0.05/0.10 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.05/0.10 % Command : tptp2X_and_run_prover9 %d %s 0.09/0.30 % Computer : n009.cluster.edu 0.09/0.30 % Model : x86_64 x86_64 0.09/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.09/0.30 % Memory : 8042.1875MB 0.09/0.30 % OS : Linux 3.10.0-693.el7.x86_64 0.09/0.30 % CPULimit : 960 0.09/0.30 % DateTime : Thu Jul 2 12:35:36 EDT 2020 0.09/0.31 % CPUTime : 2.14/2.46 ============================== Prover9 =============================== 2.14/2.46 Prover9 (32) version 2009-11A, November 2009. 2.14/2.46 Process 29110 was started by sandbox2 on n009.cluster.edu, 2.14/2.46 Thu Jul 2 12:35:38 2020 2.14/2.46 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_28957_n009.cluster.edu". 2.14/2.46 ============================== end of head =========================== 2.14/2.46 2.14/2.46 ============================== INPUT ================================= 2.14/2.46 2.14/2.46 % Reading from file /tmp/Prover9_28957_n009.cluster.edu 2.14/2.46 2.14/2.46 set(prolog_style_variables). 2.14/2.46 set(auto2). 2.14/2.46 % set(auto2) -> set(auto). 2.14/2.46 % set(auto) -> set(auto_inference). 2.14/2.46 % set(auto) -> set(auto_setup). 2.14/2.46 % set(auto_setup) -> set(predicate_elim). 2.14/2.46 % set(auto_setup) -> assign(eq_defs, unfold). 2.14/2.46 % set(auto) -> set(auto_limits). 2.14/2.46 % set(auto_limits) -> assign(max_weight, "100.000"). 2.14/2.46 % set(auto_limits) -> assign(sos_limit, 20000). 2.14/2.46 % set(auto) -> set(auto_denials). 2.14/2.46 % set(auto) -> set(auto_process). 2.14/2.46 % set(auto2) -> assign(new_constants, 1). 2.14/2.46 % set(auto2) -> assign(fold_denial_max, 3). 2.14/2.46 % set(auto2) -> assign(max_weight, "200.000"). 2.14/2.46 % set(auto2) -> assign(max_hours, 1). 2.14/2.46 % assign(max_hours, 1) -> assign(max_seconds, 3600). 2.14/2.46 % set(auto2) -> assign(max_seconds, 0). 2.14/2.46 % set(auto2) -> assign(max_minutes, 5). 2.14/2.46 % assign(max_minutes, 5) -> assign(max_seconds, 300). 2.14/2.46 % set(auto2) -> set(sort_initial_sos). 2.14/2.46 % set(auto2) -> assign(sos_limit, -1). 2.14/2.46 % set(auto2) -> assign(lrs_ticks, 3000). 2.14/2.46 % set(auto2) -> assign(max_megs, 400). 2.14/2.46 % set(auto2) -> assign(stats, some). 2.14/2.46 % set(auto2) -> clear(echo_input). 2.14/2.46 % set(auto2) -> set(quiet). 2.14/2.46 % set(auto2) -> clear(print_initial_clauses). 2.14/2.46 % set(auto2) -> clear(print_given). 2.14/2.46 assign(lrs_ticks,-1). 2.14/2.46 assign(sos_limit,10000). 2.14/2.46 assign(order,kbo). 2.14/2.46 set(lex_order_vars). 2.14/2.46 clear(print_given). 2.14/2.46 2.14/2.46 % formulas(sos). % not echoed (1200 formulas) 2.14/2.46 2.14/2.46 ============================== end of input ========================== 2.14/2.46 2.14/2.46 % From the command line: assign(max_seconds, 960). 2.14/2.46 2.14/2.46 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 2.14/2.46 2.14/2.46 % Formulas that are not ordinary clauses: 2.14/2.46 1 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_x))) # label(fact_order__refl) # label(axiom) # label(non_clause). [assumption]. 2.14/2.46 2 (all V_b all V_c all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (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)) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.46 3 (all V_x_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_RComplete_Onatceiling(V_x_2),c_Groups_Oone__class_Oone(tc_Nat_Onat)) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))) # label(fact_natceiling__le__eq__one) # label(axiom) # label(non_clause). [assumption]. 2.14/2.46 4 (all V_n 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]. 2.14/2.46 5 (all V_n_2 all V_x_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_x_2) <-> 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]. 2.14/2.46 6 (all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> 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]. 2.14/2.46 7 (all V_m 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]. 2.14/2.47 8 (all V_w all V_z (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_w,V_z) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_z,V_w))) # label(fact_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 9 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (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]. 2.14/2.47 10 (all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ouminus__class_Ouminus(T_a,V_b_2))))) # label(fact_le__minus__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 11 (all V_y_2 all V_x_2 all V_za_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_za_2) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_za_2),V_x_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_za_2),V_y_2))))) # label(fact_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 12 (all V_k_2 all V_f_2 all V_n_2 ((all B_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_i,V_n_2) -> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,hAPP(V_f_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B_i,c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(V_f_2,B_i))),c_Groups_Oone__class_Oone(tc_Int_Oint)))) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2,hAPP(V_f_2,V_n_2)) -> (exists B_i (V_k_2 = hAPP(V_f_2,B_i) & c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_i,V_n_2))))))) # label(fact_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 13 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> V_n_2 != V_m_2)) # label(fact_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 14 (all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (exists B_K (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_a all B_b c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(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(tc_RealDef_Oreal,B_a)),c_RealVector_Onorm__class_Onorm(T_a,B_b))),B_K))))))) # label(fact_scaleR_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 15 (all V_z 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),V_x),c_Groups_Oplus__class_Oplus(T_a,V_y,V_z)) = 