_Nat_OSuc,C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | B = D # label(fact_power__inject__base) # label(axiom). [clausify(289)]. 1.87/2.03 1137 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D)),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),D))) # label(fact_power__mono) # label(axiom). [clausify(291)]. 1.87/2.03 1138 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__one__le__zero) # label(axiom). [clausify(294)]. 1.87/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),C),D))). [resolve(1124,b,1125,a)]. 1.87/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),hAPP(c_Nat_OSuc,C))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),D),hAPP(c_Nat_OSuc,C)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),D)). [resolve(1124,b,1126,a)]. 1.87/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C))). [resolve(1124,b,1127,a)]. 1.87/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D))). [resolve(1124,b,1128,a)]. 1.87/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D))). [resolve(1124,b,1129,a)]. 1.87/2.03 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))))). [resolve(1124,b,1130,a)]. 1.87/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C))). [resolve(1124,b,1131,a)]. 1.87/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)). [resolve(1124,b,1133,a)]. 1.88/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D) | C = D. [resolve(1124,b,1134,a)]. 1.88/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D) | C != D. [resolve(1124,b,1135,a)]. 1.88/2.03 Derived: -class_Rings_Olinordered__idom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),hAPP(c_Nat_OSuc,C)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),D),hAPP(c_Nat_OSuc,C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | B = D. [resolve(1124,b,1136,a)]. 1.88/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),C),D))). [resolve(1124,b,1137,a)]. 1.88/2.03 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))). [resolve(1124,b,1138,a)]. 1.88/2.03 1139 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C))),c_Groups_Oone__class_Oone(A))) # label(fact_power__Suc__less__one) # label(axiom). [clausify(321)]. 1.88/2.03 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),hAPP(c_Nat_OSuc,C))),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1139,a,1124,b)]. 1.88/2.03 1140 class_Rings_Olinordered__semidom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.88/2.03 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),C))). [resolve(1140,a,1125,a)]. 1.88/2.03 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),hAPP(c_Nat_OSuc,B)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),C)). [resolve(1140,a,1126,a)]. 1.88/2.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C))). [resolve(1140,a,1128,a)]. 1.88/2.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C))). [resolve(1140,a,1129,a)]. 1.88/2.04 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)))). [resolve(1140,a,1130,a)]. 1.88/2.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(1140,a,1131,a)]. 1.88/2.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)). [resolve(1140,a,1133,a)]. 1.88/2.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C) | B = C. [resolve(1140,a,1134,a)]. 1.88/2.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C) | B != C. [resolve(1140,a,1135,a)]. 1.88/2.04 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),hAPP(c_Nat_OSuc,B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | A = C. [resolve(1140,a,1136,a)]. 1.88/2.04 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),C))). [resolve(1140,a,1137,a)]. 1.88/2.04 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1140,a,1138,a)]. 1.88/2.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B))),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(1140,a,1139,a)]. 1.88/2.04 1141 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,D),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),B))) # label(fact_power__decreasing) # label(axiom). [clausify(389)]. 1.88/2.05 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(C))),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(C),D),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(C),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(C)),D),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(C)),D),A))) | -class_Rings_Olinordered__idom(C). [resolve(1141,a,1124,b)]. 1.88/2.05 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),A))). [resolve(1141,a,1140,a)]. 1.88/2.05 1142 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) # label(fact_power__increasing__iff) # label(axiom). [clausify(447)]. 1.88/2.05 1143 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) # label(fact_power__increasing__iff) # label(axiom). [clausify(447)]. 1.88/2.05 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | -class_Rings_Olinordered__idom(A). [resolve(1143,a,1124,b)]. 1.88/2.05 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)). [resolve(1143,a,1140,a)]. 1.88/2.05 1144 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_power__Suc__less) # label(axiom). [clausify(474)]. 1.88/2.05 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1144,a,1124,b)]. 1.88/2.07 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(1144,a,1140,a)]. 1.88/2.07 1145 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),c_Groups_Oone__class_Oone(A))) # label(fact_zero__le__one) # label(axiom). [clausify(485)]. 1.88/2.07 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1145,a,1124,b)]. 1.88/2.07 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(1145,a,1140,a)]. 1.88/2.07 1146 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__one__less__zero) # label(axiom). [clausify(495)]. 1.88/2.07 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1146,a,1124,b)]. 1.88/2.07 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1146,a,1140,a)]. 1.88/2.07 1147 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_one__le__power) # label(axiom). [clausify(569)]. 1.88/2.07 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1147,a,1124,b)]. 1.88/2.07 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(1147,a,1140,a)]. 1.88/2.07 1148 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)))) # label(fact_power__less__power__Suc) # label(axiom). [clausify(570)]. 1.88/2.07 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)))) | -class_Rings_Olinordered__idom(A). [resolve(1148,a,1124,b)]. 1.88/2.07 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)))). [resolve(1148,a,1140,a)]. 1.88/2.08 1149 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),B)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C))) # label(fact_power__strict__increasing) # label(axiom). [clausify(621)]. 1.88/2.08 1150 class_Rings_Olinordered__semidom(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),C))). [resolve(1150,a,1125,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),hAPP(c_Nat_OSuc,B)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),C)). [resolve(1150,a,1126,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C))). [resolve(1150,a,1128,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C))). [resolve(1150,a,1129,a)]. 1.88/2.08 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))). [resolve(1150,a,1130,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))). [resolve(1150,a,1131,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)). [resolve(1150,a,1133,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C) | B = C. [resolve(1150,a,1134,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C) | B != C. [resolve(1150,a,1135,a)]. 1.88/2.08 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),hAPP(c_Nat_OSuc,B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | A = C. [resolve(1150,a,1136,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),C))). [resolve(1150,a,1137,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1150,a,1138,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B))),c_Groups_Oone__class_Oone(tc_Nat_Onat))). [resolve(1150,a,1139,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),c_Groups_Oone__class_Oone(tc_Nat_Onat))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),A))). [resolve(1150,a,1141,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)). [resolve(1150,a,1143,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))). [resolve(1150,a,1144,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1150,a,1146,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))). [resolve(1150,a,1147,a)]. 1.88/2.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)))). [resolve(1150,a,1148,a)]. 1.88/2.08 1151 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C))),B)) # label(fact_realpow__Suc__le__self) # label(axiom). [clausify(655)]. 1.88/2.08 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),hAPP(c_Nat_OSuc,C))),B)) | -class_Rings_Olinordered__idom(A). [resolve(1151,a,1124,b)]. 1.88/2.10 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B))),A)). [resolve(1151,a,1140,a)]. 1.88/2.10 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),c_Groups_Oone__class_Oone(tc_Nat_Onat))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B))),A)). [resolve(1151,a,1150,a)]. 1.88/2.10 1152 -class_Rings_Olinordered__semidom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | B = D # label(fact_power__eq__imp__eq__base) # label(axiom). [clausify(674)]. 1.88/2.10 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),D),C) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | B = D | -class_Rings_Olinordered__idom(A). [resolve(1152,a,1124,b)]. 1.88/2.10 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(1152,a,1140,a)]. 1.88/2.10 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(1152,a,1150,a)]. 1.88/2.10 1153 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Oone__class_Oone(A)))) # label(fact_less__add__one) # label(axiom). [clausify(717)]. 1.88/2.10 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))))) | -class_Rings_Olinordered__idom(A). [resolve(1153,a,1124,b)]. 1.88/2.10 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint)))). [resolve(1153,a,1140,a)]. 1.88/2.10 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)))). [resolve(1153,a,1150,a)]. 1.88/2.10 1154 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_less__1__mult) # label(axiom). [clausify(763)]. 1.95/2.11 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1154,a,1124,b)]. 1.95/2.11 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1154,a,1140,a)]. 1.95/2.11 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B))). [resolve(1154,a,1150,a)]. 1.95/2.11 1155 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_power__less__imp__less__base) # label(axiom). [clausify(774)]. 1.95/2.11 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),D),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),D)) | -class_Rings_Olinordered__idom(A). [resolve(1155,a,1124,b)]. 1.95/2.11 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(1155,a,1140,a)]. 1.95/2.11 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(1155,a,1150,a)]. 1.95/2.11 1156 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Groups_Oone__class_Oone(A))) # label(fact_zero__less__one) # label(axiom). [clausify(799)]. 1.95/2.11 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1156,a,1124,b)]. 1.95/2.11 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(1156,a,1140,a)]. 1.95/2.11 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))). [resolve(1156,a,1150,a)]. 1.95/2.12 1157 class_Rings_Olinordered__semidom(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B),C))). [resolve(1157,a,1125,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(c_Nat_OSuc,B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),hAPP(c_Nat_OSuc,B)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),C)). [resolve(1157,a,1126,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B))). [resolve(1157,a,1127,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C))). [resolve(1157,a,1128,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C))). [resolve(1157,a,1129,a)]. 1.95/2.12 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))). [resolve(1157,a,1130,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B))). [resolve(1157,a,1131,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)). [resolve(1157,a,1133,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C) | B = C. [resolve(1157,a,1134,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C) | B != C. [resolve(1157,a,1135,a)]. 1.95/2.12 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(c_Nat_OSuc,B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),hAPP(c_Nat_OSuc,B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | A = C. [resolve(1157,a,1136,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B),C))). [resolve(1157,a,1137,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1157,a,1138,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(c_Nat_OSuc,B))),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))). [resolve(1157,a,1139,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),A))). [resolve(1157,a,1141,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)). [resolve(1157,a,1143,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B))). [resolve(1157,a,1144,a)]. 1.95/2.12 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))). [resolve(1157,a,1145,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1157,a,1146,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B))). [resolve(1157,a,1147,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)))). [resolve(1157,a,1148,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(c_Nat_OSuc,B))),A)). [resolve(1157,a,1151,a)]. 1.95/2.12 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),B) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(1157,a,1152,a)]. 1.95/2.12 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))). [resolve(1157,a,1153,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))). [resolve(1157,a,1154,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),B))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),C)). [resolve(1157,a,1155,a)]. 1.95/2.12 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))). [resolve(1157,a,1156,a)]. 1.95/2.12 1158 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),D),B)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C))) # label(fact_power__increasing) # label(axiom). [clausify(845)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(C),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(C))),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(C),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(C)),D),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(C)),D),B))) | -class_Rings_Olinordered__idom(C). [resolve(1158,a,1124,b)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B))). [resolve(1158,a,1140,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B))). [resolve(1158,a,1150,a)]. 1.95/2.12 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),B))). [resolve(1158,a,1157,a)]. 1.95/2.13 1159 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),B))) # label(fact_power__strict__decreasing) # label(axiom). [clausify(872)]. 1.95/2.13 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(C))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(C)),D),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(C)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(C)),D),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(C)),D),A))) | -class_Rings_Olinordered__idom(C). [resolve(1159,a,1124,b)]. 1.95/2.13 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),A))). [resolve(1159,a,1140,a)]. 1.95/2.13 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),c_Groups_Oone__class_Oone(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),A))). [resolve(1159,a,1150,a)]. 1.95/2.13 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),A))). [resolve(1159,a,1157,a)]. 1.95/2.13 1160 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) # label(fact_pos__add__strict) # label(axiom). [clausify(883)]. 1.95/2.13 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D))) | -class_Rings_Olinordered__idom(A). [resolve(1160,a,1124,b)]. 1.95/2.13 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))). [resolve(1160,a,1140,a)]. 1.95/2.13 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C))). [resolve(1160,a,1150,a)]. 1.95/2.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C))). [resolve(1160,a,1157,a)]. 1.95/2.16 1161 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_zero__less__power) # label(axiom). [clausify(905)]. 1.95/2.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1161,a,1124,b)]. 1.95/2.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(1161,a,1140,a)]. 1.95/2.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B))). [resolve(1161,a,1157,a)]. 1.95/2.16 1162 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)))) # label(fact_power__gt1) # label(axiom). [clausify(952)]. 1.95/2.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),hAPP(c_Nat_OSuc,C)))) | -class_Rings_Olinordered__idom(A). [resolve(1162,a,1124,b)]. 1.95/2.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B)))). [resolve(1162,a,1140,a)]. 1.95/2.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B)))). [resolve(1162,a,1150,a)]. 1.95/2.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(c_Nat_OSuc,B)))). [resolve(1162,a,1157,a)]. 1.95/2.16 1163 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)))) # label(fact_power__gt1__lemma) # label(axiom). [clausify(991)]. 1.95/2.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)))) | -class_Rings_Olinordered__idom(A). [resolve(1163,a,1124,b)]. 2.03/2.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)))). [resolve(1163,a,1140,a)]. 2.03/2.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)))). [resolve(1163,a,1150,a)]. 2.03/2.25 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)))). [resolve(1163,a,1157,a)]. 2.03/2.25 1164 class_Rings_Olinordered__ring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__ring) # label(axiom). [assumption]. 2.03/2.25 1165 -class_Rings_Olinordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C))),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__sum__squares__lt__zero) # label(axiom). [clausify(8)]. 2.03/2.25 1166 -class_Rings_Olinordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__square__less__zero) # label(axiom). [clausify(97)]. 2.03/2.25 1167 -class_Rings_Olinordered__ring(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B))) # label(fact_zero__le__square) # label(axiom). [clausify(175)]. 2.03/2.25 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B))),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1164,a,1165,a)]. 2.03/2.25 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1164,a,1166,a)]. 2.03/2.25 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A))). [resolve(1164,a,1167,a)]. 2.03/2.25 1168 -class_Rings_Olinordered__idom(A) | class_Rings_Olinordered__ring(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Olinordered__ring) # label(axiom). [clausify(318)]. 2.03/2.25 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C))),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))). [resolve(1168,b,1165,a)]. 2.03/2.25 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))). [resolve(1168,b,1166,a)]. 2.03/2.25 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B))). [resolve(1168,b,1167,a)]. 2.25/2.46 1169 -class_Rings_Olinordered__ring(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)))) # label(fact_sum__squares__ge__zero) # label(axiom). [clausify(450)]. 2.25/2.46 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B)))). [resolve(1169,a,1164,a)]. 2.25/2.46 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C)))) | -class_Rings_Olinordered__idom(A). [resolve(1169,a,1168,b)]. 2.25/2.46 1170 class_Rings_Olinordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring) # label(axiom). [assumption]. 2.25/2.46 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1170,a,1165,a)]. 2.25/2.46 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1170,a,1166,a)]. 2.25/2.46 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A))). [resolve(1170,a,1167,a)]. 2.25/2.46 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)))). [resolve(1170,a,1169,a)]. 2.25/2.46 1171 -class_Groups_Ocancel__ab__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D) | C = D # label(fact_add__imp__eq) # label(axiom). [clausify(906)]. 2.25/2.46 1172 -class_Groups_Ocancel__comm__monoid__add(A) | class_Groups_Ocancel__ab__semigroup__add(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Ocancel__ab__semigroup__add) # label(axiom). [clausify(12)]. 2.25/2.46 1173 class_Groups_Ocancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 2.25/2.46 1174 class_Groups_Ocancel__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 2.25/2.46 1175 class_Groups_Ocancel__ab__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 2.25/2.46 1176 class_Groups_Ocancel__ab__semigroup__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 2.25/2.46 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D) | C = D | -class_Groups_Ocancel__comm__monoid__add(A). [resolve(1171,a,1172,b)]. 2.25/2.46 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C) | B = C. [resolve(1171,a,1173,a)]. 2.25/2.46 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C) | B = C. [resolve(1171,a,1174,a)]. 2.25/2.46 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C) | B = C. [resolve(1171,a,1175,a)]. 2.42/2.58 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),C) | B = C. [resolve(1171,a,1176,a)]. 2.42/2.58 1177 -class_Rings_Oordered__cancel__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_split__mult__neg__le) # label(axiom). [clausify(91)]. 2.42/2.58 1178 class_Rings_Oordered__cancel__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 2.42/2.58 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1177,a,1178,a)]. 2.42/2.58 1179 -class_Rings_Oordered__cancel__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_split__mult__neg__le) # label(axiom). [clausify(91)]. 2.42/2.58 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1179,a,1178,a)]. 2.42/2.58 1180 class_Rings_Oordered__cancel__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 2.42/2.58 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1180,a,1177,a)]. 2.42/2.58 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1180,a,1179,a)]. 2.42/2.58 1181 -class_Rings_Oordered__cancel__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__nonneg__nonpos) # label(axiom). [clausify(412)]. 2.42/2.58 1182 class_Rings_Oordered__cancel__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 2.42/2.58 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1182,a,1177,a)]. 2.47/2.64 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1182,a,1179,a)]. 2.47/2.64 1183 -class_Rings_Oordered__cancel__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__nonpos__nonneg) # label(axiom). [clausify(637)]. 2.47/2.64 1184 -class_Rings_Oordered__cancel__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__nonneg__nonpos2) # label(axiom). [clausify(913)]. 2.47/2.64 1185 -class_Rings_Olinordered__idom(A) | class_Rings_Oordered__cancel__semiring(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Oordered__cancel__semiring) # label(axiom). [clausify(955)]. 2.47/2.64 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))). [resolve(1185,b,1177,a)]. 2.47/2.64 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))). [resolve(1185,b,1179,a)]. 2.47/2.64 1186 -class_Rings_Oordered__cancel__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__nonneg__nonneg) # label(axiom). [clausify(1005)]. 2.47/2.64 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1186,a,1178,a)]. 2.47/2.64 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))). [resolve(1186,a,1180,a)]. 2.47/2.64 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B))). [resolve(1186,a,1182,a)]. 2.55/2.75 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1186,a,1185,b)]. 2.55/2.75 1187 -class_Rings_Olinordered__semiring__1__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),F)) | c_Groups_Oone__class_Oone(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),E),F) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),D))),C)) # label(fact_convex__bound__lt) # label(axiom). [clausify(350)]. 2.55/2.75 1188 class_Rings_Olinordered__semiring__1__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 2.55/2.75 1189 -class_Rings_Olinordered__idom(A) | class_Rings_Olinordered__semiring__1__strict(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Olinordered__semiring__1__strict) # label(axiom). [clausify(296)]. 2.55/2.75 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),E)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),D),E) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),C))),B)). [resolve(1187,a,1188,a)]. 2.55/2.75 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),D),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),F)) | c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),E),F) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),E),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),F),D))),C)) | -class_Rings_Olinordered__idom(A). [resolve(1187,a,1189,b)]. 2.55/2.75 1190 class_Rings_Olinordered__semiring__1__strict(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 2.55/2.75 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),E)) | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),D),E) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),E),C))),B)). [resolve(1190,a,1187,a)]. 2.72/2.89 1191 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_mult__less__imp__less__left) # label(axiom). [clausify(137)]. 2.72/2.89 1192 -class_Rings_Olinordered__idom(A) | class_Rings_Olinordered__semiring__strict(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Olinordered__semiring__strict) # label(axiom). [clausify(16)]. 2.72/2.89 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),D)) | -class_Rings_Olinordered__idom(A). [resolve(1191,a,1192,b)]. 2.72/2.89 1193 class_Rings_Olinordered__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.72/2.89 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)). [resolve(1193,a,1191,a)]. 2.72/2.89 1194 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__pos__neg) # label(axiom). [clausify(327)]. 2.72/2.89 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1194,a,1192,b)]. 2.72/2.89 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1194,a,1193,a)]. 2.72/2.89 1195 class_Rings_Olinordered__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.72/2.89 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1195,a,1194,a)]. 2.72/2.89 1196 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),D)) # label(fact_mult__right__le__imp__le) # label(axiom). [clausify(366)]. 2.74/2.90 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),D)) | -class_Rings_Olinordered__idom(A). [resolve(1196,a,1192,b)]. 2.74/2.90 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),C)). [resolve(1196,a,1193,a)]. 2.74/2.90 1197 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_zero__less__mult__pos2) # label(axiom). [clausify(455)]. 2.74/2.90 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -class_Rings_Olinordered__idom(A). [resolve(1197,a,1192,b)]. 2.74/2.90 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(1197,a,1193,a)]. 2.74/2.90 1198 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,D),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__less__le__imp__less) # label(axiom). [clausify(487)]. 2.74/2.90 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),D),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E))) | -class_Rings_Olinordered__idom(A). [resolve(1198,a,1192,b)]. 2.74/2.90 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(1198,a,1193,a)]. 2.77/2.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(1198,a,1195,a)]. 2.77/2.92 1199 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) # label(fact_mult__left__le__imp__le) # label(axiom). [clausify(525)]. 2.77/2.92 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),D)) | -class_Rings_Olinordered__idom(A). [resolve(1199,a,1192,b)]. 2.77/2.92 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),C)). [resolve(1199,a,1193,a)]. 2.77/2.92 1200 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__pos__pos) # label(axiom). [clausify(544)]. 2.77/2.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1200,a,1192,b)]. 2.77/2.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1200,a,1193,a)]. 2.77/2.92 1201 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) # label(fact_zero__less__mult__pos) # label(axiom). [clausify(617)]. 2.77/2.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | -class_Rings_Olinordered__idom(A). [resolve(1201,a,1192,b)]. 2.77/2.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)). [resolve(1201,a,1193,a)]. 2.77/2.93 1202 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__strict__mono) # label(axiom). [clausify(701)]. 2.77/2.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E))) | -class_Rings_Olinordered__idom(A). [resolve(1202,a,1192,b)]. 2.77/2.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(1202,a,1193,a)]. 2.77/2.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(1202,a,1195,a)]. 2.77/2.93 1203 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_mult__strict__left__mono) # label(axiom). [clausify(710)]. 2.77/2.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C))) | -class_Rings_Olinordered__idom(A). [resolve(1203,a,1192,b)]. 2.77/2.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))). [resolve(1203,a,1193,a)]. 2.77/2.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B))). [resolve(1203,a,1195,a)]. 2.77/2.94 1204 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__le__less__imp__less) # label(axiom). [clausify(734)]. 2.77/2.94 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E))) | -class_Rings_Olinordered__idom(A). [resolve(1204,a,1192,b)]. 2.77/2.94 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(1204,a,1193,a)]. 2.77/2.94 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(1204,a,1195,a)]. 2.77/2.94 1205 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_mult__less__imp__less__right) # label(axiom). [clausify(753)]. 2.77/2.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),D)) | -class_Rings_Olinordered__idom(A). [resolve(1205,a,1192,b)]. 2.77/2.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(1205,a,1193,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(1205,a,1195,a)]. 2.77/2.95 1206 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__neg__pos) # label(axiom). [clausify(797)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1206,a,1192,b)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1206,a,1193,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1206,a,1195,a)]. 2.77/2.95 1207 class_Rings_Olinordered__semiring__strict(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),C)). [resolve(1207,a,1191,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1207,a,1194,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),C)). [resolve(1207,a,1196,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)). [resolve(1207,a,1197,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D))). [resolve(1207,a,1198,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),C)). [resolve(1207,a,1199,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)). [resolve(1207,a,1201,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D))). [resolve(1207,a,1202,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B))). [resolve(1207,a,1203,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D))). [resolve(1207,a,1204,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),C)). [resolve(1207,a,1205,a)]. 2.77/2.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1207,a,1206,a)]. 2.77/2.98 1208 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__strict__mono_H) # label(axiom). [clausify(927)]. 2.77/2.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),D),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E))) | -class_Rings_Olinordered__idom(A). [resolve(1208,a,1192,b)]. 2.77/2.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(1208,a,1193,a)]. 2.77/2.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(1208,a,1195,a)]. 2.77/2.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D))). [resolve(1208,a,1207,a)]. 2.77/2.98 1209 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__pos__neg2) # label(axiom). [clausify(1000)]. 2.77/2.98 1210 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) # label(fact_mult__strict__right__mono) # label(axiom). [clausify(1027)]. 2.77/2.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D))) | -class_Rings_Olinordered__idom(A). [resolve(1210,a,1192,b)]. 3.08/3.23 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))). [resolve(1210,a,1193,a)]. 3.08/3.23 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))). [resolve(1210,a,1195,a)]. 3.08/3.23 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C))). [resolve(1210,a,1207,a)]. 3.08/3.23 1211 class_Rings_Oordered__comm__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 3.08/3.23 1212 -class_Rings_Oordered__comm__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_comm__mult__left__mono) # label(axiom). [clausify(19)]. 3.08/3.23 1213 class_Rings_Oordered__comm__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 3.08/3.23 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B))). [resolve(1213,a,1212,a)]. 3.08/3.23 1214 class_Rings_Oordered__comm__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 3.08/3.23 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))). [resolve(1214,a,1212,a)]. 3.08/3.23 1215 -class_Rings_Olinordered__idom(A) | class_Rings_Oordered__comm__semiring(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Oordered__comm__semiring) # label(axiom). [clausify(960)]. 3.08/3.23 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C))). [resolve(1215,b,1212,a)]. 3.08/3.23 1216 -class_Rings_Olinordered__semiring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_mult__left__less__imp__less) # label(axiom). [clausify(293)]. 3.24/3.44 1217 class_Rings_Olinordered__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring) # label(axiom). [assumption]. 3.24/3.44 1218 -class_Rings_Olinordered__idom(A) | class_Rings_Olinordered__semiring(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Olinordered__semiring) # label(axiom). [clausify(573)]. 3.24/3.44 1219 class_Rings_Olinordered__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semiring) # label(axiom). [assumption]. 3.24/3.44 1220 -class_Rings_Olinordered__semiring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_mult__right__less__imp__less) # label(axiom). [clausify(896)]. 3.24/3.44 1221 class_Rings_Olinordered__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring) # label(axiom). [assumption]. 3.24/3.44 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)). [resolve(1221,a,1216,a)]. 3.24/3.44 1222 -class_Rings_Ozero__neq__one(A) | c_Groups_Oone__class_Oone(A) != c_Groups_Ozero__class_Ozero(A) # label(fact_one__neq__zero) # label(axiom). [clausify(75)]. 3.24/3.44 1223 -class_Rings_Ocomm__semiring__1(A) | class_Rings_Ozero__neq__one(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Ozero__neq__one) # label(axiom). [clausify(26)]. 3.24/3.44 1224 class_Rings_Ozero__neq__one(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ozero__neq__one) # label(axiom). [assumption]. 3.24/3.44 Derived: c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Rings_Ocomm__semiring__1(A). [resolve(1222,a,1223,b)]. 3.24/3.44 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) != c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1222,a,1224,a)]. 3.24/3.44 1225 class_Rings_Ozero__neq__one(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Ozero__neq__one) # label(axiom). [assumption]. 3.24/3.44 Derived: c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1225,a,1222,a)]. 3.24/3.44 1226 -class_Power_Opower(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Omult__zero(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_power__eq__0__iff) # label(axiom). [clausify(785)]. 3.24/3.44 Derived: -class_Power_Opower(tc_Polynomial_Opoly(A)) | -class_Rings_Ono__zero__divisors(tc_Polynomial_Opoly(A)) | -class_Rings_Omult__zero(tc_Polynomial_Opoly(A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | -class_Rings_Ocomm__semiring__1(A). [resolve(1226,c,1223,b)]. 3.24/3.44 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(1226,c,1224,a)]. 3.24/3.44 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | -class_Rings_Ono__zero__divisors(tc_Complex_Ocomplex) | -class_Rings_Omult__zero(tc_Complex_Ocomplex) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A. [resolve(1226,c,1225,a)]. 3.24/3.47 1227 -class_Power_Opower(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Omult__zero(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != C # label(fact_power__eq__0__iff) # label(axiom). [clausify(785)]. 3.24/3.47 Derived: -class_Power_Opower(tc_Polynomial_Opoly(A)) | -class_Rings_Ono__zero__divisors(tc_Polynomial_Opoly(A)) | -class_Rings_Omult__zero(tc_Polynomial_Opoly(A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != C | -class_Rings_Ocomm__semiring__1(A). [resolve(1227,c,1223,b)]. 3.24/3.47 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(1227,c,1224,a)]. 3.24/3.47 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | -class_Rings_Ono__zero__divisors(tc_Complex_Ocomplex) | -class_Rings_Omult__zero(tc_Complex_Ocomplex) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(1227,c,1225,a)]. 3.24/3.47 1228 -class_Power_Opower(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Omult__zero(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = C # label(fact_power__eq__0__iff) # label(axiom). [clausify(785)]. 3.24/3.47 Derived: -class_Power_Opower(tc_Polynomial_Opoly(A)) | -class_Rings_Ono__zero__divisors(tc_Polynomial_Opoly(A)) | -class_Rings_Omult__zero(tc_Polynomial_Opoly(A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = C | -class_Rings_Ocomm__semiring__1(A). [resolve(1228,c,1223,b)]. 3.24/3.47 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(1228,c,1224,a)]. 3.24/3.47 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | -class_Rings_Ono__zero__divisors(tc_Complex_Ocomplex) | -class_Rings_Omult__zero(tc_Complex_Ocomplex) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(1228,c,1225,a)]. 3.24/3.47 1229 -class_Rings_Ozero__neq__one(A) | c_Groups_Oone__class_Oone(A) != c_Groups_Ozero__class_Ozero(A) # label(fact_zero__neq__one) # label(axiom). [clausify(901)]. 3.24/3.47 1230 class_Rings_Ozero__neq__one(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ozero__neq__one) # label(axiom). [assumption]. 3.24/3.47 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1230,a,1222,a)]. 3.24/3.47 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(1230,a,1226,c)]. 3.24/3.47 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(1230,a,1227,c)]. 3.24/3.47 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(1230,a,1228,c)]. 3.47/3.62 1231 class_Rings_Ozero__neq__one(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Ozero__neq__one) # label(axiom). [assumption]. 3.47/3.62 Derived: c_Groups_Oone__class_Oone(tc_RealDef_Oreal) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1231,a,1222,a)]. 3.47/3.62 Derived: -class_Power_Opower(tc_RealDef_Oreal) | -class_Rings_Ono__zero__divisors(tc_RealDef_Oreal) | -class_Rings_Omult__zero(tc_RealDef_Oreal) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1231,a,1226,c)]. 3.47/3.62 Derived: -class_Power_Opower(tc_RealDef_Oreal) | -class_Rings_Ono__zero__divisors(tc_RealDef_Oreal) | -class_Rings_Omult__zero(tc_RealDef_Oreal) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(1231,a,1227,c)]. 3.47/3.62 Derived: -class_Power_Opower(tc_RealDef_Oreal) | -class_Rings_Ono__zero__divisors(tc_RealDef_Oreal) | -class_Rings_Omult__zero(tc_RealDef_Oreal) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(1231,a,1228,c)]. 3.47/3.62 1232 -class_Groups_Omonoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Ozero__class_Ozero(A)) = B # label(fact_add__0__right) # label(axiom). [clausify(108)]. 3.47/3.62 1233 class_Groups_Omonoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__add) # label(axiom). [assumption]. 3.47/3.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(1232,a,1233,a)]. 3.47/3.62 1234 class_Groups_Omonoid__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Omonoid__add) # label(axiom). [assumption]. 3.47/3.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = A. [resolve(1234,a,1232,a)]. 3.47/3.62 1235 class_Groups_Omonoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Omonoid__add) # label(axiom). [assumption]. 3.47/3.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(1235,a,1232,a)]. 3.47/3.62 1236 -class_Groups_Ocomm__monoid__add(A) | class_Groups_Omonoid__add(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Omonoid__add) # label(axiom). [clausify(666)]. 3.47/3.62 Derived: -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) = B. [resolve(1236,b,1232,a)]. 3.47/3.62 1237 -class_Groups_Omonoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),B) = B # label(fact_add__0__left) # label(axiom). [clausify(792)]. 3.47/3.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A) = A. [resolve(1237,a,1233,a)]. 3.47/3.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A) = A. [resolve(1237,a,1234,a)]. 3.47/3.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A) = A. [resolve(1237,a,1235,a)]. 3.47/3.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B) = B | -class_Groups_Ocomm__monoid__add(A). [resolve(1237,a,1236,b)]. 3.47/3.62 1238 class_Groups_Omonoid__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Omonoid__add) # label(axiom). [assumption]. 3.47/3.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)) = A. [resolve(1238,a,1232,a)]. 3.47/3.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)),A) = A. [resolve(1238,a,1237,a)]. 3.64/3.78 1239 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,E,C) != c_Divides_Odiv__class_Omod(A,F,C) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),E),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),F),C) # label(fact_mod__diff__cong) # label(axiom). [clausify(94)]. 3.64/3.78 1240 class_Divides_Oring__div(tc_Int_Oint) # label(arity_Int__Oint__Divides_Oring__div) # label(axiom). [assumption]. 3.64/3.78 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,D,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,E,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),D),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),E),B). [resolve(1239,a,1240,a)]. 3.64/3.78 1241 -class_Fields_Ofield(A) | class_Divides_Oring__div(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Divides_Oring__div) # label(axiom). [clausify(243)]. 3.64/3.78 Derived: -class_Fields_Ofield(A) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),E,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),F,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),E),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),D),F),C). [resolve(1241,b,1239,a)]. 3.64/3.78 1242 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),c_Divides_Odiv__class_Omod(A,B,C)),c_Divides_Odiv__class_Omod(A,D,C)),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D),C) # label(fact_mod__diff__eq) # label(axiom). [clausify(258)]. 3.64/3.78 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),C),B). [resolve(1242,a,1240,a)]. 3.64/3.78 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1242,a,1241,b)]. 3.64/3.78 1243 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),c_Divides_Odiv__class_Omod(A,C,D)),D) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C),D) # label(fact_mod__diff__right__eq) # label(axiom). [clausify(567)]. 3.64/3.78 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B),C). [resolve(1243,a,1240,a)]. 3.64/3.78 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D)),D) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C),D) | -class_Fields_Ofield(A). [resolve(1243,a,1241,b)]. 3.64/3.78 1244 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D),C) # label(fact_mod__diff__left__eq) # label(axiom). [clausify(847)]. 3.78/3.94 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),C),B). [resolve(1244,a,1240,a)]. 3.78/3.94 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),D),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1244,a,1241,b)]. 3.78/3.94 1245 -class_Rings_Olinordered__idom(A) | class_Groups_Oordered__comm__monoid__add(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Oordered__comm__monoid__add) # label(axiom). [clausify(732)]. 3.78/3.94 1246 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) # label(fact_add__strict__increasing) # label(axiom). [clausify(42)]. 3.78/3.94 1247 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_add__nonpos__neg) # label(axiom). [clausify(72)]. 3.78/3.94 1248 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))) # label(fact_add__pos__pos) # label(axiom). [clausify(159)]. 3.78/3.94 1249 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) # label(fact_add__increasing) # label(axiom). [clausify(176)]. 3.78/3.94 1250 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_add__neg__neg) # label(axiom). [clausify(226)]. 3.78/3.94 1251 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))) # label(fact_add__pos__nonneg) # label(axiom). [clausify(260)]. 3.78/3.94 1252 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))) # label(fact_add__nonneg__pos) # label(axiom). [clausify(277)]. 3.78/3.94 1253 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(281)]. 3.78/3.94 1254 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(281)]. 3.78/3.94 1255 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) != C | c_Groups_Ozero__class_Ozero(A) != B # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(281)]. 3.78/3.94 1256 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) # label(fact_add__strict__increasing2) # label(axiom). [clausify(395)]. 3.78/3.94 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D))). [resolve(1245,b,1246,a)]. 3.78/3.94 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))). [resolve(1245,b,1247,a)]. 3.78/3.94 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D))). [resolve(1245,b,1249,a)]. 3.78/3.94 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))). [resolve(1245,b,1250,a)]. 3.78/3.94 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C))). [resolve(1245,b,1252,a)]. 3.78/3.94 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = C. [resolve(1245,b,1253,a)]. 3.78/3.95 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B. [resolve(1245,b,1254,a)]. 3.78/3.95 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B. [resolve(1245,b,1255,a)]. 3.78/3.95 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D))). [resolve(1245,b,1256,a)]. 3.78/3.95 1257 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_add__neg__nonpos) # label(axiom). [clausify(750)]. 3.78/3.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1257,a,1245,b)]. 3.78/3.95 1258 class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 3.78/3.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C))). [resolve(1258,a,1246,a)]. 3.78/3.95 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1258,a,1247,a)]. 3.78/3.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1258,a,1250,a)]. 3.78/3.95 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B))). [resolve(1258,a,1252,a)]. 3.78/3.96 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C))). [resolve(1258,a,1256,a)]. 3.78/3.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1258,a,1257,a)]. 3.78/3.96 1259 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B))) # label(fact_add__increasing2) # label(axiom). [clausify(807)]. 3.78/3.96 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),B))) | -class_Rings_Olinordered__idom(A). [resolve(1259,a,1245,b)]. 3.78/3.96 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),A))). [resolve(1259,a,1258,a)]. 3.78/3.96 1260 class_Groups_Oordered__comm__monoid__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 3.78/3.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C))). [resolve(1260,a,1246,a)]. 3.78/3.96 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1260,a,1247,a)]. 3.78/3.96 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C))). [resolve(1260,a,1249,a)]. 3.78/3.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1260,a,1250,a)]. 3.78/3.96 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B))). [resolve(1260,a,1252,a)]. 3.78/3.98 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B. [resolve(1260,a,1253,a)]. 3.78/3.98 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1260,a,1254,a)]. 3.78/3.98 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1260,a,1255,a)]. 3.78/3.98 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C))). [resolve(1260,a,1256,a)]. 3.78/3.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1260,a,1257,a)]. 3.78/3.98 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),A))). [resolve(1260,a,1259,a)]. 3.78/3.98 1261 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))) # label(fact_add__nonneg__nonneg) # label(axiom). [clausify(887)]. 3.78/3.98 1262 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_add__nonpos__nonpos) # label(axiom). [clausify(946)]. 3.78/3.98 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1262,a,1245,b)]. 3.78/3.98 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(1262,a,1258,a)]. 3.85/3.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1262,a,1260,a)]. 3.85/3.99 1263 class_Groups_Oordered__comm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 3.85/3.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))). [resolve(1263,a,1246,a)]. 3.85/3.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1263,a,1247,a)]. 3.85/3.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))). [resolve(1263,a,1249,a)]. 3.85/3.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1263,a,1250,a)]. 3.85/3.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B))). [resolve(1263,a,1252,a)]. 3.85/3.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(1263,a,1253,a)]. 3.85/3.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(1263,a,1254,a)]. 3.85/3.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(1263,a,1255,a)]. 3.85/3.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))). [resolve(1263,a,1256,a)]. 3.99/4.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1263,a,1257,a)]. 3.99/4.15 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A))). [resolve(1263,a,1259,a)]. 3.99/4.15 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1263,a,1262,a)]. 3.99/4.15 1264 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) | B = D # label(fact_add__right__cancel) # label(axiom). [clausify(158)]. 3.99/4.15 1265 class_Groups_Ocancel__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 3.99/4.15 1266 class_Groups_Ocancel__semigroup__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 3.99/4.15 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B) | A = C. [resolve(1264,a,1265,a)]. 3.99/4.15 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),C),B) | A = C. [resolve(1264,a,1266,a)]. 3.99/4.15 1267 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) | B != D # label(fact_add__right__cancel) # label(axiom). [clausify(158)]. 3.99/4.15 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B) | A != C. [resolve(1267,a,1265,a)]. 3.99/4.15 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),C),B) | A != C. [resolve(1267,a,1266,a)]. 3.99/4.15 1268 -class_Groups_Ocancel__comm__monoid__add(A) | class_Groups_Ocancel__semigroup__add(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Ocancel__semigroup__add) # label(axiom). [clausify(165)]. 3.99/4.15 Derived: -class_Groups_Ocancel__comm__monoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),C) | B = D. [resolve(1268,b,1264,a)]. 3.99/4.15 Derived: -class_Groups_Ocancel__comm__monoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),C) | B != D. [resolve(1268,b,1267,a)]. 3.99/4.15 1269 class_Groups_Ocancel__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 3.99/4.15 1270 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) | B = D # label(fact_add__right__imp__eq) # label(axiom). [clausify(837)]. 3.99/4.15 1271 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D) | C = D # label(fact_add__left__imp__eq) # label(axiom). [clausify(874)]. 3.99/4.15 1272 -class_Groups_Ocancel__semigroup__add(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) # label(fact_add__left__cancel) # label(axiom). [clausify(900)]. 4.08/4.27 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B). [resolve(1272,a,1265,a)]. 4.08/4.27 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),C),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),C),B). [resolve(1272,a,1266,a)]. 4.08/4.27 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(C)),D),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(C)),D),B) | -class_Groups_Ocancel__comm__monoid__add(C). [resolve(1272,a,1268,b)]. 4.08/4.27 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B). [resolve(1272,a,1269,a)]. 4.08/4.27 1273 -class_Groups_Ocancel__semigroup__add(A) | B = C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) # label(fact_add__left__cancel) # label(axiom). [clausify(900)]. 4.08/4.27 1274 class_Groups_Ocancel__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 4.08/4.27 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),B) | A = C. [resolve(1274,a,1264,a)]. 4.08/4.27 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),B) | A != C. [resolve(1274,a,1267,a)]. 4.08/4.27 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),B). [resolve(1274,a,1272,a)]. 4.08/4.27 1275 class_Rings_Oordered__ring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oordered__ring) # label(axiom). [assumption]. 4.08/4.27 1276 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)),E)),F))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)),F))) # label(fact_less__add__iff2) # label(axiom). [clausify(55)]. 4.08/4.27 1277 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)),E)),F))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)),F))) # label(fact_less__add__iff2) # label(axiom). [clausify(55)]. 4.08/4.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),B),C)),D)),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D)),E))). [resolve(1275,a,1276,a)]. 4.08/4.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),B),C)),D)),E))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D)),E))). [resolve(1275,a,1277,a)]. 4.08/4.28 1278 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__left__mono__neg) # label(axiom). [clausify(162)]. 4.08/4.28 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A))). [resolve(1278,a,1275,a)]. 4.08/4.28 1279 -class_Rings_Olinordered__idom(A) | class_Rings_Oordered__ring(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Oordered__ring) # label(axiom). [clausify(234)]. 4.08/4.28 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),C),D)),E)),F))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E)),F))). [resolve(1279,b,1276,a)]. 4.08/4.28 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),C),D)),E)),F))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E)),F))). [resolve(1279,b,1277,a)]. 4.08/4.28 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),D),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B))). [resolve(1279,b,1278,a)]. 4.08/4.28 1280 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)),E)),F))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)),F))) # label(fact_le__add__iff2) # label(axiom). [clausify(257)]. 4.08/4.28 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),B),C)),D)),E))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D)),E))). [resolve(1280,a,1275,a)]. 4.15/4.28 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),C),D)),E)),F))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E)),F))) | -class_Rings_Olinordered__idom(A). [resolve(1280,a,1279,b)]. 