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)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 16 (exists B_K (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Complex_OIm(B_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B_x)),B_K))))) # label(fact_Im_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 17 (all V_b_H all V_b 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_Oplus__class_Oplus(T_a,V_b,V_b_H)) = 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)))) # label(fact_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 18 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b))))) # label(fact_norm__triangle__ineq4) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 19 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_m_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_n_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m_2,V_k_2),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n_2,V_k_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2))))) # label(fact_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 20 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 21 (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),c_Nat_OSuc(V_n)))) # label(fact_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 22 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_a) -> -c_Orderings_Oord__class_Oless(T_a,V_a,V_b)))) # label(fact_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 23 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> 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]. 2.14/2.47 24 (all V_x all V_r c_Complex_Ocomplex_OComplex(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),c_Complex_ORe(V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),c_Complex_OIm(V_x))) = c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,V_r,V_x)) # label(fact_complex__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 25 (all V_y all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,V_y) -> 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]. 2.14/2.47 26 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b))))) # label(fact_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 27 (all V_m c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_m) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 28 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2) = c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_d_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2))))) # label(fact_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 29 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)))))) # label(fact_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 30 (all V_i all V_n (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,c_Nat_OSuc(V_i)),V_n))) # label(fact_diff__Suc__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 31 (all V_a all V_n (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_a) -> (exists B_r (hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B_r),V_n) = V_a & c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_r)))))) # label(fact_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 32 (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),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)),V_m) = 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]. 2.14/2.47 33 (all V_r_H all V_q_H all V_b_H all V_r all V_q all V_b (c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q),V_r) = 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_Orderings_Oord__class_Oless(tc_Int_Oint,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)) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_r,V_b) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_r_H) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b_H) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_b_H,V_b) -> 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]. 2.14/2.47 34 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 35 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> c_Orderings_Oord__class_Oless(T_a,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]. 2.14/2.47 36 (all V_z all V_w (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_w,V_z) -> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,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]. 2.14/2.47 37 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a),V_b)) = V_b)) # label(fact_add__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 38 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_b) = V_a)) # label(fact_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 39 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a = V_b -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a))))) # label(fact_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 40 (all V_n c_RComplete_Onatfloor(c_RealDef_Oreal(tc_Nat_Onat,V_n)) = V_n) # label(fact_natfloor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 41 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) <-> V_x_2 = V_y_2 | c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)))) # label(fact_order__le__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 42 (all V_n_2 all V_k_2 all V_m_2 (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)) <-> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)))) # label(fact_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 43 (all T_a (class_Rings_Olinordered__semidom(T_a) -> 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]. 2.14/2.47 44 (all V_b all V_a all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.47 45 (all V_a all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> c_RComplete_Onatceiling(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,c_RealDef_Oreal(tc_Nat_Onat,V_a))) = 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]. 