4.15/4.28 1281 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)),E)),F))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)),F))) # label(fact_le__add__iff2) # label(axiom). [clausify(257)]. 4.15/4.28 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),B),C)),D)),E))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D)),E))). [resolve(1281,a,1275,a)]. 4.15/4.28 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),C),D)),E)),F))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E)),F))) | -class_Rings_Olinordered__idom(A). [resolve(1281,a,1279,b)]. 4.15/4.28 1282 class_Rings_Oordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__ring) # label(axiom). [assumption]. 4.15/4.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)),D)),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)),E))). [resolve(1282,a,1276,a)]. 4.15/4.28 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)),D)),E))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)),E))). [resolve(1282,a,1277,a)]. 4.15/4.28 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(1282,a,1278,a)]. 4.15/4.29 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)),D)),E))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)),E))). [resolve(1282,a,1280,a)]. 4.15/4.29 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)),D)),E))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)),E))). [resolve(1282,a,1281,a)]. 4.15/4.29 1283 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),D)),E)),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),F))) # label(fact_le__add__iff1) # label(axiom). [clausify(356)]. 4.15/4.29 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),B)),C)),D)),E)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)),E))). [resolve(1283,a,1275,a)]. 4.15/4.29 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C)),D)),E)),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),F))) | -class_Rings_Olinordered__idom(A). [resolve(1283,a,1279,b)]. 4.15/4.29 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),C)),D)),E)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),E))). [resolve(1283,a,1282,a)]. 4.15/4.29 1284 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),D)),E)),F)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),F))) # label(fact_le__add__iff1) # label(axiom). [clausify(356)]. 4.15/4.33 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),B)),C)),D)),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)),E))). [resolve(1284,a,1275,a)]. 4.15/4.33 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C)),D)),E)),F)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),F))) | -class_Rings_Olinordered__idom(A). [resolve(1284,a,1279,b)]. 4.15/4.33 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),C)),D)),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),E))). [resolve(1284,a,1282,a)]. 4.15/4.33 1285 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__right__mono__neg) # label(axiom). [clausify(635)]. 4.15/4.33 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))). [resolve(1285,a,1275,a)]. 4.15/4.33 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),D),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D))) | -class_Rings_Olinordered__idom(A). [resolve(1285,a,1279,b)]. 4.15/4.33 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))). [resolve(1285,a,1282,a)]. 4.15/4.33 1286 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),D)),E)),F)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),F))) # label(fact_less__add__iff1) # label(axiom). [clausify(854)]. 4.15/4.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),B)),C)),D)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)),E))). [resolve(1286,a,1275,a)]. 4.15/4.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C)),D)),E)),F)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),F))) | -class_Rings_Olinordered__idom(A). [resolve(1286,a,1279,b)]. 4.15/4.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),C)),D)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),E))). [resolve(1286,a,1282,a)]. 4.15/4.33 1287 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),D)),E)),F)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),F))) # label(fact_less__add__iff1) # label(axiom). [clausify(854)]. 4.15/4.33 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),B)),C)),D)),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)),E))). [resolve(1287,a,1275,a)]. 4.15/4.33 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C)),D)),E)),F)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),F))) | -class_Rings_Olinordered__idom(A). [resolve(1287,a,1279,b)]. 4.15/4.33 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),C)),D)),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),E))). [resolve(1287,a,1282,a)]. 4.30/4.45 1288 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__nonpos__nonpos) # label(axiom). [clausify(951)]. 4.30/4.45 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))). [resolve(1288,a,1275,a)]. 4.30/4.45 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1288,a,1279,b)]. 4.30/4.45 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1288,a,1282,a)]. 4.30/4.45 1289 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B))) # label(fact_split__mult__pos__le) # label(axiom). [clausify(996)]. 4.30/4.45 1290 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_split__mult__pos__le) # label(axiom). [clausify(996)]. 4.30/4.45 1291 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),c_RealVector_Onorm__class_Onorm(A,C))),f5(A)))) # label(fact_mult_Ononneg__bounded) # label(axiom). [clausify(85)]. 4.30/4.45 1292 class_RealVector_Oreal__normed__algebra(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra) # label(axiom). [assumption]. 4.30/4.45 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B))),f5(tc_Complex_Ocomplex)))). [resolve(1291,a,1292,a)]. 4.30/4.45 1293 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f5(A))) # label(fact_mult_Ononneg__bounded) # label(axiom). [clausify(85)]. 4.30/4.46 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f5(tc_Complex_Ocomplex))). [resolve(1293,a,1292,a)]. 4.30/4.46 1294 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),C) # label(fact_mult__left_Oadd) # label(axiom). [clausify(95)]. 4.30/4.46 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),C)),B). [resolve(1294,a,1292,a)]. 4.30/4.46 1295 -class_RealVector_Oreal__normed__algebra(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,D)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),E))) # label(fact_norm__mult__less) # label(axiom). [clausify(204)]. 4.30/4.46 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D))). [resolve(1295,a,1292,a)]. 4.30/4.46 1296 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),f16(C,A)))) # label(fact_mult__left_Obounded) # label(axiom). [clausify(223)]. 4.30/4.46 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),f16(B,tc_Complex_Ocomplex)))). [resolve(1296,a,1292,a)]. 4.30/4.46 1297 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),C) # label(fact_mult_Oadd__left) # label(axiom). [clausify(256)]. 4.30/4.46 1298 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ozero__class_Ozero(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_mult_Ozero__right) # label(axiom). [clausify(269)]. 4.30/4.46 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1298,a,1292,a)]. 4.30/4.46 1299 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_mult__right_Oadd) # label(axiom). [clausify(295)]. 4.30/4.46 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),B),C)). [resolve(1299,a,1292,a)]. 4.30/4.48 1300 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_mult_Oadd__right) # label(axiom). [clausify(382)]. 4.30/4.48 1301 class_RealVector_Oreal__normed__algebra(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Oreal__normed__algebra) # label(axiom). [assumption]. 4.30/4.48 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B))),f5(tc_RealDef_Oreal)))). [resolve(1301,a,1291,a)]. 4.30/4.48 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f5(tc_RealDef_Oreal))). [resolve(1301,a,1293,a)]. 4.30/4.48 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D))). [resolve(1301,a,1295,a)]. 4.30/4.48 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),f16(B,tc_RealDef_Oreal)))). [resolve(1301,a,1296,a)]. 4.30/4.48 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1301,a,1298,a)]. 4.30/4.48 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),C)). [resolve(1301,a,1299,a)]. 4.30/4.48 1302 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ozero__class_Ozero(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_mult__right_Ozero) # label(axiom). [clausify(480)]. 4.30/4.48 1303 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D)),C) # label(fact_mult__left_Odiff) # label(axiom). [clausify(484)]. 4.30/4.48 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),A),C)),B). [resolve(1303,a,1292,a)]. 4.30/4.48 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),C)),B). [resolve(1303,a,1301,a)]. 4.30/4.50 1304 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f38(B,A))) # label(fact_mult__right_Ononneg__bounded) # label(axiom). [clausify(501)]. 4.30/4.50 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f38(A,tc_Complex_Ocomplex))). [resolve(1304,a,1292,a)]. 4.30/4.50 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f38(A,tc_RealDef_Oreal))). [resolve(1304,a,1301,a)]. 4.30/4.50 1305 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,C)),f38(B,A)))) # label(fact_mult__right_Ononneg__bounded) # label(axiom). [clausify(501)]. 4.38/4.52 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B)),f38(A,tc_Complex_Ocomplex)))). [resolve(1305,a,1292,a)]. 4.38/4.52 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B)),f38(A,tc_RealDef_Oreal)))). [resolve(1305,a,1301,a)]. 4.38/4.52 1306 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ozero__class_Ozero(A)),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_mult_Ozero__left) # label(axiom). [clausify(646)]. 4.38/4.52 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)),A) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1306,a,1292,a)]. 4.38/4.52 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1306,a,1301,a)]. 4.38/4.52 1307 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),c_RealVector_Onorm__class_Onorm(A,C)))) # label(fact_norm__mult__ineq) # label(axiom). [clausify(661)]. 4.38/4.52 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B)))). [resolve(1307,a,1292,a)]. 4.38/4.52 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B)))). [resolve(1307,a,1301,a)]. 4.38/4.52 1308 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),c_RealVector_Onorm__class_Onorm(A,C))),f49(A)))) # label(fact_mult_Opos__bounded) # label(axiom). [clausify(716)]. 4.38/4.52 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B))),f49(tc_Complex_Ocomplex)))). [resolve(1308,a,1292,a)]. 4.38/4.52 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B))),f49(tc_RealDef_Oreal)))). [resolve(1308,a,1301,a)]. 4.38/4.52 1309 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f49(A))) # label(fact_mult_Opos__bounded) # label(axiom). [clausify(716)]. 4.38/4.52 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f49(tc_Complex_Ocomplex))). [resolve(1309,a,1292,a)]. 4.38/4.52 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f49(tc_RealDef_Oreal))). [resolve(1309,a,1301,a)]. 4.38/4.52 1310 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ozero__class_Ozero(A)),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_mult__left_Ozero) # label(axiom). [clausify(823)]. 4.38/4.52 1311 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)) # label(fact_mult_Odiff__right) # label(axiom). [clausify(827)]. 4.38/4.52 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),B),C)). [resolve(1311,a,1292,a)]. 4.38/4.52 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),B),C)). [resolve(1311,a,1301,a)]. 4.38/4.52 1312 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f58(B,A))) # label(fact_mult__left_Ononneg__bounded) # label(axiom). [clausify(873)]. 4.38/4.52 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f58(A,tc_Complex_Ocomplex))). [resolve(1312,a,1292,a)]. 4.38/4.52 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f58(A,tc_RealDef_Oreal))). [resolve(1312,a,1301,a)]. 4.38/4.52 1313 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),f58(C,A)))) # label(fact_mult__left_Ononneg__bounded) # label(axiom). [clausify(873)]. 4.38/4.52 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),f58(B,tc_Complex_Ocomplex)))). [resolve(1313,a,1292,a)]. 4.38/4.54 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),f58(B,tc_RealDef_Oreal)))). [resolve(1313,a,1301,a)]. 4.38/4.54 1314 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D)),C) # label(fact_mult_Odiff__left) # label(axiom). [clausify(889)]. 4.38/4.54 1315 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,C)),f67(B,A)))) # label(fact_mult__right_Opos__bounded) # label(axiom). [clausify(909)]. 4.38/4.54 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B)),f67(A,tc_Complex_Ocomplex)))). [resolve(1315,a,1292,a)]. 4.38/4.54 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B)),f67(A,tc_RealDef_Oreal)))). [resolve(1315,a,1301,a)]. 4.38/4.54 1316 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f67(B,A))) # label(fact_mult__right_Opos__bounded) # label(axiom). [clausify(909)]. 4.38/4.54 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f67(A,tc_Complex_Ocomplex))). [resolve(1316,a,1292,a)]. 4.38/4.54 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f67(A,tc_RealDef_Oreal))). [resolve(1316,a,1301,a)]. 4.38/4.54 1317 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),f77(C,A)))) # label(fact_mult__left_Opos__bounded) # label(axiom). [clausify(984)]. 4.38/4.54 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),f77(B,tc_Complex_Ocomplex)))). [resolve(1317,a,1292,a)]. 4.38/4.54 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),f77(B,tc_RealDef_Oreal)))). [resolve(1317,a,1301,a)]. 4.38/4.54 1318 -class_RealVector_Oreal__normed__algebra(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f77(B,A))) # label(fact_mult__left_Opos__bounded) # label(axiom). [clausify(984)]. 4.38/4.54 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f77(A,tc_Complex_Ocomplex))). [resolve(1318,a,1292,a)]. 4.38/4.54 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f77(A,tc_RealDef_Oreal))). [resolve(1318,a,1301,a)]. 4.68/4.83 1319 -class_RealVector_Oreal__normed__algebra(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)) # label(fact_mult__right_Odiff) # label(axiom). [clausify(1032)]. 4.68/4.83 1320 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 4.68/4.83 1321 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) # label(fact_add__le__imp__le__left) # label(axiom). [clausify(65)]. 4.68/4.83 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),C)). [resolve(1320,a,1321,a)]. 4.68/4.83 1322 -class_Rings_Olinordered__idom(A) | class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [clausify(333)]. 4.68/4.83 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),D)). [resolve(1322,b,1321,a)]. 4.68/4.83 1323 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 4.68/4.83 1324 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) # label(fact_add__le__cancel__left) # label(axiom). [clausify(438)]. 4.68/4.83 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)). [resolve(1324,a,1323,a)]. 4.68/4.83 1325 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) # label(fact_add__le__cancel__left) # label(axiom). [clausify(438)]. 4.68/4.83 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),D)) | -class_Rings_Olinordered__idom(A). [resolve(1325,a,1322,b)]. 4.68/4.83 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)). [resolve(1325,a,1323,a)]. 4.68/4.83 1326 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 4.68/4.83 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),C)). [resolve(1326,a,1321,a)]. 4.68/4.83 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),C)). [resolve(1326,a,1325,a)]. 4.68/4.84 1327 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D))) # label(fact_add__less__cancel__right) # label(axiom). [clausify(529)]. 4.68/4.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),D))) | -class_Rings_Olinordered__idom(A). [resolve(1327,a,1322,b)]. 4.68/4.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C))). [resolve(1327,a,1323,a)]. 4.68/4.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),C))). [resolve(1327,a,1326,a)]. 4.68/4.84 1328 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D))) # label(fact_add__less__cancel__right) # label(axiom). [clausify(529)]. 4.68/4.84 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C))). [resolve(1328,a,1320,a)]. 4.68/4.84 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),D))) | -class_Rings_Olinordered__idom(A). [resolve(1328,a,1322,b)]. 4.68/4.84 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C))). [resolve(1328,a,1323,a)]. 4.68/4.84 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),C))). [resolve(1328,a,1326,a)]. 4.68/4.84 1329 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__less__cancel__left) # label(axiom). [clausify(542)]. 4.68/4.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))). [resolve(1329,a,1320,a)]. 4.68/4.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),C))) | -class_Rings_Olinordered__idom(A). [resolve(1329,a,1322,b)]. 4.68/4.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),B))). [resolve(1329,a,1326,a)]. 4.75/4.90 1330 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__less__cancel__left) # label(axiom). [clausify(542)]. 4.75/4.90 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))). [resolve(1330,a,1320,a)]. 4.75/4.90 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),C))) | -class_Rings_Olinordered__idom(A). [resolve(1330,a,1322,b)]. 4.75/4.90 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),B))). [resolve(1330,a,1326,a)]. 4.75/4.90 1331 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_add__less__imp__less__right) # label(axiom). [clausify(771)]. 4.75/4.90 1332 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_add__less__imp__less__left) # label(axiom). [clausify(828)]. 4.75/4.90 1333 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),D)) # label(fact_add__le__cancel__right) # label(axiom). [clausify(973)]. 4.75/4.90 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),C)). [resolve(1333,a,1320,a)]. 4.75/4.90 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),C))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),D)) | -class_Rings_Olinordered__idom(A). [resolve(1333,a,1322,b)]. 4.75/4.90 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),C)). [resolve(1333,a,1323,a)]. 4.75/4.90 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),B))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),C)). [resolve(1333,a,1326,a)]. 4.75/4.90 1334 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),D)) # label(fact_add__le__cancel__right) # label(axiom). [clausify(973)]. 4.75/4.90 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),C)). [resolve(1334,a,1320,a)]. 4.86/5.02 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),D)) | -class_Rings_Olinordered__idom(A). [resolve(1334,a,1322,b)]. 4.86/5.02 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),C)). [resolve(1334,a,1323,a)]. 4.86/5.02 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),B))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),C)). [resolve(1334,a,1326,a)]. 4.86/5.02 1335 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),D)) # label(fact_add__le__imp__le__right) # label(axiom). [clausify(1011)]. 4.86/5.02 1336 -class_Rings_Ocomm__semiring__0(A) | class_Rings_Osemiring__0(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Osemiring__0) # label(axiom). [clausify(93)]. 4.86/5.02 1337 -class_Power_Opower(A) | -class_Rings_Osemiring__0(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Nat_OSuc,B)) = c_Groups_Ozero__class_Ozero(A) # label(fact_power__0__Suc) # label(axiom). [clausify(70)]. 4.86/5.02 Derived: -class_Rings_Ocomm__semiring__0(A) | -class_Power_Opower(tc_Polynomial_Opoly(A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(c_Nat_OSuc,B)) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)). [resolve(1336,b,1337,b)]. 4.86/5.02 1338 class_Rings_Osemiring__0(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Osemiring__0) # label(axiom). [assumption]. 4.86/5.02 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)),hAPP(c_Nat_OSuc,A)) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1338,a,1337,b)]. 4.86/5.02 1339 -class_Power_Opower(A) | -class_Rings_Osemiring__0(A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_power__0__left) # label(axiom). [clausify(174)]. 4.86/5.02 Derived: -class_Power_Opower(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B | c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B) | -class_Rings_Ocomm__semiring__0(A). [resolve(1339,b,1336,b)]. 4.86/5.02 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)),A). [resolve(1339,b,1338,a)]. 4.86/5.02 1340 -class_Power_Opower(A) | -class_Rings_Osemiring__0(A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_power__0__left) # label(axiom). [clausify(174)]. 4.86/5.02 Derived: -class_Power_Opower(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Rings_Ocomm__semiring__0(A). [resolve(1340,b,1336,b)]. 5.04/5.20 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)),A) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1340,b,1338,a)]. 5.04/5.20 1341 class_Rings_Osemiring__0(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Osemiring__0) # label(axiom). [assumption]. 5.04/5.20 Derived: -class_Power_Opower(tc_RealDef_Oreal) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(c_Nat_OSuc,A)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1341,a,1337,b)]. 5.04/5.20 Derived: -class_Power_Opower(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A). [resolve(1341,a,1339,b)]. 5.04/5.21 Derived: -class_Power_Opower(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1341,a,1340,b)]. 5.04/5.21 1342 class_Rings_Osemiring__0(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring__0) # label(axiom). [assumption]. 5.04/5.21 Derived: -class_Power_Opower(tc_Nat_Onat) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_OSuc,A)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1342,a,1337,b)]. 5.04/5.21 Derived: -class_Power_Opower(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A). [resolve(1342,a,1339,b)]. 5.04/5.21 Derived: -class_Power_Opower(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(1342,a,1340,b)]. 5.04/5.21 1343 class_Rings_Osemiring__0(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring__0) # label(axiom). [assumption]. 5.04/5.21 Derived: -class_Power_Opower(tc_Int_Oint) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_OSuc,A)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1343,a,1337,b)]. 5.04/5.21 Derived: -class_Power_Opower(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(1343,a,1339,b)]. 5.04/5.21 Derived: -class_Power_Opower(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1343,a,1340,b)]. 5.04/5.21 1344 -class_RealVector_Oreal__normed__algebra__1(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),C))) # label(fact_norm__power__ineq) # label(axiom). [clausify(465)]. 5.04/5.21 1345 class_RealVector_Oreal__normed__algebra__1(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Oreal__normed__algebra__1) # label(axiom). [assumption]. 5.04/5.21 1346 class_RealVector_Oreal__normed__algebra__1(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra__1) # label(axiom). [assumption]. 5.04/5.21 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),B))). [resolve(1344,a,1345,a)]. 5.04/5.21 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),B))). [resolve(1344,a,1346,a)]. 5.20/5.39 1347 -class_RealVector_Oreal__normed__algebra__1(A) | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = c_RealVector_Onorm__class_Onorm(A,c_Groups_Oone__class_Oone(A)) # label(fact_norm__one) # label(axiom). [clausify(890)]. 5.20/5.39 Derived: c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)). [resolve(1347,a,1345,a)]. 5.20/5.39 Derived: c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)). [resolve(1347,a,1346,a)]. 5.20/5.39 1348 -class_Groups_Olinordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_double__zero__sym) # label(axiom). [clausify(222)]. 5.20/5.39 1349 -class_Rings_Olinordered__idom(A) | class_Groups_Olinordered__ab__group__add(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Olinordered__ab__group__add) # label(axiom). [clausify(77)]. 5.20/5.39 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),B) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | -class_Rings_Olinordered__idom(A). [resolve(1348,a,1349,b)]. 5.20/5.39 1350 -class_Groups_Olinordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_double__zero__sym) # label(axiom). [clausify(222)]. 5.20/5.39 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | -class_Rings_Olinordered__idom(A). [resolve(1350,a,1349,b)]. 5.20/5.39 1351 class_Groups_Olinordered__ab__group__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 5.20/5.39 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),A) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1351,a,1348,a)]. 5.20/5.39 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1351,a,1350,a)]. 5.20/5.39 1352 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(360)]. 5.20/5.39 1353 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(360)]. 5.20/5.39 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),B)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1353,a,1349,b)]. 5.20/5.39 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1353,a,1351,a)]. 5.20/5.39 1354 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(430)]. 5.27/5.42 1355 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(430)]. 5.27/5.42 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),B))) | -class_Rings_Olinordered__idom(A). [resolve(1355,a,1349,b)]. 5.27/5.42 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),A))). [resolve(1355,a,1351,a)]. 5.27/5.42 1356 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(777)]. 5.27/5.42 1357 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(777)]. 5.27/5.42 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),B))) | -class_Rings_Olinordered__idom(A). [resolve(1357,a,1349,b)]. 5.27/5.42 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),A))). [resolve(1357,a,1351,a)]. 5.27/5.42 1358 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(817)]. 5.27/5.42 1359 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(817)]. 5.27/5.42 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),B)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1359,a,1349,b)]. 5.27/5.42 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1359,a,1351,a)]. 5.35/5.56 1360 class_Groups_Olinordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 5.35/5.56 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(1360,a,1348,a)]. 5.35/5.56 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(1360,a,1350,a)]. 5.35/5.56 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1360,a,1353,a)]. 5.35/5.56 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A))). [resolve(1360,a,1355,a)]. 5.35/5.56 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A))). [resolve(1360,a,1357,a)]. 5.35/5.56 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1360,a,1359,a)]. 5.35/5.56 1361 -class_Groups_Olinordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_double__eq__0__iff) # label(axiom). [clausify(1007)]. 5.35/5.56 1362 -class_Groups_Olinordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_double__eq__0__iff) # label(axiom). [clausify(1007)]. 5.35/5.56 1363 class_Rings_Oordered__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__semiring) # label(axiom). [assumption]. 5.35/5.56 1364 -class_Rings_Oordered__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,D),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__mono_H) # label(axiom). [clausify(90)]. 