2.14/2.47 46 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j,V_k) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_k)))) # label(fact_le__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 47 (all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> c_RComplete_Onatfloor(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_RComplete_Onatfloor(V_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_natfloor__add__one) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 48 (all V_b all V_a_H all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b),c_RealVector_OscaleR__class_OscaleR(T_a,V_a_H,V_b)) = c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_a,V_a_H),V_b))) # label(fact_scaleR_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 49 (all V_i ((c_Orderings_Oord__class_Oless(tc_Int_Oint,V_i,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) -> c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,V_i) = c_Groups_Oabs__class_Oabs(tc_Int_Oint,V_i)) & (-c_Orderings_Oord__class_Oless(tc_Int_Oint,V_i,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) -> c_Groups_Oabs__class_Oabs(tc_Int_Oint,V_i) = V_i))) # label(fact_zabs__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 50 (all V_y_2 all V_x_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x_2) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_y_2) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Transcendental_Oln(V_x_2),c_Transcendental_Oln(V_y_2)))))) # label(fact_ln__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 51 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a),V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),V_b))))) # label(fact_abs__leI) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 52 (all V_c all V_a all V_b all T_a (class_Rings_Oordered__ring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.47 53 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a),c_Groups_Ouminus__class_Ouminus(T_a,V_b)))) # label(fact_minus__add__distrib) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 54 (all V_c_2 all V_b_2 all V_a_2 all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (V_b_2 = V_c_2 <-> c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_b_2)))) # label(fact_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 55 (all V_y_2 all V_x_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (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) <-> c_Groups_Ozero__class_Ozero(T_a) = V_x_2 & V_y_2 = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 56 (all V_x all V_r c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,V_r,c_Complex_OIm(V_x)) = c_Complex_OIm(c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,V_r,V_x))) # label(fact_Im_OscaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 57 (all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_x 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]. 2.14/2.47 58 (all V_b all V_r all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_r,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b)) = c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_r,V_b)))) # label(fact_scaleR_OscaleR__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 59 (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]. 2.14/2.47 60 (all V_c_2 all V_a_2 all V_b_2 all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2) = c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_a_2) <-> V_c_2 = V_b_2))) # label(fact_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 61 (all V_m all V_n all V_k (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,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]. 2.14/2.47 62 (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),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),V_n) = 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)))) # label(fact_power__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 63 (all V_y all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K all B_x 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]. 2.14/2.47 64 (all V_n all V_a all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ouminus__class_Ouminus(T_a,V_a)),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Oone__class_Oone(T_a))),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 65 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a)) # label(fact_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 66 (all V_x (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> 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]. 2.14/2.47 67 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_z))))) # label(fact_order__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 68 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b))),c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b))))) # label(fact_norm__triangle__ineq3) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 69 (all V_y all V_x (c_Nat_OSuc(V_x) = c_Nat_OSuc(V_y) -> V_y = V_x)) # label(fact_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 70 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b))),c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b))))) # label(fact_abs__triangle__ineq3) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 71 (all V_n all V_m c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n),V_m)) # label(fact_diff__le__self) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 72 (all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b_2,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)) <-> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ouminus__class_Ouminus(T_a,V_b_2))))) # label(fact_less__minus__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 73 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__comm__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.47 74 (all V_n all V_m c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_m),c_Nat_OSuc(V_n))) # label(fact_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 75 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 76 (all V_x all V_u