5.35/5.56 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(1363,a,1364,a)]. 5.35/5.56 1365 -class_Rings_Oordered__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) # label(fact_mult__right__mono) # label(axiom). [clausify(286)]. 5.46/5.61 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))). [resolve(1365,a,1363,a)]. 5.46/5.61 1366 -class_Rings_Olinordered__idom(A) | class_Rings_Oordered__semiring(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Oordered__semiring) # label(axiom). [clausify(338)]. 5.46/5.61 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),D),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E))). [resolve(1366,b,1364,a)]. 5.46/5.61 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D))). [resolve(1366,b,1365,a)]. 5.46/5.61 1367 -class_Rings_Oordered__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,D),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__mono) # label(axiom). [clausify(593)]. 5.46/5.61 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(1367,a,1363,a)]. 5.46/5.61 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),D),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E))) | -class_Rings_Olinordered__idom(A). [resolve(1367,a,1366,b)]. 5.46/5.61 1368 -class_Rings_Oordered__semiring(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_mult__left__mono) # label(axiom). [clausify(727)]. 5.46/5.61 1369 class_Rings_Oordered__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__semiring) # label(axiom). [assumption]. 5.64/5.77 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(1369,a,1364,a)]. 5.64/5.77 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))). [resolve(1369,a,1365,a)]. 5.64/5.77 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(1369,a,1367,a)]. 5.64/5.77 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B))). [resolve(1369,a,1368,a)]. 5.64/5.77 1370 class_Rings_Oordered__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oordered__semiring) # label(axiom). [assumption]. 5.64/5.77 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D))). [resolve(1370,a,1364,a)]. 5.64/5.77 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C))). [resolve(1370,a,1365,a)]. 5.64/5.77 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D))). [resolve(1370,a,1367,a)]. 5.64/5.77 1371 class_Rings_Odivision__ring(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Odivision__ring) # label(axiom). [assumption]. 5.64/5.77 1372 -class_Rings_Odivision__ring(A) | c_Rings_Oinverse__class_Oinverse(A,B) != c_Rings_Oinverse__class_Oinverse(A,C) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | B = C # label(fact_nonzero__inverse__eq__imp__eq) # label(axiom). [clausify(96)]. 5.64/5.79 1373 -class_Rings_Odivision__ring(A) | c_Rings_Oinverse__class_Oinverse(A,B) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_inverse__zero__imp__zero) # label(axiom). [clausify(235)]. 5.64/5.79 1374 -class_Rings_Odivision__ring(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Rings_Oinverse__class_Oinverse(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Rings_Oinverse__class_Oinverse(A,B)),C) # label(fact_nonzero__power__inverse) # label(axiom). [clausify(247)]. 5.64/5.79 1375 -class_Rings_Odivision__ring(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Rings_Oinverse__class_Oinverse(A,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Rings_Oinverse__class_Oinverse(A,B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),B))),c_Rings_Oinverse__class_Oinverse(A,C)) # label(fact_division__ring__inverse__diff) # label(axiom). [clausify(557)]. 5.64/5.79 1376 -class_Rings_Odivision__ring(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Rings_Oinverse__class_Oinverse(A,B)) # label(fact_right__inverse) # label(axiom). [clausify(602)]. 5.64/5.79 1377 -class_Rings_Odivision__ring(A) | c_Rings_Oinverse__class_Oinverse(A,c_Groups_Oone__class_Oone(A)) = c_Groups_Oone__class_Oone(A) # label(fact_inverse__1) # label(axiom). [clausify(606)]. 5.64/5.79 Derived: c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A) != c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B | A = B. [resolve(1371,a,1372,a)]. 5.64/5.79 Derived: c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A. [resolve(1371,a,1373,a)]. 5.64/5.79 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)),B). [resolve(1371,a,1374,a)]. 5.64/5.79 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B | hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),B),A))),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B)). [resolve(1371,a,1375,a)]. 5.64/5.79 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)). [resolve(1371,a,1376,a)]. 5.64/5.79 Derived: c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)) = c_Groups_Oone__class_Oone(tc_Complex_Ocomplex). [resolve(1371,a,1377,a)]. 5.64/5.79 1378 -class_Rings_Odivision__ring(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Rings_Oinverse__class_Oinverse(A,B)),B) # label(fact_left__inverse) # label(axiom). [clausify(721)]. 5.64/5.79 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)),A). [resolve(1378,a,1371,a)]. 5.64/5.79 1379 -class_Rings_Odivision__ring(A) | c_Groups_Oone__class_Oone(A) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) | c_Rings_Oinverse__class_Oinverse(A,B) = C # label(fact_inverse__unique) # label(axiom). [clausify(794)]. 5.64/5.82 Derived: c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B) | c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A) = B. [resolve(1379,a,1371,a)]. 5.64/5.82 1380 class_Rings_Odivision__ring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Odivision__ring) # label(axiom). [assumption]. 5.64/5.82 Derived: c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A) != c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B | A = B. [resolve(1380,a,1372,a)]. 5.64/5.82 Derived: c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1380,a,1373,a)]. 5.64/5.82 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),B). [resolve(1380,a,1374,a)]. 5.64/5.82 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B | hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),B),A))),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)). [resolve(1380,a,1375,a)]. 5.64/5.82 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)). [resolve(1380,a,1376,a)]. 5.64/5.82 Derived: c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal). [resolve(1380,a,1377,a)]. 5.64/5.82 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),A). [resolve(1380,a,1378,a)]. 5.64/5.82 Derived: c_Groups_Oone__class_Oone(tc_RealDef_Oreal) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B) | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A) = B. [resolve(1380,a,1379,a)]. 5.64/5.82 1381 -class_Rings_Odivision__ring(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Rings_Oinverse__class_Oinverse(A,B) != c_Groups_Ozero__class_Ozero(A) # label(fact_nonzero__imp__inverse__nonzero) # label(axiom). [clausify(961)]. 5.64/5.82 1382 -class_Rings_Odivision__ring(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Rings_Oinverse__class_Oinverse(A,c_Rings_Oinverse__class_Oinverse(A,B)) = B # label(fact_nonzero__inverse__inverse__eq) # label(axiom). [clausify(1003)]. 5.64/5.82 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)) = A. [resolve(1382,a,1371,a)]. 5.64/5.82 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)) = A. [resolve(1382,a,1380,a)]. 5.64/5.82 1383 -class_Rings_Odivision__ring(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Rings_Oinverse__class_Oinverse(A,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Rings_Oinverse__class_Oinverse(A,B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))),c_Rings_Oinverse__class_Oinverse(A,C)) # label(fact_division__ring__inverse__add) # label(axiom). [clausify(1004)]. 5.64/5.82 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B))),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B)). [resolve(1383,a,1371,a)]. 5.79/5.93 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B))),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)). [resolve(1383,a,1380,a)]. 5.79/5.93 1384 -class_Rings_Odivision__ring(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | c_Rings_Oinverse__class_Oinverse(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Rings_Oinverse__class_Oinverse(A,C)),c_Rings_Oinverse__class_Oinverse(A,B)) # label(fact_nonzero__inverse__mult__distrib) # label(axiom). [clausify(1039)]. 5.79/5.93 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B | c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B)),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)). [resolve(1384,a,1371,a)]. 5.79/5.93 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)). [resolve(1384,a,1380,a)]. 5.79/5.93 1385 class_Fields_Ofield(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Fields_Ofield) # label(axiom). [assumption]. 5.79/5.93 1386 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | -c_Polynomial_Opdivmod__rel(A,B,C,F,V6) | E = V6 # label(fact_pdivmod__rel__unique__mod) # label(axiom). [clausify(100)]. 5.79/5.93 1387 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Polynomial_Opdivmod__rel(A,c_Polynomial_Osmult(A,F,B),C,c_Polynomial_Osmult(A,F,D),c_Polynomial_Osmult(A,F,E)) # label(fact_pdivmod__rel__smult__left) # label(axiom). [clausify(130)]. 5.79/5.93 1388 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Rings_Oinverse__class_Oinverse(A,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Rings_Oinverse__class_Oinverse(A,B))),c_Rings_Oinverse__class_Oinverse(A,C)) # label(fact_inverse__add) # label(axiom). [clausify(153)]. 5.79/5.93 1389 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,C) = D # label(fact_div__poly__eq) # label(axiom). [clausify(168)]. 5.79/5.93 1390 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | -c_Polynomial_Opdivmod__rel(A,D,F,V6,V7) | c_Polynomial_Opdivmod__rel(A,B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),F),V6,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),V7)),E)) # label(fact_pdivmod__rel__mult) # label(axiom). [clausify(177)]. 5.79/5.94 1391 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) # label(fact_pdivmod__rel__0) # label(axiom). [clausify(180)]. 5.79/5.94 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,E,F) | D = F. [resolve(1385,a,1386,a)]. 5.79/5.94 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,c_Polynomial_Osmult(tc_Complex_Ocomplex,E,A),B,c_Polynomial_Osmult(tc_Complex_Ocomplex,E,C),c_Polynomial_Osmult(tc_Complex_Ocomplex,E,D)). [resolve(1385,a,1387,a)]. 5.79/5.94 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B)),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A))),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B)). [resolve(1385,a,1388,a)]. 5.79/5.94 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B) = C. [resolve(1385,a,1389,a)]. 5.79/5.94 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,C,E,F,V6) | c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),E),F,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),V6)),D)). [resolve(1385,a,1390,a)]. 5.79/5.94 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex))). [resolve(1385,a,1391,a)]. 5.79/5.94 1392 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) = E # label(fact_mod__poly__eq) # label(axiom). [clausify(264)]. 5.79/5.94 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B) = D. [resolve(1392,a,1385,a)]. 5.79/5.94 1393 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Polynomial_Opoly__gcd(A,B,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B)) = c_Polynomial_Opoly__gcd(A,C,B) # label(fact_poly__gcd_Osimps_I2_J) # label(axiom). [clausify(278)]. 5.79/5.94 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A)) = c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,B,A). [resolve(1393,a,1385,a)]. 5.79/5.94 1394 -class_Fields_Ofield(A) | c_Polynomial_Osmult(A,B,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D)) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),c_Polynomial_Osmult(A,B,C),D) # label(fact_mod__smult__left) # label(axiom). [clausify(362)]. 5.79/5.94 Derived: c_Polynomial_Osmult(tc_Complex_Ocomplex,A,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,C)) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),c_Polynomial_Osmult(tc_Complex_Ocomplex,A,B),C). [resolve(1394,a,1385,a)]. 5.79/5.94 1395 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,B,C,c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,C),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)) # label(fact_pdivmod__rel) # label(axiom). [clausify(434)]. 5.79/5.94 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)). [resolve(1395,a,1385,a)]. 5.79/5.95 1396 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) # label(fact_poly__gcd__0__0) # label(axiom). [clausify(479)]. 5.79/5.95 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1396,a,1385,a)]. 5.79/5.95 1397 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C | hAPP(c_Polynomial_Ocoeff(A,c_Polynomial_Opoly__gcd(A,B,C)),c_Polynomial_Odegree(A,c_Polynomial_Opoly__gcd(A,B,C))) = c_Groups_Ozero__class_Ozero(A) # label(fact_poly__gcd__monic) # label(axiom). [clausify(490)]. 5.79/5.95 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != A | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != B | hAPP(c_Polynomial_Ocoeff(tc_Complex_Ocomplex,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,B)),c_Polynomial_Odegree(tc_Complex_Ocomplex,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,B))) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1397,a,1385,a)]. 5.79/5.95 1398 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Groups_Oone__class_Oone(A) = hAPP(c_Polynomial_Ocoeff(A,c_Polynomial_Opoly__gcd(A,C,B)),c_Polynomial_Odegree(A,c_Polynomial_Opoly__gcd(A,C,B))) # label(fact_poly__gcd__monic) # label(axiom). [clausify(490)]. 5.79/5.95 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = hAPP(c_Polynomial_Ocoeff(tc_Complex_Ocomplex,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,B,A)),c_Polynomial_Odegree(tc_Complex_Ocomplex,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,B,A))). [resolve(1398,a,1385,a)]. 5.79/5.95 1399 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Groups_Oone__class_Oone(A) = hAPP(c_Polynomial_Ocoeff(A,c_Polynomial_Opoly__gcd(A,B,C)),c_Polynomial_Odegree(A,c_Polynomial_Opoly__gcd(A,B,C))) # label(fact_poly__gcd__monic) # label(axiom). [clausify(490)]. 5.79/5.95 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = hAPP(c_Polynomial_Ocoeff(tc_Complex_Ocomplex,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,B)),c_Polynomial_Odegree(tc_Complex_Ocomplex,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,B))). [resolve(1399,a,1385,a)]. 5.79/5.95 1400 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(A,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B))),c_Polynomial_Odegree(A,B))) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) # label(fact_degree__mod__less) # label(axiom). [clausify(498)]. 5.79/5.95 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_Complex_Ocomplex,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A))),c_Polynomial_Odegree(tc_Complex_Ocomplex,A))) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1400,a,1385,a)]. 5.79/5.95 1401 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | -c_Polynomial_Opdivmod__rel(A,B,C,F,V6) | D = F # label(fact_pdivmod__rel__unique) # label(axiom). [clausify(517)]. 5.79/5.95 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,E,F) | C = E. [resolve(1401,a,1385,a)]. 5.79/5.95 1402 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | -c_Polynomial_Opdivmod__rel(A,B,C,F,V6) | E = V6 # label(fact_pdivmod__rel__unique) # label(axiom). [clausify(517)]. 5.79/5.97 1403 -class_Fields_Ofield(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(A,B)),c_Polynomial_Odegree(A,C))) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) = B # label(fact_mod__poly__less) # label(axiom). [clausify(530)]. 5.79/5.97 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_Complex_Ocomplex,A)),c_Polynomial_Odegree(tc_Complex_Ocomplex,B))) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B) = A. [resolve(1403,a,1385,a)]. 5.79/5.97 1404 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)),B) = c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)) # label(fact_poly__gcd__1__left) # label(axiom). [clausify(535)]. 5.79/5.97 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A) = c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1404,a,1385,a)]. 5.79/5.97 1405 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Polynomial_Opoly__gcd(A,C,B) = c_Polynomial_Osmult(A,c_Rings_Oinverse__class_Oinverse(A,hAPP(c_Polynomial_Ocoeff(A,C),c_Polynomial_Odegree(A,C))),C) # label(fact_poly__gcd__code) # label(axiom). [clausify(562)]. 5.79/5.97 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != A | c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,B,A) = c_Polynomial_Osmult(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(c_Polynomial_Ocoeff(tc_Complex_Ocomplex,B),c_Polynomial_Odegree(tc_Complex_Ocomplex,B))),B). [resolve(1405,a,1385,a)]. 5.79/5.97 1406 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Polynomial_Opoly__gcd(A,B,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B)) = c_Polynomial_Opoly__gcd(A,C,B) # label(fact_poly__gcd__code) # label(axiom). [clausify(562)]. 5.79/5.97 1407 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,c_Polynomial_Opoly__gcd(A,B,C),D) = c_Polynomial_Opoly__gcd(A,B,c_Polynomial_Opoly__gcd(A,C,D)) # label(fact_poly__gcd_Oassoc) # label(axiom). [clausify(589)]. 5.79/5.97 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,B),C) = c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,B,C)). [resolve(1407,a,1385,a)]. 5.79/5.97 1408 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) = c_Polynomial_Osmult(A,c_Rings_Oinverse__class_Oinverse(A,hAPP(c_Polynomial_Ocoeff(A,B),c_Polynomial_Odegree(A,B))),B) # label(fact_poly__gcd_Osimps_I1_J) # label(axiom). [clausify(647)]. 5.79/5.97 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex))) = c_Polynomial_Osmult(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(c_Polynomial_Ocoeff(tc_Complex_Ocomplex,A),c_Polynomial_Odegree(tc_Complex_Ocomplex,A))),A). [resolve(1408,a,1385,a)]. 5.79/5.97 1409 -class_Fields_Ofield(A) | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),c_Polynomial_Osmult(A,B,C),D) = c_Polynomial_Osmult(A,B,c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,D)) # label(fact_div__smult__left) # label(axiom). [clausify(685)]. 5.79/5.97 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),c_Polynomial_Osmult(tc_Complex_Ocomplex,A,B),C) = c_Polynomial_Osmult(tc_Complex_Ocomplex,A,c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,C)). [resolve(1409,a,1385,a)]. 5.79/5.97 1410 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Rings_Oinverse__class_Oinverse(A,B)),B) # label(fact_field__inverse) # label(axiom). [clausify(714)]. 5.79/5.97 1411 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,B,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) = c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)) # label(fact_poly__gcd__1__right) # label(axiom). [clausify(720)]. 5.79/5.98 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_Complex_Ocomplex))) = c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1411,a,1385,a)]. 5.79/5.98 1412 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,c_Polynomial_Osmult(A,B,D)) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D) # label(fact_mod__smult__right) # label(axiom). [clausify(724)]. 5.79/5.98 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,c_Polynomial_Osmult(tc_Complex_Ocomplex,A,C)) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,C). [resolve(1412,a,1385,a)]. 5.79/5.98 1413 -class_Fields_Ofield(A) | c_Groups_Ozero__class_Ozero(A) = B | -c_Polynomial_Opdivmod__rel(A,C,D,E,F) | c_Polynomial_Opdivmod__rel(A,C,c_Polynomial_Osmult(A,B,D),c_Polynomial_Osmult(A,c_Rings_Oinverse__class_Oinverse(A,B),E),F) # label(fact_pdivmod__rel__smult__right) # label(axiom). [clausify(725)]. 5.79/5.98 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,B,C,D,E) | c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,B,c_Polynomial_Osmult(tc_Complex_Ocomplex,A,C),c_Polynomial_Osmult(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A),D),E). [resolve(1413,a,1385,a)]. 5.79/5.98 1414 -class_Fields_Ofield(A) | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,C),D) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)) # label(fact_poly__div__mult__right) # label(axiom). [clausify(761)]. 5.79/5.98 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B),C) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),C)). [resolve(1414,a,1385,a)]. 5.79/5.98 1415 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | -c_Polynomial_Opdivmod__rel(A,B,C,F,V6) | D = F # label(fact_pdivmod__rel__unique__div) # label(axiom). [clausify(775)]. 5.79/5.98 1416 class_Fields_Ofield(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Fields_Ofield) # label(axiom). [assumption]. 5.79/5.98 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,E,F) | D = F. [resolve(1416,a,1386,a)]. 5.79/5.98 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,c_Polynomial_Osmult(tc_RealDef_Oreal,E,A),B,c_Polynomial_Osmult(tc_RealDef_Oreal,E,C),c_Polynomial_Osmult(tc_RealDef_Oreal,E,D)). [resolve(1416,a,1387,a)]. 5.79/5.98 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)). [resolve(1416,a,1388,a)]. 5.79/5.98 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B) = C. [resolve(1416,a,1389,a)]. 5.79/5.98 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,C,E,F,V6) | c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),E),F,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),V6)),D)). [resolve(1416,a,1390,a)]. 5.79/5.99 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal))). [resolve(1416,a,1391,a)]. 5.79/5.99 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B) = D. [resolve(1416,a,1392,a)]. 5.79/5.99 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A)) = c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,B,A). [resolve(1416,a,1393,a)]. 5.79/5.99 Derived: c_Polynomial_Osmult(tc_RealDef_Oreal,A,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,C)) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),c_Polynomial_Osmult(tc_RealDef_Oreal,A,B),C). [resolve(1416,a,1394,a)]. 5.79/5.99 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)). [resolve(1416,a,1395,a)]. 5.79/5.99 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1416,a,1396,a)]. 5.79/5.99 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != A | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != B | hAPP(c_Polynomial_Ocoeff(tc_RealDef_Oreal,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,B)),c_Polynomial_Odegree(tc_RealDef_Oreal,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,B))) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1416,a,1397,a)]. 5.79/5.99 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(c_Polynomial_Ocoeff(tc_RealDef_Oreal,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,B,A)),c_Polynomial_Odegree(tc_RealDef_Oreal,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,B,A))). [resolve(1416,a,1398,a)]. 5.79/5.99 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(c_Polynomial_Ocoeff(tc_RealDef_Oreal,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,B)),c_Polynomial_Odegree(tc_RealDef_Oreal,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,B))). [resolve(1416,a,1399,a)]. 5.79/5.99 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_RealDef_Oreal,c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A))),c_Polynomial_Odegree(tc_RealDef_Oreal,A))) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1416,a,1400,a)]. 5.79/5.99 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,E,F) | C = E. [resolve(1416,a,1401,a)]. 5.79/5.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_RealDef_Oreal,A)),c_Polynomial_Odegree(tc_RealDef_Oreal,B))) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B) = A. [resolve(1416,a,1403,a)]. 5.79/5.99 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A) = c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1416,a,1404,a)]. 5.79/5.99 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != A | c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,B,A) = c_Polynomial_Osmult(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,hAPP(c_Polynomial_Ocoeff(tc_RealDef_Oreal,B),c_Polynomial_Odegree(tc_RealDef_Oreal,B))),B). [resolve(1416,a,1405,a)]. 5.79/5.99 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,B),C) = c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,B,C)). [resolve(1416,a,1407,a)]. 5.79/5.99 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal))) = c_Polynomial_Osmult(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,hAPP(c_Polynomial_Ocoeff(tc_RealDef_Oreal,A),c_Polynomial_Odegree(tc_RealDef_Oreal,A))),A). [resolve(1416,a,1408,a)]. 5.79/5.99 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),c_Polynomial_Osmult(tc_RealDef_Oreal,A,B),C) = c_Polynomial_Osmult(tc_RealDef_Oreal,A,c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,C)). [resolve(1416,a,1409,a)]. 5.79/5.99 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_RealDef_Oreal))) = c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1416,a,1411,a)]. 5.79/5.99 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,c_Polynomial_Osmult(tc_RealDef_Oreal,A,C)) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,C). [resolve(1416,a,1412,a)]. 5.79/5.99 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,B,C,D,E) | c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,B,c_Polynomial_Osmult(tc_RealDef_Oreal,A,C),c_Polynomial_Osmult(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A),D),E). [resolve(1416,a,1413,a)]. 5.79/5.99 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B),C) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),C)). [resolve(1416,a,1414,a)]. 5.79/5.99 1417 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),C,D) | D = B # label(fact_pdivmod__rel__by__0__iff) # label(axiom). [clausify(830)]. 5.79/5.99 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B,C) | C = A. [resolve(1417,a,1385,a)]. 5.79/5.99 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B,C) | C = A. [resolve(1417,a,1416,a)]. 5.79/5.99 1418 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = C # label(fact_pdivmod__rel__by__0__iff) # label(axiom). [clausify(830)]. 5.79/5.99 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B,C) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = B. [resolve(1418,a,1385,a)]. 5.79/5.99 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B,C) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = B. [resolve(1418,a,1416,a)]. 5.79/5.99 1419 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),C,D) | D != B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C # label(fact_pdivmod__rel__by__0__iff) # label(axiom). [clausify(830)]. 5.79/5.99 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B,C) | C != A | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != B. [resolve(1419,a,1385,a)]. 5.79/5.99 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B,C) | C != A | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != B. [resolve(1419,a,1416,a)]. 5.79/5.99 1420 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = D # label(fact_pdivmod__rel__0__iff) # label(axiom). [clausify(832)]. 5.79/6.00 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A,B,C) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = C. [resolve(1420,a,1385,a)]. 5.79/6.00 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A,B,C) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = C. [resolve(1420,a,1416,a)]. 5.79/6.00 1421 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = C # label(fact_pdivmod__rel__0__iff) # label(axiom). [clausify(832)]. 5.79/6.00 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A,B,C) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = B. [resolve(1421,a,1385,a)]. 5.79/6.00 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A,B,C) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = B. [resolve(1421,a,1416,a)]. 5.79/6.00 1422 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != D | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C # label(fact_pdivmod__rel__0__iff) # label(axiom). [clausify(832)]. 5.79/6.00 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A,B,C) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != B. [resolve(1422,a,1385,a)]. 5.79/6.00 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A,B,C) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != B. [resolve(1422,a,1416,a)]. 5.79/6.00 1423 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,B,C) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B # label(fact_poly__gcd__zero__iff) # label(axiom). [clausify(876)]. 5.79/6.00 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,B) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A. [resolve(1423,a,1385,a)]. 5.79/6.00 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,B) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A. [resolve(1423,a,1416,a)]. 5.79/6.00 1424 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,B,C) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = C # label(fact_poly__gcd__zero__iff) # label(axiom). [clausify(876)]. 5.79/6.00 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,B) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = B. [resolve(1424,a,1385,a)]. 5.79/6.00 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,B) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = B. [resolve(1424,a,1416,a)]. 5.79/6.00 1425 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,B,C) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C # label(fact_poly__gcd__zero__iff) # label(axiom). [clausify(876)]. 5.79/6.00 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != A | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != B. [resolve(1425,a,1385,a)]. 5.87/6.02 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != A | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != B. [resolve(1425,a,1416,a)]. 5.87/6.02 1426 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,B,C) = c_Polynomial_Opoly__gcd(A,C,B) # label(fact_poly__gcd_Ocommute) # label(axiom). [clausify(924)]. 5.87/6.02 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,B) = c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,B,A). [resolve(1426,a,1385,a)]. 5.87/6.02 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,B) = c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,B,A). [resolve(1426,a,1416,a)]. 5.87/6.02 1427 -class_Fields_Ofield(A) | c_Polynomial_Opoly__gcd(A,B,c_Polynomial_Opoly__gcd(A,C,D)) = c_Polynomial_Opoly__gcd(A,C,c_Polynomial_Opoly__gcd(A,B,D)) # label(fact_poly__gcd_Oleft__commute) # label(axiom). [clausify(947)]. 5.87/6.02 Derived: c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,B,C)) = c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,B,c_Polynomial_Opoly__gcd(tc_Complex_Ocomplex,A,C)). [resolve(1427,a,1385,a)]. 5.87/6.02 Derived: c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,B,C)) = c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,B,c_Polynomial_Opoly__gcd(tc_RealDef_Oreal,A,C)). [resolve(1427,a,1416,a)]. 5.87/6.02 1428 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),B) # label(fact_pdivmod__rel__by__0) # label(axiom). [clausify(1021)]. 5.87/6.02 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A). [resolve(1428,a,1385,a)]. 5.87/6.02 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A). [resolve(1428,a,1416,a)]. 5.87/6.02 1429 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = D # label(fact_pdivmod__rel__def) # label(axiom). [clausify(1030)]. 5.87/6.02 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = C. [resolve(1429,a,1385,a)]. 5.87/6.02 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = C. [resolve(1429,a,1416,a)]. 5.87/6.02 1430 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(A,E)),c_Polynomial_Odegree(A,C))) # label(fact_pdivmod__rel__def) # label(axiom). [clausify(1030)]. 5.87/6.02 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = D | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_Complex_Ocomplex,D)),c_Polynomial_Odegree(tc_Complex_Ocomplex,B))). [resolve(1430,a,1385,a)]. 5.87/6.02 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = D | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_RealDef_Oreal,D)),c_Polynomial_Odegree(tc_RealDef_Oreal,B))). [resolve(1430,a,1416,a)]. 5.87/6.02 1431 -class_Fields_Ofield(A) | -c_Polynomial_Opdivmod__rel(A,B,C,D,E) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),E) = B # label(fact_pdivmod__rel__def) # label(axiom). [clausify(1030)]. 5.87/6.03 Derived: -c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),B)),D) = A. [resolve(1431,a,1385,a)]. 5.87/6.03 Derived: -c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),B)),D) = A. [resolve(1431,a,1416,a)]. 5.87/6.03 1432 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),E) != B # label(fact_pdivmod__rel__def) # label(axiom). [clausify(1030)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),B)),D) != A. [resolve(1432,a,1385,a)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),B)),D) != A. [resolve(1432,a,1416,a)]. 5.87/6.03 1433 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(A,E)),c_Polynomial_Odegree(A,C))) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),E) != B # label(fact_pdivmod__rel__def) # label(axiom). [clausify(1030)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_Complex_Ocomplex,D)),c_Polynomial_Odegree(tc_Complex_Ocomplex,B))) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),B)),D) != A. [resolve(1433,a,1385,a)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_RealDef_Oreal,D)),c_Polynomial_Odegree(tc_RealDef_Oreal,B))) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),B)),D) != A. [resolve(1433,a,1416,a)]. 5.87/6.03 1434 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != D | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),E) != B # label(fact_pdivmod__rel__def) # label(axiom). [clausify(1030)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),B)),D) != A. [resolve(1434,a,1385,a)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),B)),D) != A. [resolve(1434,a,1416,a)]. 5.87/6.03 1435 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != D | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),E) != B # label(fact_pdivmod__rel__def) # label(axiom). [clausify(1030)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),B)),D) != A. [resolve(1435,a,1385,a)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != C | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),B)),D) != A. [resolve(1435,a,1416,a)]. 5.87/6.03 1436 -class_Fields_Ofield(A) | c_Polynomial_Opdivmod__rel(A,B,C,D,E) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != D | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(A,E)),c_Polynomial_Odegree(A,C))) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),E) != B # label(fact_pdivmod__rel__def) # label(axiom). [clausify(1030)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_Complex_Ocomplex,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_Complex_Ocomplex,D)),c_Polynomial_Odegree(tc_Complex_Ocomplex,B))) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),B)),D) != A. [resolve(1436,a,1385,a)]. 5.87/6.03 Derived: c_Polynomial_Opdivmod__rel(tc_RealDef_Oreal,A,B,C,D) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Polynomial_Odegree(tc_RealDef_Oreal,D)),c_Polynomial_Odegree(tc_RealDef_Oreal,B))) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),B)),D) != A. [resolve(1436,a,1416,a)]. 5.87/6.03 1437 -class_Fields_Ofield(A) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),E,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),F,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),E),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),F),C). [resolve(1042,b,1043,a)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),C,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),D,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),E,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),D),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),E),B). [resolve(1437,a,1385,a)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),C,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),D,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),E,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),D),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),E),B). [resolve(1437,a,1416,a)]. 5.87/6.03 1438 -class_Fields_Ofield(A) | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)). [resolve(1042,b,1044,a)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1438,a,1385,a)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1438,a,1416,a)]. 5.87/6.03 1439 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1045,a,1042,b)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1439,b,1385,a)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1439,b,1416,a)]. 5.87/6.03 1440 c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1046,a,1042,b)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1440,b,1385,a)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1440,b,1416,a)]. 5.87/6.03 1441 c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) = B | -class_Fields_Ofield(A). [resolve(1047,a,1042,b)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_Complex_Ocomplex))) = A. [resolve(1441,b,1385,a)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_RealDef_Oreal))) = A. [resolve(1441,b,1416,a)]. 5.87/6.03 1442 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1048,a,1042,b)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1442,b,1385,a)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1442,b,1416,a)]. 5.87/6.03 1443 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),D),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1049,a,1042,b)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C),B). [resolve(1443,b,1385,a)]. 5.87/6.03 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C),B). [resolve(1443,b,1416,a)]. 5.87/6.03 1444 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,D) | -class_Fields_Ofield(A). [resolve(1050,a,1042,b)]. 5.87/6.03 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),A)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,C). [resolve(1444,c,1385,a)]. 5.87/6.03 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),A)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,C). [resolve(1444,c,1416,a)]. 5.87/6.03 1445 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),B),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) | -class_Fields_Ofield(A). [resolve(1051,a,1042,b)]. 5.87/6.03 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),A),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_Complex_Ocomplex))). [resolve(1445,c,1385,a)]. 5.87/6.03 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),A),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_RealDef_Oreal))). [resolve(1445,c,1416,a)]. 5.87/6.04 1446 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1052,a,1042,b)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),C,B)),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C),B). [resolve(1446,b,1385,a)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),C,B)),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C),B). [resolve(1446,b,1416,a)]. 5.87/6.04 1447 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) | -class_Fields_Ofield(A). [resolve(1053,a,1042,b)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B). [resolve(1447,b,1385,a)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B). [resolve(1447,b,1416,a)]. 5.87/6.04 1448 c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C),C) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1054,a,1042,b)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1448,b,1385,a)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1448,b,1416,a)]. 5.87/6.04 1449 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),E,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),F,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),E),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),F),C) | -class_Fields_Ofield(A). [resolve(1056,a,1042,b)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),C,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),D,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),E,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),D),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),E),B). [resolve(1449,d,1385,a)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),C,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),D,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),E,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),D),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),E),B). [resolve(1449,d,1416,a)]. 5.87/6.04 1450 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1057,a,1042,b)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1450,b,1385,a)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,A) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1450,b,1416,a)]. 5.87/6.04 1451 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1058,a,1042,b)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_Complex_Ocomplex))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1451,b,1385,a)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_RealDef_Oreal))) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1451,b,1416,a)]. 5.87/6.04 1452 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1059,a,1042,b)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),C,B)),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C),B). [resolve(1452,b,1385,a)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),C,B)),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C),B). [resolve(1452,b,1416,a)]. 5.87/6.04 1453 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B),B) = C | -class_Fields_Ofield(A). [resolve(1060,a,1042,b)]. 5.87/6.04 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),A),A) = B. [resolve(1453,c,1385,a)]. 5.87/6.04 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),A),A) = B. [resolve(1453,c,1416,a)]. 5.87/6.04 1454 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),D),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1061,a,1042,b)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C),B). [resolve(1454,b,1385,a)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C),B). [resolve(1454,b,1416,a)]. 5.87/6.04 1455 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B))),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B)) = C | -class_Fields_Ofield(A). [resolve(1062,a,1042,b)]. 5.87/6.04 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A))),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A)) = B. [resolve(1455,b,1385,a)]. 5.87/6.04 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A))),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A)) = B. [resolve(1455,b,1416,a)]. 5.87/6.04 1456 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B))),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),D) | -class_Fields_Ofield(A). [resolve(1063,a,1042,b)]. 5.87/6.04 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A))),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),C). [resolve(1456,b,1385,a)]. 5.87/6.04 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A))),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),C). [resolve(1456,b,1416,a)]. 5.87/6.04 1457 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) | -class_Fields_Ofield(A). [resolve(1064,a,1042,b)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B). [resolve(1457,b,1385,a)]. 5.87/6.04 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B). [resolve(1457,b,1416,a)]. 5.87/6.05 1458 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B)) | -class_Fields_Ofield(A). [resolve(1065,a,1042,b)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),A)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A)). [resolve(1458,c,1385,a)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),A)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A)). [resolve(1458,c,1416,a)]. 5.87/6.05 1459 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) = B | -class_Fields_Ofield(A). [resolve(1067,a,1042,b)]. 5.87/6.05 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex))) = A. [resolve(1459,b,1385,a)]. 5.87/6.05 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal))) = A. [resolve(1459,b,1416,a)]. 5.87/6.05 1460 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),C) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1068,a,1042,b)]. 5.87/6.05 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1460,b,1385,a)]. 5.87/6.05 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1460,b,1416,a)]. 5.87/6.05 1461 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B)) | -class_Fields_Ofield(A). [resolve(1069,a,1042,b)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A)). [resolve(1461,c,1385,a)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A)). [resolve(1461,c,1416,a)]. 5.87/6.05 1462 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,C)),C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D) | -class_Fields_Ofield(A). [resolve(1070,a,1042,b)]. 5.87/6.05 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C). [resolve(1462,b,1385,a)]. 5.87/6.05 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C). [resolve(1462,b,1416,a)]. 5.87/6.05 1463 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,D) | -class_Fields_Ofield(A). [resolve(1071,a,1042,b)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,C). [resolve(1463,c,1385,a)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,C). [resolve(1463,c,1416,a)]. 5.87/6.05 1464 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),B) = C | -class_Fields_Ofield(A). [resolve(1072,a,1042,b)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),A) = B. [resolve(1464,c,1385,a)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),A) = B. [resolve(1464,c,1416,a)]. 5.87/6.05 1465 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,D)),C) | -class_Fields_Ofield(A). [resolve(1073,a,1042,b)]. 5.87/6.05 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,C)),B). [resolve(1465,b,1385,a)]. 5.87/6.05 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,C)),B). [resolve(1465,b,1416,a)]. 5.87/6.05 1466 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) | -class_Fields_Ofield(A). [resolve(1075,a,1042,b)]. 5.87/6.05 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),C)),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B). [resolve(1466,b,1385,a)]. 5.87/6.05 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),C)),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B). [resolve(1466,b,1416,a)]. 5.87/6.05 1467 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,B) | -class_Fields_Ofield(A). [resolve(1076,a,1042,b)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,A). [resolve(1467,c,1385,a)]. 5.87/6.05 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,A). [resolve(1467,c,1416,a)]. 5.87/6.05 1468 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,C)),C)) = B | -class_Fields_Ofield(A). [resolve(1077,a,1042,b)]. 5.87/6.05 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),B)) = A. [resolve(1468,b,1385,a)]. 5.87/6.05 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),B)) = A. [resolve(1468,b,1416,a)]. 5.87/6.05 1469 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D)),D) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C),D) | -class_Fields_Ofield(A). [resolve(1078,a,1042,b)]. 5.87/6.05 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),C). [resolve(1469,b,1385,a)]. 5.87/6.06 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),C). [resolve(1469,b,1416,a)]. 5.87/6.06 1470 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),C,B)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) | -class_Fields_Ofield(A). [resolve(1079,a,1042,b)]. 5.87/6.06 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_Complex_Ocomplex))). [resolve(1470,c,1385,a)]. 5.87/6.06 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) = A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(tc_RealDef_Oreal))). [resolve(1470,c,1416,a)]. 5.87/6.06 1471 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D)),D) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),D) | -class_Fields_Ofield(A). [resolve(1082,a,1042,b)]. 5.87/6.06 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),C). [resolve(1471,b,1385,a)]. 5.87/6.06 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),C). [resolve(1471,b,1416,a)]. 5.87/6.06 1472 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D)) | -class_Fields_Ofield(A). [resolve(1083,a,1042,b)]. 5.87/6.06 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,C)). [resolve(1472,b,1385,a)]. 5.87/6.06 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,C)). [resolve(1472,b,1416,a)]. 5.87/6.06 1473 c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Fields_Ofield(A). [resolve(1084,a,1042,b)]. 5.87/6.06 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)) != A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C)) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_Complex_Ocomplex)). [resolve(1473,c,1385,a)]. 5.87/6.06 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)) != A | c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C)) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(tc_RealDef_Oreal)). [resolve(1473,c,1416,a)]. 5.87/6.06 1474 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,B) | -class_Fields_Ofield(A). [resolve(1086,a,1042,b)]. 5.87/6.06 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),A) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,A). [resolve(1474,b,1385,a)]. 5.87/6.06 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),A) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,A). [resolve(1474,b,1416,a)]. 5.87/6.06 1475 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(A),B,C)),C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)) = B | -class_Fields_Ofield(A). [resolve(1087,a,1042,b)]. 5.87/6.06 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)) = A. [resolve(1475,b,1385,a)]. 5.87/6.06 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Odiv(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)) = A. [resolve(1475,b,1416,a)]. 5.87/6.06 1476 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),D) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,D) | -class_Fields_Ofield(A). [resolve(1088,a,1042,b)]. 5.87/6.06 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),B),C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,C). [resolve(1476,b,1385,a)]. 5.87/6.06 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(tc_RealDef_Oreal)),B),C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,C). [resolve(1476,b,1416,a)]. 5.95/6.10 1477 -class_Fields_Ofield(A) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),E,C) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),F,C) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),E),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),D),F),C). [resolve(1241,b,1239,a)]. 5.95/6.10 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),C,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),D,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),E,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),D),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),C),E),B). [resolve(1477,a,1385,a)]. 5.95/6.10 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),C,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),D,B) != c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),E,B) | c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),D),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),C),E),B). [resolve(1477,a,1416,a)]. 5.95/6.10 1478 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),D,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1242,a,1241,b)]. 5.95/6.10 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),C,B)),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C),B). [resolve(1478,b,1385,a)]. 5.95/6.10 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),C,B)),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C),B). [resolve(1478,b,1416,a)]. 5.95/6.10 1479 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),C,D)),D) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C),D) | -class_Fields_Ofield(A). [resolve(1243,a,1241,b)]. 5.95/6.10 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),B,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),B),C). [resolve(1479,b,1385,a)]. 6.17/6.37 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),B,C)),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),B),C). [resolve(1479,b,1416,a)]. 6.17/6.37 1480 c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),B,C)),D),C) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),D),C) | -class_Fields_Ofield(A). [resolve(1244,a,1241,b)]. 6.17/6.37 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_Complex_Ocomplex)),A),C),B). [resolve(1480,b,1385,a)]. 6.17/6.37 Derived: c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Polynomial_Opoly(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(tc_RealDef_Oreal)),A),C),B). [resolve(1480,b,1416,a)]. 6.17/6.37 1481 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != c_Groups_Ozero__class_Ozero(A) # label(fact_field__power__not__zero) # label(axiom). [clausify(911)]. 6.17/6.37 1482 -class_Rings_Oidom(A) | class_Rings_Oring__1__no__zero__divisors(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Oring__1__no__zero__divisors) # label(axiom). [clausify(101)]. 6.17/6.37 1483 class_Rings_Oring__1__no__zero__divisors(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 6.17/6.37 1484 class_Rings_Oring__1__no__zero__divisors(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 6.17/6.37 1485 class_Rings_Oring__1__no__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 6.17/6.37 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Rings_Oidom(A). [resolve(1481,a,1482,b)]. 6.17/6.37 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1481,a,1483,a)]. 6.17/6.37 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1481,a,1484,a)]. 6.17/6.37 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1481,a,1485,a)]. 6.17/6.37 1486 -class_Groups_Oab__semigroup__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom). [clausify(139)]. 6.17/6.37 1487 -class_Rings_Ocomm__semiring__0(A) | class_Groups_Oab__semigroup__mult(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Oab__semigroup__mult) # label(axiom). [clausify(107)]. 6.47/6.62 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)) | -class_Rings_Ocomm__semiring__0(A). [resolve(1486,a,1487,b)]. 6.47/6.62 1488 class_Groups_Oab__semigroup__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 6.47/6.62 1489 class_Groups_Oab__semigroup__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 6.47/6.62 1490 class_Groups_Oab__semigroup__mult(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 6.47/6.62 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)). [resolve(1490,a,1486,a)]. 6.47/6.62 1491 class_Groups_Oab__semigroup__mult(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 6.47/6.62 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),C)). [resolve(1491,a,1486,a)]. 6.47/6.62 1492 class_Fields_Olinordered__field(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Fields_Olinordered__field) # label(axiom). [assumption]. 6.47/6.62 1493 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_positive__imp__inverse__positive) # label(axiom). [clausify(111)]. 6.47/6.62 1494 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Ozero__class_Ozero(A))) | c_Groups_Ozero__class_Ozero(A) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_inverse__negative__imp__negative) # label(axiom). [clausify(141)]. 6.47/6.62 1495 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Rings_Oinverse__class_Oinverse(A,C)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_le__imp__inverse__le) # label(axiom). [clausify(207)]. 6.47/6.62 1496 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_one__le__inverse) # label(axiom). [clausify(242)]. 6.47/6.62 1497 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Rings_Oinverse__class_Oinverse(A,C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_inverse__less__imp__less__neg) # label(axiom). [clausify(328)]. 6.47/6.62 1498 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_one__less__inverse) # label(axiom). [clausify(329)]. 6.47/6.62 1499 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_negative__imp__inverse__negative) # label(axiom). [clausify(408)]. 6.47/6.62 1500 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Rings_Oinverse__class_Oinverse(A,B)),c_Rings_Oinverse__class_Oinverse(A,C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_inverse__le__imp__le) # label(axiom). [clausify(428)]. 6.47/6.62 1501 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Rings_Oinverse__class_Oinverse(A,B)),c_Rings_Oinverse__class_Oinverse(A,C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_inverse__le__imp__le__neg) # label(axiom). [clausify(492)]. 6.47/6.62 1502 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,C)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_less__imp__inverse__less) # label(axiom). [clausify(548)]. 6.47/6.62 1503 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Rings_Oinverse__class_Oinverse(A,C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_inverse__less__imp__less) # label(axiom). [clausify(612)]. 6.47/6.62 1504 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Rings_Oinverse__class_Oinverse(A,C)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_le__imp__inverse__le__neg) # label(axiom). [clausify(697)]. 6.47/6.62 1505 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,C)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_less__imp__inverse__less__neg) # label(axiom). [clausify(738)]. 6.47/6.62 1506 -class_Fields_Olinordered__field(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Rings_Oinverse__class_Oinverse(A,B))) | c_Groups_Ozero__class_Ozero(A) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_inverse__positive__imp__positive) # label(axiom). [clausify(860)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1492,a,1493,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1492,a,1494,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1492,a,1495,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1492,a,1496,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),A)). [resolve(1492,a,1497,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1492,a,1498,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1492,a,1499,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),A)). [resolve(1492,a,1500,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),A)). [resolve(1492,a,1501,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1492,a,1502,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),A)). [resolve(1492,a,1503,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1492,a,1504,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1492,a,1505,a)]. 6.47/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)). [resolve(1492,a,1506,a)]. 6.55/6.76 1507 class_Fields_Olinordered__field__inverse__zero(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Fields_Olinordered__field__inverse__zero) # label(axiom). [assumption]. 6.55/6.76 1508 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),B)) # label(fact_inverse__le__1__iff) # label(axiom). [clausify(118)]. 6.55/6.76 1509 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Oone__class_Oone(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) # label(fact_inverse__le__1__iff) # label(axiom). [clausify(118)]. 6.55/6.76 1510 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Oone__class_Oone(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),B)) # label(fact_inverse__le__1__iff) # label(axiom). [clausify(118)]. 6.55/6.76 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)). [resolve(1507,a,1508,a)]. 6.55/6.76 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1507,a,1509,a)]. 6.55/6.76 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)). [resolve(1507,a,1510,a)]. 6.55/6.76 1511 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),f30(B,C,A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_field__le__mult__one__interval) # label(axiom). [clausify(364)]. 6.55/6.76 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),f30(A,B,tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),A)). [resolve(1511,a,1507,a)]. 6.55/6.76 1512 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f30(B,C,A)),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_field__le__mult__one__interval) # label(axiom). [clausify(364)]. 6.55/6.76 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),f30(A,B,tc_RealDef_Oreal)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),A)). [resolve(1512,a,1507,a)]. 6.55/6.76 1513 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),f30(B,C,A)),C)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_field__le__mult__one__interval) # label(axiom). [clausify(364)]. 6.55/6.76 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),f30(A,B,tc_RealDef_Oreal)),B)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),A)). [resolve(1513,a,1507,a)]. 6.55/6.77 1514 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_one__le__inverse__iff) # label(axiom). [clausify(400)]. 6.55/6.77 1515 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_one__le__inverse__iff) # label(axiom). [clausify(400)]. 6.55/6.77 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1515,a,1507,a)]. 6.55/6.77 1516 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Oone__class_Oone(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_one__le__inverse__iff) # label(axiom). [clausify(400)]. 6.55/6.77 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1516,a,1507,a)]. 6.55/6.77 1517 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) # label(fact_inverse__less__1__iff) # label(axiom). [clausify(478)]. 6.55/6.77 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1517,a,1507,a)]. 6.55/6.77 1518 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Oone__class_Oone(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) # label(fact_inverse__less__1__iff) # label(axiom). [clausify(478)]. 6.55/6.77 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A)). [resolve(1518,a,1507,a)]. 6.55/6.77 1519 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Oone__class_Oone(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) # label(fact_inverse__less__1__iff) # label(axiom). [clausify(478)]. 6.55/6.77 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1519,a,1507,a)]. 6.55/6.77 1520 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_inverse__negative__iff__negative) # label(axiom). [clausify(505)]. 6.66/6.79 1521 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_inverse__negative__iff__negative) # label(axiom). [clausify(505)]. 6.66/6.79 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1521,a,1507,a)]. 6.66/6.79 1522 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_inverse__positive__iff__positive) # label(axiom). [clausify(614)]. 6.66/6.79 1523 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_inverse__positive__iff__positive) # label(axiom). [clausify(614)]. 6.66/6.79 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1523,a,1507,a)]. 6.66/6.79 1524 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_inverse__nonnegative__iff__nonnegative) # label(axiom). [clausify(836)]. 6.66/6.79 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1524,a,1507,a)]. 6.66/6.79 1525 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),c_Rings_Oinverse__class_Oinverse(A,B))) # label(fact_inverse__nonnegative__iff__nonnegative) # label(axiom). [clausify(836)]. 6.66/6.79 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1525,a,1507,a)]. 6.66/6.79 1526 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),c_Rings_Oinverse__class_Oinverse(A,B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_one__less__inverse__iff) # label(axiom). [clausify(848)]. 6.66/6.79 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)). [resolve(1526,a,1507,a)]. 6.66/6.79 1527 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),c_Rings_Oinverse__class_Oinverse(A,B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Oone__class_Oone(A))) # label(fact_one__less__inverse__iff) # label(axiom). [clausify(848)]. 6.77/6.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))). [resolve(1527,a,1507,a)]. 6.77/6.96 1528 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),c_Rings_Oinverse__class_Oinverse(A,B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Oone__class_Oone(A))) # label(fact_one__less__inverse__iff) # label(axiom). [clausify(848)]. 6.77/6.96 1529 -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) # label(fact_inverse__nonpositive__iff__nonpositive) # label(axiom). [clausify(950)]. 6.77/6.96 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1529,a,1507,a)]. 6.77/6.96 1530 -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Rings_Oinverse__class_Oinverse(A,B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) # label(fact_inverse__nonpositive__iff__nonpositive) # label(axiom). [clausify(950)]. 6.77/6.96 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1530,a,1507,a)]. 6.77/6.96 1531 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__le__cancel__left__neg) # label(axiom). [clausify(164)]. 6.77/6.96 1532 class_Rings_Olinordered__ring__strict(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 6.77/6.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))). [resolve(1531,a,1532,a)]. 6.77/6.96 1533 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__le__cancel__left__neg) # label(axiom). [clausify(164)]. 6.77/6.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))). [resolve(1533,a,1532,a)]. 6.77/6.96 1534 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(337)]. 6.77/6.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))). [resolve(1534,a,1532,a)]. 6.77/6.96 1535 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(337)]. 6.77/6.96 1536 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(337)]. 6.77/6.96 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))). [resolve(1536,a,1532,a)]. 6.77/6.96 1537 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(337)]. 6.77/6.96 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))). [resolve(1537,a,1532,a)]. 6.77/6.96 1538 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(337)]. 6.77/6.96 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A))). [resolve(1538,a,1532,a)]. 6.77/6.96 1539 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(337)]. 6.77/6.96 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A))). [resolve(1539,a,1532,a)]. 6.77/6.96 1540 -class_Rings_Olinordered__idom(A) | class_Rings_Olinordered__ring__strict(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Olinordered__ring__strict) # label(axiom). [clausify(367)]. 6.77/6.96 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))). [resolve(1540,b,1531,a)]. 6.77/6.96 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))). [resolve(1540,b,1533,a)]. 6.77/6.96 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),D),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D))). [resolve(1540,b,1534,a)]. 6.77/6.96 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D))). [resolve(1540,b,1536,a)]. 6.77/6.96 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D))). [resolve(1540,b,1537,a)]. 6.77/6.96 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B))). [resolve(1540,b,1538,a)]. 6.77/6.96 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B))). [resolve(1540,b,1539,a)]. 6.77/6.97 1541 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__strict__right__mono__neg) # label(axiom). [clausify(391)]. 6.77/6.97 1542 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)))) # label(fact_sum__squares__gt__zero__iff) # label(axiom). [clausify(401)]. 6.77/6.97 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B)))). [resolve(1542,a,1532,a)]. 6.77/6.97 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C)))) | -class_Rings_Olinordered__idom(A). [resolve(1542,a,1540,b)]. 6.77/6.97 1543 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)))) # label(fact_sum__squares__gt__zero__iff) # label(axiom). [clausify(401)]. 6.77/6.97 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)))). [resolve(1543,a,1532,a)]. 6.77/6.97 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)))) | -class_Rings_Olinordered__idom(A). [resolve(1543,a,1540,b)]. 6.77/6.97 1544 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)))) # label(fact_sum__squares__gt__zero__iff) # label(axiom). [clausify(401)]. 6.77/6.97 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B)))). [resolve(1544,a,1532,a)]. 6.77/6.97 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C)))) | -class_Rings_Olinordered__idom(A). [resolve(1544,a,1540,b)]. 6.77/6.98 1545 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(454)]. 6.77/6.98 1546 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(454)]. 6.77/6.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A))). [resolve(1546,a,1532,a)]. 6.77/6.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),D),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B))) | -class_Rings_Olinordered__idom(A). [resolve(1546,a,1540,b)]. 6.77/6.98 1547 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(454)]. 6.77/6.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B))). [resolve(1547,a,1532,a)]. 6.77/6.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C))) | -class_Rings_Olinordered__idom(A). [resolve(1547,a,1540,b)]. 6.77/6.98 1548 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(454)]. 6.77/6.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B))). [resolve(1548,a,1532,a)]. 6.77/6.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),D),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C))) | -class_Rings_Olinordered__idom(A). [resolve(1548,a,1540,b)]. 6.86/6.98 1549 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(454)]. 6.86/6.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))). [resolve(1549,a,1532,a)]. 6.86/6.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1549,a,1540,b)]. 6.86/6.98 1550 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(454)]. 6.86/6.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))). [resolve(1550,a,1532,a)]. 6.86/6.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D))) | -class_Rings_Olinordered__idom(A). [resolve(1550,a,1540,b)]. 6.86/6.98 1551 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__squares__le__zero__iff) # label(axiom). [clausify(507)]. 6.86/6.98 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A))),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1551,a,1532,a)]. 6.86/6.99 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B))),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1551,a,1540,b)]. 6.86/6.99 1552 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__squares__le__zero__iff) # label(axiom). [clausify(507)]. 6.86/6.99 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A))),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1552,a,1532,a)]. 6.86/6.99 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B))),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1552,a,1540,b)]. 6.86/6.99 1553 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__squares__le__zero__iff) # label(axiom). [clausify(507)]. 6.86/6.99 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B))),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1553,a,1532,a)]. 6.86/6.99 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C))),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1553,a,1540,b)]. 6.86/6.99 1554 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_mult__le__0__iff) # label(axiom). [clausify(524)]. 6.86/6.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)). [resolve(1554,a,1532,a)]. 6.86/6.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -class_Rings_Olinordered__idom(A). [resolve(1554,a,1540,b)]. 6.86/6.99 1555 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(524)]. 6.86/6.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1555,a,1540,b)]. 6.86/6.99 1556 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_mult__le__0__iff) # label(axiom). [clausify(524)]. 6.86/6.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -class_Rings_Olinordered__idom(A). [resolve(1556,a,1540,b)]. 6.86/6.99 1557 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(524)]. 6.86/6.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1557,a,1532,a)]. 6.86/6.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1557,a,1540,b)]. 6.86/6.99 1558 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(524)]. 6.86/6.99 1559 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(524)]. 6.86/7.00 1560 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(543)]. 6.86/7.00 1561 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(543)]. 6.86/7.00 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))). [resolve(1561,a,1532,a)]. 6.86/7.00 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1561,a,1540,b)]. 6.86/7.00 1562 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(558)]. 6.86/7.00 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),C)). [resolve(1562,a,1532,a)]. 6.86/7.00 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),D)) | -class_Rings_Olinordered__idom(A). [resolve(1562,a,1540,b)]. 6.86/7.00 1563 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(558)]. 6.86/7.00 1564 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(610)]. 6.86/7.00 1565 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(610)]. 6.86/7.00 1566 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(610)]. 6.86/7.00 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B))) | -class_Rings_Olinordered__idom(A). [resolve(1566,a,1540,b)]. 6.86/7.00 1567 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(610)]. 6.86/7.00 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A))). [resolve(1567,a,1532,a)]. 6.86/7.00 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B))) | -class_Rings_Olinordered__idom(A). [resolve(1567,a,1540,b)]. 6.86/7.00 1568 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(610)]. 6.86/7.00 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))). [resolve(1568,a,1532,a)]. 6.86/7.00 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1568,a,1540,b)]. 6.86/7.01 1569 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(610)]. 6.86/7.01 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1569,a,1540,b)]. 6.86/7.01 1570 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)) = c_Groups_Ozero__class_Ozero(A) # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(631)]. 6.86/7.01 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1570,a,1532,a)]. 6.86/7.01 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Rings_Olinordered__idom(A). [resolve(1570,a,1540,b)]. 6.86/7.01 1571 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)) != c_Groups_Ozero__class_Ozero(A) # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(631)]. 6.86/7.01 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1571,a,1532,a)]. 6.86/7.01 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Rings_Olinordered__idom(A). [resolve(1571,a,1540,b)]. 6.86/7.01 1572 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)) != c_Groups_Ozero__class_Ozero(A) # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(631)]. 6.86/7.01 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),B)) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1572,a,1532,a)]. 6.86/7.01 Derived: c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),C)) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) | -class_Rings_Olinordered__idom(A). [resolve(1572,a,1540,b)]. 6.86/7.02 1573 class_Rings_Olinordered__ring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 6.86/7.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1573,a,1531,a)]. 6.86/7.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1573,a,1533,a)]. 6.86/7.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))). [resolve(1573,a,1534,a)]. 6.86/7.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))). [resolve(1573,a,1536,a)]. 6.86/7.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))). [resolve(1573,a,1537,a)]. 6.86/7.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(1573,a,1538,a)]. 6.86/7.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(1573,a,1539,a)]. 6.86/7.02 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)))). [resolve(1573,a,1542,a)]. 6.86/7.02 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)))). [resolve(1573,a,1543,a)]. 6.86/7.02 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)))). [resolve(1573,a,1544,a)]. 6.86/7.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(1573,a,1546,a)]. 6.86/7.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))). [resolve(1573,a,1547,a)]. 6.86/7.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))). [resolve(1573,a,1548,a)]. 6.86/7.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1573,a,1549,a)]. 6.86/7.02 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))). [resolve(1573,a,1550,a)]. 6.86/7.02 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1573,a,1551,a)]. 6.86/7.02 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1573,a,1552,a)]. 6.86/7.02 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1573,a,1553,a)]. 6.86/7.02 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(1573,a,1554,a)]. 6.86/7.02 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1573,a,1555,a)]. 6.86/7.03 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(1573,a,1556,a)]. 6.86/7.03 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1573,a,1557,a)]. 6.86/7.03 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1573,a,1561,a)]. 6.86/7.03 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)). [resolve(1573,a,1562,a)]. 6.86/7.03 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A))). [resolve(1573,a,1566,a)]. 6.86/7.03 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A))). [resolve(1573,a,1567,a)]. 6.86/7.03 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1573,a,1568,a)]. 6.86/7.03 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1573,a,1569,a)]. 6.86/7.03 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1573,a,1570,a)]. 6.86/7.03 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)) != c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1573,a,1571,a)]. 6.86/7.03 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)) != c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1573,a,1572,a)]. 6.86/7.05 1574 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__strict__left__mono__neg) # label(axiom). [clausify(886)]. 6.86/7.05 1575 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) # label(fact_mult__le__cancel__left__pos) # label(axiom). [clausify(920)]. 6.86/7.05 1576 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),D)) # label(fact_mult__le__cancel__left__pos) # label(axiom). [clausify(920)]. 6.86/7.05 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),C)). [resolve(1576,a,1532,a)]. 6.86/7.05 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),D)) | -class_Rings_Olinordered__idom(A). [resolve(1576,a,1540,b)]. 6.86/7.05 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),C)). [resolve(1576,a,1573,a)]. 6.86/7.05 1577 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__neg__neg) # label(axiom). [clausify(938)]. 6.86/7.05 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B))). [resolve(1577,a,1532,a)]. 6.86/7.05 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C))) | -class_Rings_Olinordered__idom(A). [resolve(1577,a,1540,b)]. 6.86/7.05 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(1577,a,1573,a)]. 7.00/7.21 1578 class_Orderings_Olinorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Olinorder) # label(axiom). [assumption]. 7.00/7.21 1579 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | B != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(121)]. 7.00/7.21 1580 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | B = C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(121)]. 7.00/7.21 1581 -class_Orderings_Olinorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_linorder__le__cases) # label(axiom). [clausify(124)]. 7.00/7.21 1582 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(172)]. 7.00/7.21 1583 -class_Orderings_Olinorder(A) | B != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(172)]. 7.00/7.21 1584 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | C = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(172)]. 7.00/7.21 1585 -class_Orderings_Olinorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_not__leE) # label(axiom). [clausify(273)]. 7.00/7.21 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | A = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(1578,a,1580,a)]. 7.00/7.21 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),A)). [resolve(1578,a,1581,a)]. 7.00/7.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),A)). [resolve(1578,a,1582,a)]. 7.00/7.21 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),A)). [resolve(1578,a,1585,a)]. 7.00/7.21 1586 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | B != C # label(fact_linorder__neq__iff) # label(axiom). [clausify(297)]. 7.00/7.21 1587 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | C != B # label(fact_linorder__neq__iff) # label(axiom). [clausify(297)]. 7.00/7.21 1588 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) | B = C # label(fact_linorder__neq__iff) # label(axiom). [clausify(297)]. 7.00/7.21 1589 -class_Orderings_Olinorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__le__less__linear) # label(axiom). [clausify(396)]. 7.00/7.21 1590 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | B = C # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(398)]. 7.00/7.21 1591 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | B != C # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(398)]. 7.00/7.21 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | A != B. [resolve(1591,a,1578,a)]. 7.09/7.24 1592 -class_Orderings_Olinorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_linorder__linear) # label(axiom). [clausify(429)]. 7.09/7.24 1593 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) | C = B # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(432)]. 7.09/7.24 1594 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) | C != B # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(432)]. 7.09/7.24 1595 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | C = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__cases) # label(axiom). [clausify(534)]. 7.09/7.24 1596 class_Orderings_Olinorder(tc_Int_Oint) # label(arity_Int__Oint__Orderings_Olinorder) # label(axiom). [assumption]. 7.09/7.24 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(1596,a,1579,a)]. 7.09/7.24 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | A = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(1596,a,1580,a)]. 7.09/7.24 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B),A)). [resolve(1596,a,1581,a)]. 7.09/7.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)). [resolve(1596,a,1582,a)]. 7.09/7.24 Derived: A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(1596,a,1583,a)]. 7.09/7.24 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | B = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)). [resolve(1596,a,1584,a)]. 7.09/7.24 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)). [resolve(1596,a,1585,a)]. 7.09/7.24 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | A != B. [resolve(1596,a,1591,a)]. 7.09/7.24 1597 -class_Orderings_Olinorder(A) | B = C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_linorder__neqE) # label(axiom). [clausify(682)]. 7.09/7.24 1598 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__not__less) # label(axiom). [clausify(703)]. 7.09/7.24 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),A)). [resolve(1598,a,1578,a)]. 7.09/7.24 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)). [resolve(1598,a,1596,a)]. 7.09/7.24 1599 -class_Orderings_Olinorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__not__less) # label(axiom). [clausify(703)]. 7.09/7.24 1600 class_Orderings_Olinorder(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Orderings_Olinorder) # label(axiom). [assumption]. 7.09/7.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),A)). [resolve(1600,a,1582,a)]. 7.09/7.24 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | B = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),A)). [resolve(1600,a,1584,a)]. 7.09/7.24 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),A)). [resolve(1600,a,1585,a)]. 7.09/7.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | B != A. [resolve(1600,a,1587,a)]. 7.31/7.43 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | A != B. [resolve(1600,a,1591,a)]. 7.31/7.43 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),B),A)). [resolve(1600,a,1598,a)]. 7.31/7.43 1601 -class_Orderings_Olinorder(A) | B = C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__less__linear) # label(axiom). [clausify(723)]. 7.31/7.43 1602 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_leI) # label(axiom). [clausify(850)]. 7.31/7.43 1603 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_leD) # label(axiom). [clausify(864)]. 7.31/7.43 1604 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_linorder__not__le) # label(axiom). [clausify(878)]. 7.31/7.43 1605 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,C),B)) # label(fact_linorder__not__le) # label(axiom). [clausify(878)]. 7.31/7.43 1606 -class_Rings_Olinordered__idom(A) | class_Orderings_Olinorder(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Orderings_Olinorder) # label(axiom). [clausify(979)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | B != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)). [resolve(1606,b,1579,a)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | B = C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)). [resolve(1606,b,1580,a)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),C),B)). [resolve(1606,b,1581,a)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),B)). [resolve(1606,b,1582,a)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | B != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)). [resolve(1606,b,1583,a)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | C = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),B)). [resolve(1606,b,1584,a)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),B)). [resolve(1606,b,1585,a)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | C != B. [resolve(1606,b,1587,a)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | B != C. [resolve(1606,b,1591,a)]. 7.31/7.43 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),C),B)). [resolve(1606,b,1598,a)]. 7.31/7.43 1607 -class_Rings_Olinordered__idom(A) | class_Rings_Olinordered__semiring__1(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Olinordered__semiring__1) # label(axiom). [clausify(185)]. 7.47/7.62 1608 -class_Rings_Olinordered__semiring__1(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,D),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A)),F)) | c_Groups_Oone__class_Oone(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),E),F) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),D))),C)) # label(fact_convex__bound__le) # label(axiom). [clausify(131)]. 7.47/7.62 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),D),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),E)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))),F)) | c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),E),F) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),E),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),F),D))),C)). [resolve(1607,b,1608,a)]. 7.47/7.62 1609 class_Rings_Olinordered__semiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 7.47/7.62 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),E)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),D),E) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),C))),B)). [resolve(1609,a,1608,a)]. 7.47/7.62 1610 class_Rings_Olinordered__semiring__1(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 7.47/7.62 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),D)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),E)) | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),D),E) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),E),C))),B)). [resolve(1610,a,1608,a)]. 7.47/7.62 1611 -class_Rings_Olinordered__idom(A) | class_Groups_Oordered__ab__group__add(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Oordered__ab__group__add) # label(axiom). [clausify(509)]. 7.47/7.62 1612 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(132)]. 7.47/7.62 1613 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(132)]. 7.47/7.64 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)). [resolve(1611,b,1612,a)]. 7.47/7.64 Derived: -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)). [resolve(1611,b,1613,a)]. 7.47/7.64 1614 class_Groups_Oordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 7.47/7.64 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)). [resolve(1614,a,1612,a)]. 7.47/7.64 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)). [resolve(1614,a,1613,a)]. 7.47/7.64 1615 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(815)]. 7.47/7.64 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1615,a,1611,b)]. 7.47/7.64 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1615,a,1614,a)]. 7.47/7.64 1616 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(815)]. 7.47/7.64 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),C)),c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)))) | -class_Rings_Olinordered__idom(A). [resolve(1616,a,1611,b)]. 7.47/7.64 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(1616,a,1614,a)]. 7.47/7.64 1617 class_Groups_Oordered__ab__group__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 7.47/7.64 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)). [resolve(1617,a,1612,a)]. 7.47/7.64 Derived: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)). [resolve(1617,a,1613,a)]. 7.47/7.64 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1617,a,1615,a)]. 7.70/7.83 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),B)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))). [resolve(1617,a,1616,a)]. 7.70/7.83 1618 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = B # label(fact_power__one__right) # label(axiom). [clausify(187)]. 7.70/7.83 1619 class_Groups_Omonoid__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Omonoid__mult) # label(axiom). [assumption]. 7.70/7.83 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1618,a,1619,a)]. 7.70/7.83 1620 class_Groups_Omonoid__mult(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Omonoid__mult) # label(axiom). [assumption]. 7.70/7.83 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1620,a,1618,a)]. 7.70/7.83 1621 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D)) # label(fact_power__add) # label(axiom). [clausify(407)]. 7.70/7.83 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),C)). [resolve(1621,a,1620,a)]. 7.70/7.83 1622 class_Groups_Omonoid__mult(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Omonoid__mult) # label(axiom). [assumption]. 7.70/7.83 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1622,a,1618,a)]. 7.70/7.83 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C)). [resolve(1622,a,1621,a)]. 7.70/7.83 1623 -class_Rings_Ocomm__semiring__1(A) | class_Groups_Omonoid__mult(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Omonoid__mult) # label(axiom). [clausify(616)]. 7.70/7.83 Derived: -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = B. [resolve(1623,b,1618,a)]. 7.70/7.83 Derived: -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D)). [resolve(1623,b,1621,a)]. 7.70/7.83 1624 class_Groups_Omonoid__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__mult) # label(axiom). [assumption]. 7.70/7.83 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1624,a,1618,a)]. 7.70/7.83 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C)). [resolve(1624,a,1621,a)]. 7.70/7.83 1625 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),D) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),D)) # label(fact_power__mult) # label(axiom). [clausify(820)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(1625,a,1619,a)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(1625,a,1620,a)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(1625,a,1622,a)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),D) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),D)) | -class_Rings_Ocomm__semiring__1(A). [resolve(1625,a,1623,b)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(1625,a,1624,a)]. 7.70/7.84 1626 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),B) # label(fact_power__Suc2) # label(axiom). [clausify(852)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),A). [resolve(1626,a,1619,a)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B)),A). [resolve(1626,a,1620,a)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),A). [resolve(1626,a,1622,a)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),hAPP(c_Nat_OSuc,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),B) | -class_Rings_Ocomm__semiring__1(A). [resolve(1626,a,1623,b)]. 7.70/7.84 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),A). [resolve(1626,a,1624,a)]. 7.70/7.84 1627 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oone__class_Oone(A)),B) = B # label(fact_mult__1__left) # label(axiom). [clausify(863)]. 7.70/7.84 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A) = A. [resolve(1627,a,1619,a)]. 7.70/7.84 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)),A) = A. [resolve(1627,a,1620,a)]. 7.70/7.84 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A) = A. [resolve(1627,a,1622,a)]. 7.70/7.84 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B) = B | -class_Rings_Ocomm__semiring__1(A). [resolve(1627,a,1623,b)]. 7.70/7.84 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A) = A. [resolve(1627,a,1624,a)]. 7.70/7.84 1628 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Oone__class_Oone(A)) = B # label(fact_mult__1__right) # label(axiom). [clausify(933)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A. [resolve(1628,a,1619,a)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)) = A. [resolve(1628,a,1620,a)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) = A. [resolve(1628,a,1622,a)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) = B | -class_Rings_Ocomm__semiring__1(A). [resolve(1628,a,1623,b)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1628,a,1624,a)]. 7.87/8.00 1629 -class_Groups_Omonoid__mult(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Oone__class_Oone(A)),B) # label(fact_power__one) # label(axiom). [clausify(1008)]. 7.87/8.00 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A). [resolve(1629,a,1619,a)]. 7.87/8.00 Derived: c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)),A). [resolve(1629,a,1620,a)]. 7.87/8.00 Derived: c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A). [resolve(1629,a,1622,a)]. 7.87/8.00 Derived: c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))),B) | -class_Rings_Ocomm__semiring__1(A). [resolve(1629,a,1623,b)]. 7.87/8.00 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A). [resolve(1629,a,1624,a)]. 7.87/8.00 1630 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_power__commutes) # label(axiom). [clausify(1014)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)). [resolve(1630,a,1619,a)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B)). [resolve(1630,a,1620,a)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)). [resolve(1630,a,1622,a)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),C)) | -class_Rings_Ocomm__semiring__1(A). [resolve(1630,a,1623,b)]. 7.87/8.00 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)). [resolve(1630,a,1624,a)]. 7.87/8.00 1631 -class_Int_Oring__char__0(A) | -class_Rings_Oidom(A) | B != C | c_Polynomial_Opoly(A,B) = c_Polynomial_Opoly(A,C) # label(fact_poly__eq__iff) # label(axiom). [clausify(288)]. 7.87/8.00 1632 -class_Rings_Olinordered__idom(A) | class_Int_Oring__char__0(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Int_Oring__char__0) # label(axiom). [clausify(169)]. 8.06/8.20 Derived: -class_Rings_Oidom(tc_Polynomial_Opoly(A)) | B != C | c_Polynomial_Opoly(tc_Polynomial_Opoly(A),B) = c_Polynomial_Opoly(tc_Polynomial_Opoly(A),C) | -class_Rings_Olinordered__idom(A). [resolve(1631,a,1632,b)]. 8.06/8.20 1633 -class_Int_Oring__char__0(A) | -class_Rings_Oidom(A) | B = C | c_Polynomial_Opoly(A,B) != c_Polynomial_Opoly(A,C) # label(fact_poly__eq__iff) # label(axiom). [clausify(288)]. 8.06/8.20 Derived: -class_Rings_Oidom(tc_Polynomial_Opoly(A)) | B = C | c_Polynomial_Opoly(tc_Polynomial_Opoly(A),B) != c_Polynomial_Opoly(tc_Polynomial_Opoly(A),C) | -class_Rings_Olinordered__idom(A). [resolve(1633,a,1632,b)]. 8.06/8.20 1634 -class_Rings_Oidom(A) | -class_Int_Oring__char__0(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | c_Polynomial_Opoly(A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) = c_Polynomial_Opoly(A,B) # label(fact_poly__zero) # label(axiom). [clausify(650)]. 8.06/8.20 1635 -class_Rings_Oidom(A) | -class_Int_Oring__char__0(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Polynomial_Opoly(A,c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A))) != c_Polynomial_Opoly(A,B) # label(fact_poly__zero) # label(axiom). [clausify(650)]. 8.06/8.20 1636 class_Int_Oring__char__0(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Int_Oring__char__0) # label(axiom). [assumption]. 8.06/8.20 Derived: -class_Rings_Oidom(tc_Complex_Ocomplex) | A != B | c_Polynomial_Opoly(tc_Complex_Ocomplex,A) = c_Polynomial_Opoly(tc_Complex_Ocomplex,B). [resolve(1636,a,1631,a)]. 8.06/8.20 Derived: -class_Rings_Oidom(tc_Complex_Ocomplex) | A = B | c_Polynomial_Opoly(tc_Complex_Ocomplex,A) != c_Polynomial_Opoly(tc_Complex_Ocomplex,B). [resolve(1636,a,1633,a)]. 8.06/8.20 1637 class_Int_Oring__char__0(tc_Int_Oint) # label(arity_Int__Oint__Int_Oring__char__0) # label(axiom). [assumption]. 8.06/8.20 Derived: -class_Rings_Oidom(tc_Int_Oint) | A != B | c_Polynomial_Opoly(tc_Int_Oint,A) = c_Polynomial_Opoly(tc_Int_Oint,B). [resolve(1637,a,1631,a)]. 8.06/8.20 Derived: -class_Rings_Oidom(tc_Int_Oint) | A = B | c_Polynomial_Opoly(tc_Int_Oint,A) != c_Polynomial_Opoly(tc_Int_Oint,B). [resolve(1637,a,1633,a)]. 8.06/8.20 1638 class_Int_Oring__char__0(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Int_Oring__char__0) # label(axiom). [assumption]. 8.06/8.20 Derived: -class_Rings_Oidom(tc_RealDef_Oreal) | A != B | c_Polynomial_Opoly(tc_RealDef_Oreal,A) = c_Polynomial_Opoly(tc_RealDef_Oreal,B). [resolve(1638,a,1631,a)]. 8.06/8.20 Derived: -class_Rings_Oidom(tc_RealDef_Oreal) | A = B | c_Polynomial_Opoly(tc_RealDef_Oreal,A) != c_Polynomial_Opoly(tc_RealDef_Oreal,B). [resolve(1638,a,1633,a)]. 8.06/8.20 1639 -class_RealVector_Oreal__normed__div__algebra(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B),c_RealVector_Onorm__class_Onorm(A,C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Rings_Oinverse__class_Oinverse(A,C))),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B))) # label(fact_Bseq__inverse__lemma) # label(axiom). [clausify(181)]. 8.06/8.20 1640 class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra) # label(axiom). [assumption]. 8.06/8.20 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B))),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1639,a,1640,a)]. 8.06/8.20 1641 -class_RealVector_Oreal__normed__div__algebra(A) | c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),c_RealVector_Onorm__class_Onorm(A,C)) # label(fact_norm__mult) # label(axiom). [clausify(190)]. 8.06/8.24 Derived: c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B)). [resolve(1641,a,1640,a)]. 8.06/8.24 1642 class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra) # label(axiom). [assumption]. 8.06/8.24 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B))),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A))). [resolve(1642,a,1639,a)]. 8.06/8.24 Derived: c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B)). [resolve(1642,a,1641,a)]. 8.06/8.24 1643 -class_RealVector_Oreal__normed__div__algebra(A) | c_RealVector_Onorm__class_Onorm(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(A,B)),C) # label(fact_norm__power) # label(axiom). [clausify(572)]. 8.06/8.24 Derived: c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)),B). [resolve(1643,a,1640,a)]. 8.06/8.24 Derived: c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)),B). [resolve(1643,a,1642,a)]. 8.06/8.24 1644 -class_Rings_Odivision__ring__inverse__zero(A) | -class_RealVector_Oreal__normed__div__algebra(A) | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,B)) = c_RealVector_Onorm__class_Onorm(A,c_Rings_Oinverse__class_Oinverse(A,B)) # label(fact_norm__inverse) # label(axiom). [clausify(796)]. 8.06/8.24 Derived: -class_Rings_Odivision__ring__inverse__zero(tc_Complex_Ocomplex) | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)) = c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)). [resolve(1644,b,1640,a)]. 8.06/8.24 Derived: -class_Rings_Odivision__ring__inverse__zero(tc_RealDef_Oreal) | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)) = c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)). [resolve(1644,b,1642,a)]. 8.06/8.24 1645 -class_RealVector_Oreal__normed__div__algebra(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,B)) = c_RealVector_Onorm__class_Onorm(A,c_Rings_Oinverse__class_Oinverse(A,B)) # label(fact_nonzero__norm__inverse) # label(axiom). [clausify(897)]. 8.06/8.24 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A)) = c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)). [resolve(1645,a,1640,a)]. 8.06/8.24 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A)) = c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)). [resolve(1645,a,1642,a)]. 8.45/8.58 1646 -class_Rings_Osemiring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D)),E) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),E)) # label(fact_combine__common__factor) # label(axiom). [clausify(884)]. 8.45/8.58 1647 class_Rings_Osemiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring) # label(axiom). [assumption]. 8.45/8.58 1648 class_Rings_Osemiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Osemiring) # label(axiom). [assumption]. 8.45/8.58 1649 class_Rings_Osemiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring) # label(axiom). [assumption]. 8.45/8.58 1650 class_Rings_Osemiring(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Osemiring) # label(axiom). [assumption]. 8.45/8.58 1651 -class_Rings_Ocomm__semiring__0(A) | class_Rings_Osemiring(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Osemiring) # label(axiom). [clausify(806)]. 8.45/8.58 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),D)). [resolve(1646,a,1647,a)]. 8.45/8.58 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)),D)). [resolve(1646,a,1648,a)]. 8.45/8.58 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)),D)). [resolve(1646,a,1649,a)]. 8.45/8.58 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),C)),D)). [resolve(1646,a,1650,a)]. 8.45/8.58 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),D)),E) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)),E)) | -class_Rings_Ocomm__semiring__0(A). [resolve(1646,a,1651,b)]. 8.45/8.58 1652 class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 8.45/8.58 1653 -class_Groups_Ocomm__monoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D)),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),D)) # label(fact_power__mult__distrib) # label(axiom). [clausify(262)]. 8.45/8.58 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),C)). [resolve(1652,a,1653,a)]. 8.67/8.78 1654 -class_Groups_Ocomm__monoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Oone__class_Oone(A)) = B # label(fact_mult_Ocomm__neutral) # label(axiom). [clausify(574)]. 8.67/8.78 1655 class_Groups_Ocomm__monoid__mult(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 8.67/8.78 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B),C)). [resolve(1655,a,1653,a)]. 8.67/8.78 1656 -class_Rings_Ocomm__semiring__1(A) | class_Groups_Ocomm__monoid__mult(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Ocomm__monoid__mult) # label(axiom). [clausify(669)]. 8.67/8.78 Derived: -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Polynomial_Opoly(A)),C),D)). [resolve(1656,b,1653,a)]. 8.67/8.78 1657 -class_Groups_Ocomm__monoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oone__class_Oone(A)),B) = B # label(fact_mult__1) # label(axiom). [clausify(861)]. 8.67/8.78 1658 class_Groups_Ocomm__monoid__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 8.67/8.78 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),C)). [resolve(1658,a,1653,a)]. 8.67/8.78 1659 class_Groups_Ocomm__monoid__mult(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 8.67/8.78 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),B),C)). [resolve(1659,a,1653,a)]. 8.67/8.78 1660 class_Groups_Oordered__cancel__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 8.67/8.78 1661 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D))) # label(fact_add__strict__right__mono) # label(axiom). [clausify(263)]. 8.67/8.78 1662 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,D),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),E))) # label(fact_add__less__le__mono) # label(axiom). [clausify(439)]. 8.67/8.78 1663 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__strict__left__mono) # label(axiom). [clausify(680)]. 8.67/8.78 1664 class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 8.67/8.78 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))). [resolve(1664,a,1662,a)]. 8.67/8.81 1665 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),E))) # label(fact_add__strict__mono) # label(axiom). [clausify(748)]. 8.67/8.81 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),D))). [resolve(1665,a,1660,a)]. 8.67/8.81 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))). [resolve(1665,a,1664,a)]. 8.67/8.81 1666 -class_Rings_Olinordered__idom(A) | class_Groups_Oordered__cancel__ab__semigroup__add(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [clausify(787)]. 8.67/8.81 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),D),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),E))). [resolve(1666,b,1662,a)]. 8.67/8.81 Derived: -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),D),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),E))). [resolve(1666,b,1665,a)]. 8.67/8.81 1667 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),E))) # label(fact_add__le__less__mono) # label(axiom). [clausify(870)]. 8.67/8.81 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),D))). [resolve(1667,a,1660,a)]. 8.67/8.81 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))). [resolve(1667,a,1664,a)]. 8.67/8.81 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),D),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),E))) | -class_Rings_Olinordered__idom(A). [resolve(1667,a,1666,b)]. 8.67/8.81 1668 class_Groups_Oordered__cancel__ab__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 8.82/8.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),D))). [resolve(1668,a,1662,a)]. 8.82/8.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),D))). [resolve(1668,a,1665,a)]. 8.82/8.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),D))). [resolve(1668,a,1667,a)]. 8.82/8.99 1669 -class_Rings_Olinordered__idom(A) | class_Groups_Oordered__ab__semigroup__add(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Oordered__ab__semigroup__add) # label(axiom). [clausify(522)]. 8.82/8.99 1670 -class_Groups_Oordered__ab__semigroup__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D))) # label(fact_add__right__mono) # label(axiom). [clausify(287)]. 8.82/8.99 1671 -class_Groups_Oordered__ab__semigroup__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__left__mono) # label(axiom). [clausify(556)]. 8.82/8.99 1672 class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 8.82/8.99 1673 class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 8.82/8.99 1674 class_Groups_Oordered__ab__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 8.82/8.99 1675 -class_Groups_Oordered__ab__semigroup__add(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,D),E)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),E))) # label(fact_add__mono) # label(axiom). [clausify(846)]. 8.82/8.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),B),C)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),D),E)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Polynomial_Opoly(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),E))) | -class_Rings_Olinordered__idom(A). [resolve(1675,a,1669,b)]. 8.82/8.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))). [resolve(1675,a,1672,a)]. 8.82/8.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),D))). [resolve(1675,a,1673,a)]. 8.82/8.99 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),B)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),D))). [resolve(1675,a,1674,a)]. 9.20/9.33 1676 class_Rings_Ocomm__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ocomm__semiring) # label(axiom). [assumption]. 9.20/9.33 1677 -class_Rings_Ocomm__semiring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),C) # label(fact_comm__semiring__class_Odistrib) # label(axiom). [clausify(303)]. 9.20/9.33 1678 class_Rings_Ocomm__semiring(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Ocomm__semiring) # label(axiom). [assumption]. 9.20/9.33 1679 -class_Rings_Ocomm__semiring__0(A) | class_Rings_Ocomm__semiring(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Ocomm__semiring) # label(axiom). [clausify(555)]. 9.20/9.33 1680 class_Rings_Ocomm__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Ocomm__semiring) # label(axiom). [assumption]. 9.20/9.33 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),C)),B). [resolve(1680,a,1677,a)]. 9.20/9.33 1681 class_Rings_Ocomm__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__semiring) # label(axiom). [assumption]. 9.20/9.33 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),B). [resolve(1681,a,1677,a)]. 9.20/9.33 1682 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)) # label(fact_crossproduct__eq) # label(axiom). [clausify(363)]. 9.20/9.33 1683 class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [assumption]. 9.20/9.33 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),D),A)). [resolve(1682,a,1683,a)]. 9.20/9.33 1684 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_crossproduct__eq) # label(axiom). [clausify(363)]. 9.20/9.33 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),C)). [resolve(1684,a,1683,a)]. 9.20/9.33 1685 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B = C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)) # label(fact_crossproduct__eq) # label(axiom). [clausify(363)]. 9.20/9.36 Derived: A = B | C = D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),D),B)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),D),A)). [resolve(1685,a,1683,a)]. 9.20/9.36 1686 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B = C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)) # label(fact_crossproduct__noteq) # label(axiom). [clausify(678)]. 9.20/9.36 1687 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)) # label(fact_crossproduct__noteq) # label(axiom). [clausify(678)]. 9.20/9.36 1688 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_crossproduct__noteq) # label(axiom). [clausify(678)]. 9.20/9.36 1689 class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [assumption]. 9.20/9.36 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),A)). [resolve(1689,a,1682,a)]. 9.20/9.36 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)). [resolve(1689,a,1684,a)]. 9.20/9.36 Derived: A = B | C = D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),A)). [resolve(1689,a,1685,a)]. 9.20/9.36 1690 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Ozero__class_Ozero(A) = B | C != D | E = F | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),F)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)) # label(fact_add__scale__eq__noteq) # label(axiom). [clausify(811)]. 9.20/9.36 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | B != C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),D)). [resolve(1690,a,1683,a)]. 9.20/9.37 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | B != C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),D)). [resolve(1690,a,1689,a)]. 9.20/9.37 1691 class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Int_Oint) # label(arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [assumption]. 9.20/9.37 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),A)). [resolve(1691,a,1682,a)]. 9.20/9.37 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)). [resolve(1691,a,1684,a)]. 9.20/9.37 Derived: A = B | C = D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),A)). [resolve(1691,a,1685,a)]. 9.20/9.37 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | B != C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)). [resolve(1691,a,1690,a)]. 9.20/9.37 1692 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != B | c_Groups_Ozero__class_Ozero(A) = C # label(fact_add__0__iff) # label(axiom). [clausify(898)]. 9.20/9.37 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B) != A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B. [resolve(1692,a,1683,a)]. 9.20/9.37 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B. [resolve(1692,a,1689,a)]. 9.20/9.37 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(1692,a,1691,a)]. 9.20/9.37 1693 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) = B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_add__0__iff) # label(axiom). [clausify(898)]. 9.20/9.37 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != B. [resolve(1693,a,1683,a)]. 9.20/9.37 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B. [resolve(1693,a,1689,a)]. 9.20/9.37 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(1693,a,1691,a)]. 9.20/9.37 1694 -class_Rings_Oidom(A) | class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [clausify(915)]. 9.20/9.37 Derived: -class_Rings_Oidom(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),E),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),E),B)). [resolve(1694,b,1682,a)]. 9.27/9.38 Derived: -class_Rings_Oidom(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),D)). [resolve(1694,b,1684,a)]. 9.27/9.38 Derived: -class_Rings_Oidom(A) | B = C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),E),C)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),E),B)). [resolve(1694,b,1685,a)]. 9.27/9.38 Derived: -class_Rings_Oidom(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | C != D | E = F | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),F)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),E)). [resolve(1694,b,1690,a)]. 9.27/9.38 Derived: -class_Rings_Oidom(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C) != B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = C. [resolve(1694,b,1692,a)]. 9.27/9.38 Derived: -class_Rings_Oidom(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C) = B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != C. [resolve(1694,b,1693,a)]. 9.27/9.38 1695 class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Nat_Onat) # label(arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [assumption]. 9.27/9.38 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),D),A)). [resolve(1695,a,1682,a)]. 9.27/9.38 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(1695,a,1684,a)]. 9.27/9.38 Derived: A = B | C = D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),D),B)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),D),A)). [resolve(1695,a,1685,a)]. 9.27/9.38 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | B != C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)). [resolve(1695,a,1690,a)]. 9.27/9.38 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(1695,a,1692,a)]. 9.27/9.38 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(1695,a,1693,a)]. 9.57/9.75 1696 -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(D,C)),hAPP(D,B))) | -c_SEQ_Odecseq(A,D) # label(fact_decseq__def) # label(axiom). [clausify(316)]. 9.57/9.75 1697 -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f22(B,A)),f23(B,A))) | c_SEQ_Odecseq(A,B) # label(fact_decseq__def) # label(axiom). [clausify(316)]. 9.57/9.75 1698 -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f23(B,A))),hAPP(B,f22(B,A)))) | c_SEQ_Odecseq(A,B) # label(fact_decseq__def) # label(axiom). [clausify(316)]. 9.57/9.75 Derived: -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(D,C)),hAPP(D,B))) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f22(D,A)),f23(D,A))). [resolve(1696,d,1697,c)]. 9.57/9.75 Derived: -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B),C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(D,C)),hAPP(D,B))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(D,f23(D,A))),hAPP(D,f22(D,A)))). [resolve(1696,d,1698,c)]. 9.57/9.75 1699 -class_Orderings_Oorder(A) | -c_SEQ_Odecseq(A,B) | c_SEQ_Omonoseq(A,B) # label(fact_decseq__imp__monoseq) # label(axiom). [clausify(577)]. 9.57/9.75 Derived: -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f22(B,A)),f23(B,A))). [resolve(1699,b,1697,c)]. 9.57/9.75 Derived: -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f23(B,A))),hAPP(B,f22(B,A)))). [resolve(1699,b,1698,c)]. 9.57/9.75 1700 -class_Fields_Ofield__inverse__zero(A) | c_Rings_Oinverse__class_Oinverse(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Rings_Oinverse__class_Oinverse(A,B)),c_Rings_Oinverse__class_Oinverse(A,C)) # label(fact_inverse__mult__distrib) # label(axiom). [clausify(340)]. 9.57/9.75 1701 class_Fields_Ofield__inverse__zero(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Fields_Ofield__inverse__zero) # label(axiom). [assumption]. 9.57/9.75 Derived: c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A)),c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,B)). [resolve(1700,a,1701,a)]. 9.57/9.75 1702 -class_Fields_Ofield__inverse__zero(A) | c_Rings_Oinverse__class_Oinverse(A,B) != c_Groups_Oone__class_Oone(A) | c_Groups_Oone__class_Oone(A) = B # label(fact_inverse__eq__1__iff) # label(axiom). [clausify(435)]. 9.57/9.75 Derived: c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A) != c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) | c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = A. [resolve(1702,a,1701,a)]. 9.57/9.75 1703 -class_Fields_Ofield__inverse__zero(A) | c_Rings_Oinverse__class_Oinverse(A,B) = c_Groups_Oone__class_Oone(A) | c_Groups_Oone__class_Oone(A) != B # label(fact_inverse__eq__1__iff) # label(axiom). [clausify(435)]. 9.57/9.75 Derived: c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,A) = c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) | c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) != A. [resolve(1703,a,1701,a)]. 9.57/9.75 1704 -class_Fields_Ofield__inverse__zero(A) | c_Rings_Oinverse__class_Oinverse(A,c_Groups_Ozero__class_Ozero(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_field__inverse__zero) # label(axiom). [clausify(802)]. 9.57/9.75 Derived: c_Rings_Oinverse__class_Oinverse(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1704,a,1701,a)]. 9.57/9.75 1705 class_Fields_Ofield__inverse__zero(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Fields_Ofield__inverse__zero) # label(axiom). [assumption]. 10.01/10.12 Derived: c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A)),c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,B)). [resolve(1705,a,1700,a)]. 10.01/10.12 Derived: c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A) != c_Groups_Oone__class_Oone(tc_RealDef_Oreal) | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = A. [resolve(1705,a,1702,a)]. 10.01/10.12 Derived: c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,A) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) != A. [resolve(1705,a,1703,a)]. 10.01/10.12 Derived: c_Rings_Oinverse__class_Oinverse(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1705,a,1704,a)]. 10.01/10.12 1706 -class_Rings_Olinordered__comm__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_comm__mult__strict__left__mono) # label(axiom). [clausify(493)]. 10.01/10.12 1707 class_Rings_Olinordered__comm__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 10.01/10.12 1708 class_Rings_Olinordered__comm__semiring__strict(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 10.01/10.12 1709 -class_Rings_Olinordered__idom(A) | class_Rings_Olinordered__comm__semiring__strict(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Olinordered__comm__semiring__strict) # label(axiom). [clausify(700)]. 10.01/10.12 1710 class_Rings_Olinordered__comm__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 10.01/10.12 1711 -class_Rings_Oring__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) != B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) = c_Groups_Ozero__class_Ozero(A) # label(fact_mult__eq__0__iff) # label(axiom). [clausify(755)]. 10.01/10.12 1712 class_Rings_Oring__no__zero__divisors(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 10.01/10.12 1713 class_Rings_Oring__no__zero__divisors(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 10.01/10.12 1714 class_Rings_Oring__no__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 10.01/10.12 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1711,a,1712,a)]. 10.01/10.12 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1711,a,1713,a)]. 10.01/10.12 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1711,a,1714,a)]. 10.01/10.12 1715 -class_Rings_Oring__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) != B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_mult__eq__0__iff) # label(axiom). [clausify(755)]. 10.01/10.12 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1715,a,1712,a)]. 10.01/10.12 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),A) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1715,a,1713,a)]. 10.01/10.12 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1715,a,1714,a)]. 10.18/10.30 1716 -class_Rings_Oring__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) != c_Groups_Ozero__class_Ozero(A) # label(fact_mult__eq__0__iff) # label(axiom). [clausify(755)]. 10.18/10.30 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1716,a,1712,a)]. 10.18/10.30 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1716,a,1713,a)]. 10.18/10.30 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B) != c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1716,a,1714,a)]. 10.18/10.30 1717 -class_Rings_Oidom(A) | class_Rings_Oring__no__zero__divisors(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Oring__no__zero__divisors) # label(axiom). [clausify(844)]. 10.18/10.30 Derived: -class_Rings_Oidom(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)). [resolve(1717,b,1711,a)]. 10.18/10.30 Derived: -class_Rings_Oidom(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) != B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),C),B) = c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)). [resolve(1717,b,1715,a)]. 10.18/10.30 Derived: -class_Rings_Oidom(A) | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = B | c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)) = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),C) != c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(A)). [resolve(1717,b,1716,a)]. 10.18/10.30 1718 -class_Rings_Oring__1(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),c_Groups_Oone__class_Oone(A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Oone__class_Oone(A))),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),c_Groups_Oone__class_Oone(A))) # label(fact_real__squared__diff__one__factored) # label(axiom). [clausify(564)]. 10.18/10.30 1719 -class_Rings_Ocomm__ring__1(A) | class_Rings_Oring__1(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Rings_Oring__1) # label(axiom). [clausify(378)]. 10.18/10.30 1720 class_Rings_Oring__1(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oring__1) # label(axiom). [assumption]. 10.18/10.30 1721 class_Rings_Oring__1(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Oring__1) # label(axiom). [assumption]. 10.18/10.30 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),B),B)),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)))),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Polynomial_Opoly(A)),B),c_Groups_Oone__class_Oone(tc_Polynomial_Opoly(A)))) | -class_Rings_Ocomm__ring__1(A). [resolve(1718,a,1719,b)]. 10.18/10.30 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A)),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))). [resolve(1718,a,1720,a)]. 10.18/10.30 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),A)),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex),A),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))). [resolve(1718,a,1721,a)]. 10.57/10.68 1722 class_Rings_Oring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1) # label(axiom). [assumption]. 10.57/10.68 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),c_Groups_Oone__class_Oone(tc_Int_Oint)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(1722,a,1718,a)]. 10.57/10.68 1723 -class_Orderings_Owellorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Orderings_Oord__class_OLeast(A,C))) | -hBOOL(hAPP(C,B)) # label(fact_not__less__Least) # label(axiom). [clausify(469)]. 10.57/10.68 1724 class_Orderings_Owellorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Owellorder) # label(axiom). [assumption]. 10.57/10.68 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Orderings_Oord__class_OLeast(tc_Nat_Onat,B))) | -hBOOL(hAPP(B,A)). [resolve(1723,a,1724,a)]. 10.57/10.68 1725 -class_Orderings_Owellorder(A) | -hBOOL(hAPP(B,C)) | hBOOL(hAPP(B,c_Orderings_Oord__class_OLeast(A,B))) # label(fact_LeastI__ex) # label(axiom). [clausify(515)]. 10.57/10.68 Derived: -hBOOL(hAPP(A,B)) | hBOOL(hAPP(A,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,A))). [resolve(1725,a,1724,a)]. 10.57/10.68 1726 -class_Orderings_Owellorder(A) | -hBOOL(hAPP(B,C)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,c_Orderings_Oord__class_OLeast(A,B)),C)) # label(fact_Least__le) # label(axiom). [clausify(604)]. 10.57/10.68 Derived: -hBOOL(hAPP(A,B)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,A)),B)). [resolve(1726,a,1724,a)]. 10.57/10.68 1727 -class_Orderings_Owellorder(A) | -hBOOL(hAPP(B,C)) | hBOOL(hAPP(B,c_Orderings_Oord__class_OLeast(A,B))) # label(fact_LeastI) # label(axiom). [clausify(662)]. 10.57/10.68 1728 -class_Groups_Oab__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(767)]. 10.57/10.68 1729 class_Groups_Oab__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 10.57/10.68 1730 -class_Groups_Ocomm__monoid__add(A) | class_Groups_Oab__semigroup__add(tc_Polynomial_Opoly(A)) # label(arity_Polynomial__Opoly__Groups_Oab__semigroup__add) # label(axiom). [clausify(560)]. 10.57/10.68 1731 class_Groups_Oab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 10.57/10.68 1732 class_Groups_Oab__semigroup__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 10.57/10.68 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal),B),C)). [resolve(1728,a,1729,a)]. 10.57/10.68 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Polynomial_Opoly(A)),C),D)) | -class_Groups_Ocomm__monoid__add(A). [resolve(1728,a,1730,b)]. 10.57/10.68 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)). [resolve(1728,a,1731,a)]. 10.93/11.04 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex),B),C)). [resolve(1728,a,1732,a)]. 10.93/11.04 1733 class_Groups_Oab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 10.93/11.04 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)). [resolve(1733,a,1728,a)]. 10.93/11.04 1734 -class_Orderings_Oorder(A) | c_SEQ_Oincseq(A,B) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f75(B,A)),f76(B,A))) # label(fact_incseq__def) # label(axiom). [clausify(942)]. 10.93/11.04 1735 -class_Orderings_Oorder(A) | -c_SEQ_Oincseq(A,B) | c_SEQ_Omonoseq(A,B) # label(fact_incseq__imp__monoseq) # label(axiom). [clausify(475)]. 10.93/11.04 1736 -class_Orderings_Oorder(A) | -c_SEQ_Oincseq(A,B) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,D))) # label(fact_incseq__def) # label(axiom). [clausify(942)]. 10.93/11.04 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f75(B,A)),f76(B,A))) | -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B). [resolve(1734,b,1735,b)]. 10.93/11.04 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f75(B,A)),f76(B,A))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,D))). [resolve(1734,b,1736,b)]. 10.93/11.04 1737 -class_Orderings_Oorder(A) | c_SEQ_Oincseq(A,B) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f75(B,A))),hAPP(B,f76(B,A)))) # label(fact_incseq__def) # label(axiom). [clausify(942)]. 10.93/11.04 Derived: -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f75(B,A))),hAPP(B,f76(B,A)))) | -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B). [resolve(1737,b,1735,b)]. 10.93/11.04 Derived: -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f75(B,A))),hAPP(B,f76(B,A)))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,D))). [resolve(1737,b,1736,b)]. 10.93/11.04 1738 -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f39(B,A)),f40(B,A))) # label(fact_monoseq__def) # label(axiom). [clausify(551)]. 10.93/11.04 1739 -class_Orderings_Oorder(A) | -c_SEQ_Omonoseq(A,B) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D)),hAPP(B,C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,E),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,E)),hAPP(B,F))) # label(fact_monoseq__def) # label(axiom). [clausify(551)]. 10.93/11.04 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f39(B,A)),f40(B,A))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D)),hAPP(B,C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,E),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,E)),hAPP(B,F))). [resolve(1738,b,1739,b)]. 10.93/11.04 1740 -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f40(B,A))),hAPP(B,f39(B,A)))) # label(fact_monoseq__def) # label(axiom). [clausify(551)]. 10.93/11.04 Derived: -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f40(B,A))),hAPP(B,f39(B,A)))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D)),hAPP(B,C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,E),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,E)),hAPP(B,F))). [resolve(1740,b,1739,b)]. 10.93/11.06 1741 -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f41(B,A)),f42(B,A))) # label(fact_monoseq__def) # label(axiom). [clausify(551)]. 10.93/11.06 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f41(B,A)),f42(B,A))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D)),hAPP(B,C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,E),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,E)),hAPP(B,F))). [resolve(1741,b,1739,b)]. 10.93/11.06 1742 -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f41(B,A))),hAPP(B,f42(B,A)))) # label(fact_monoseq__def) # label(axiom). [clausify(551)]. 10.93/11.06 Derived: -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f41(B,A))),hAPP(B,f42(B,A)))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D)),hAPP(B,C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,E),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,E)),hAPP(B,F))). [resolve(1742,b,1739,b)]. 10.93/11.06 1743 -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f55(B,A))),hAPP(B,hAPP(c_Nat_OSuc,f55(B,A))))) | c_SEQ_Omonoseq(A,B) # label(fact_monoseq__Suc) # label(axiom). [clausify(857)]. 10.93/11.06 Derived: -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f55(B,A))),hAPP(B,hAPP(c_Nat_OSuc,f55(B,A))))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D)),hAPP(B,C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,E),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,E)),hAPP(B,F))). [resolve(1743,c,1739,b)]. 10.93/11.06 1744 -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,f56(B,A)))),hAPP(B,f56(B,A)))) | c_SEQ_Omonoseq(A,B) # label(fact_monoseq__Suc) # label(axiom). [clausify(857)]. 10.93/11.06 Derived: -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,f56(B,A)))),hAPP(B,f56(B,A)))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D)),hAPP(B,C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,E),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,E)),hAPP(B,F))). [resolve(1744,c,1739,b)]. 10.93/11.06 1745 -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))) | -c_SEQ_Omonoseq(A,B) # label(fact_monoseq__Suc) # label(axiom). [clausify(857)]. 10.93/11.06 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f39(B,A)),f40(B,A))). [resolve(1745,d,1738,b)]. 10.93/11.06 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f40(B,A))),hAPP(B,f39(B,A)))). [resolve(1745,d,1740,b)]. 10.93/11.06 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f41(B,A)),f42(B,A))). [resolve(1745,d,1741,b)]. 11.07/11.20 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f41(B,A))),hAPP(B,f42(B,A)))). [resolve(1745,d,1742,b)]. 11.07/11.20 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f55(B,A))),hAPP(B,hAPP(c_Nat_OSuc,f55(B,A))))). [resolve(1745,d,1743,c)]. 11.07/11.20 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,f56(B,A)))),hAPP(B,f56(B,A)))). [resolve(1745,d,1744,c)]. 11.07/11.20 1746 -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | c_SEQ_Omonoseq(tc_RealDef_Oreal,hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A)) # label(fact_monoseq__realpow) # label(axiom). [clausify(993)]. 11.07/11.20 Derived: -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A)) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) | -class_Orderings_Oorder(tc_RealDef_Oreal) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(c_Nat_OSuc,B)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),hAPP(c_Nat_OSuc,C))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),C))). [resolve(1746,c,1745,d)]. 11.07/11.20 1747 -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f22(B,A)),f23(B,A))). [resolve(1699,b,1697,c)]. 11.07/11.20 Derived: -class_Orderings_Oorder(A) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f22(B,A)),f23(B,A))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D)),hAPP(B,C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,E),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,E)),hAPP(B,F))). [resolve(1747,b,1739,b)]. 11.07/11.20 Derived: -class_Orderings_Oorder(A) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f22(B,A)),f23(B,A))) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))). [resolve(1747,b,1745,d)]. 11.07/11.20 1748 -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f23(B,A))),hAPP(B,f22(B,A)))). [resolve(1699,b,1698,c)]. 11.07/11.20 Derived: -class_Orderings_Oorder(A) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f23(B,A))),hAPP(B,f22(B,A)))) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C),D)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D)),hAPP(B,C))) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,E),F)) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,E)),hAPP(B,F))). [resolve(1748,b,1739,b)]. 11.07/11.20 Derived: -class_Orderings_Oorder(A) | -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f23(B,A))),hAPP(B,f22(B,A)))) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))). [resolve(1748,b,1745,d)]. 12.12/12.22 1749 -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f75(B,A)),f76(B,A))) | -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B). [resolve(1734,b,1735,b)]. 12.12/12.22 Derived: -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f75(B,A)),f76(B,A))) | -class_Orderings_Oorder(A) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))). [resolve(1749,d,1745,d)]. 12.12/12.22 1750 -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f75(B,A))),hAPP(B,f76(B,A)))) | -class_Orderings_Oorder(A) | c_SEQ_Omonoseq(A,B). [resolve(1737,b,1735,b)]. 12.12/12.22 Derived: -class_Orderings_Oorder(A) | -hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f75(B,A))),hAPP(B,f76(B,A)))) | -class_Orderings_Oorder(A) | -class_Orderings_Oorder(A) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,C)),hAPP(B,hAPP(c_Nat_OSuc,C)))) | hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(A,hAPP(B,hAPP(c_Nat_OSuc,D))),hAPP(B,D))). [resolve(1750,d,1745,d)]. 12.12/12.22 12.12/12.22 ============================== end predicate elimination ============= 12.12/12.22 12.12/12.22 Auto_denials: (non-Horn, no changes). 12.12/12.22 12.12/12.22 Term ordering decisions: 12.12/12.22 Function symbol KB weights: tc_RealDef_Oreal=1. tc_Int_Oint=1. tc_Nat_Onat=1. tc_Complex_Ocomplex=1. c_Nat_OSuc=1. tc_HOL_Obool=1. v_q____=1. v_pa____=1. c_fequal=1. c_fimplies=1. v_c____=1. v_p=1. c1=1. c2=1. c3=1. c4=1. hAPP=1. c_Orderings_Oord__class_Oless__eq=1. c_RealVector_Onorm__class_Onorm=1. c_Rings_Oinverse__class_Oinverse=1. tc_fun=1. c_RealDef_Oreal=1. c_Polynomial_Odegree=1. c_Polynomial_Opoly=1. c_Polynomial_Ocoeff=1. c_Orderings_Oord__class_OLeast=1. c_Fundamental__Theorem__Algebra__Mirabelle_Opsize=1. c_Polynomial_OAbs__poly=1. f1=1. f3=1. f8=1. f9=1. f10=1. f11=1. f14=1. f16=1. f17=1. f19=1. f21=1. f22=1. f23=1. f27=1. f28=1. f32=1. f33=1. f38=1. f39=1. f40=1. f41=1. f42=1. f45=1. f50=1. f51=1. f52=1. f53=1. f55=1. f56=1. f58=1. f60=1. f61=1. f67=1. f70=1. f74=1. f75=1. f76=1. f77=1. f78=1. c_Groups_Ozero__class_Ozero=1. tc_Polynomial_Opoly=1. c_Orderings_Oord__class_Oless=1. c_Groups_Otimes__class_Otimes=1. c_Groups_Oplus__class_Oplus=1. c_Groups_Oone__class_Oone=1. c_Power_Opower__class_Opower=1. c_Groups_Ominus__class_Ominus=1. c_Transcendental_Oln=1. c_RComplete_Onatfloor=1. c_RComplete_Onatceiling=1. c_HOL_OAll=1. c_If=1. f4=1. f5=1. f31=1. f49=1. f57=1. f73=1. c_Divides_Odiv__class_Omod=1. c_Divides_Odiv__class_Odiv=1. c_Polynomial_Osmult=1. c_Polynomial_Opoly__gcd=1. c_Polynomial_Omonom=1. c_COMBS=1. c_Power_Opower_Opower=1. c_Polynomial_Oorder=1. c_Polynomial_Opcompose=1. c_Polynomial_Osynthetic__div=1. c_COMBK=1. f2=1. f6=1. f7=1. f12=1. f13=1. f15=1. f18=1. f24=1. f25=1. f26=1. f29=1. f30=1. f34=1. f35=1. f36=1. f37=1. f43=1. f44=1. f47=1. f54=1. f59=1. f62=1. f63=1. f64=1. f65=1. f66=1. f68=1. f69=1. f71=1. f72=1. c_COMBB=1. f20=1. f46=1. f48=1. c_COMBC=1. 12.12/12.22 12.12/12.22 ============================== end of process initial clauses ======== 12.12/12.22 12.12/12.22 ============================== CLAUSES FOR SEARCH ==================== 12.12/12.22 12.12/12.22 ============================== end of clauses for search ============= 12.12/12.22 12.12/12.22 ============================== SEARCH ================================ 12.12/12.22 12.12/12.22 % Starting search at 10.15 seconds. 12.12/12.22 12.12/12.22 Low Water (keep): wt=62.000, iters=3351 12.12/12.22 12.12/12.22 Low Water (keep): wt=60.000, iters=3383 12.12/12.22 12.12/12.22 Low Water (keep): wt=58.000, iters=3347 12.12/12.22 12.12/12.22 Low Water (keep): wt=56.000, iters=3349 12.12/12.22 12.12/12.22 Low Water (keep): wt=55.000, iters=3344 12.12/12.22 12.12/12.22 Low Water (keep): wt=54.000, iters=3378 12.12/12.22 12.12/12.22 Low Water (keep): wt=53.000, iters=3412 12.12/12.22 12.12/12.22 Low Water (keep): wt=52.000, iters=3424 12.12/12.22 12.12/12.22 Low Water (keep): wt=51.000, iters=3350 12.12/12.22 12.12/12.22 Low Water (keep): wt=50.000Cputime limit exceeded (core dumped) 300.09/300.21 EOF