c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_u),V_u)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_x))) # label(fact_real__minus__mult__self__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 77 (all V_r_2 all V_a_2 all V_x_2 all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x_2,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_r_2)) & c_Orderings_Oord__class_Oless(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_r_2),V_x_2) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_x_2,V_a_2)),V_r_2)))) # label(fact_abs__diff__less__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 78 (all V_b all V_a all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> -c_Orderings_Oord__class_Oless(T_a,V_b,V_a)))) # label(fact_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 79 (all V_n_2 ((exists B_m c_Nat_OSuc(B_m) = V_n_2) <-> 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]. 2.14/2.47 80 (all V_y_2 all V_x_2 all T_a (class_RealVector_Ometric__space(T_a) -> (c_RealVector_Odist__class_Odist(T_a,V_x_2,V_y_2) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) <-> V_x_2 = V_y_2))) # label(fact_dist__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 81 (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]. 2.14/2.47 82 (all V_n all V_k all V_j (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_j,V_k) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_n),V_k))) # label(fact_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 83 (all V_d all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,V_c,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]. 2.14/2.47 84 (all V_b all V_a all V_c all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b)) -> 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]. 2.14/2.47 85 (all V_x_2 all V_y_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) <-> V_y_2 = V_x_2)))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 86 (all V_wa_2 all V_za_2 (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_za_2,V_wa_2) & V_wa_2 != V_za_2 <-> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_za_2,V_wa_2))) # label(fact_zless__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 87 (all V_x all V_n all T_a (class_Groups_Omonoid__mult(T_a) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,c_Groups_Oone__class_Oone(tc_Nat_Onat)))),V_x) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)))) # label(fact_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 88 (all V_a all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_x),V_a) -> 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]. 2.14/2.47 89 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b),c_Groups_Ouminus__class_Ouminus(T_a,V_a)))) # label(fact_minus__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 90 (all V_y all V_x c_Complex_Ocomplex_OComplex(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Complex_ORe(V_x)),c_Complex_ORe(V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Complex_OIm(V_x)),c_Complex_OIm(V_y))),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Complex_ORe(V_x)),c_Complex_OIm(V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Complex_OIm(V_x)),c_Complex_ORe(V_y)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),V_x),V_y)) # label(fact_complex__mult__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 91 (all V_x (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) -> 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]. 2.14/2.47 92 (all V_x all V_n (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_n),V_x) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,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_Onatfloor(V_x) = V_n))) # label(fact_natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 93 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> V_a = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_add__0__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 94 (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]. 2.14/2.47 95 (all V_b all V_c all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (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)) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.47 96 (all V_z3 all V_z2 all V_z1 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)) = 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)) # label(fact_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 97 (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 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_n c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(V_X_2,B_n)),B_K))))))) # label(fact_BseqD) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 98 (all V_r (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_r) -> (exists B_n (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_r,c_RealDef_Oreal(tc_Nat_Onat,B_n)) & c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,B_n,c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_r))))) # label(fact_reals__Archimedean6) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 99 (all V_x all V_r c_Complex_ORe(c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,V_r,V_x)) = c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,V_r,c_Complex_ORe(V_x))) # label(fact_Re_OscaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 100 (all T_a (class_Power_Opower(T_a) -> c_Power_Opower_Opower(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Otimes__class_Otimes(T_a)) = c_Power_Opower__class_Opower(T_a))) # label(fact_power__power__power) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 101 (all V_z all V_y all V_x 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) = 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))) # label(fact_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 102 (all V_x c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,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]. 2.14/2.47 103 (all V_w all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_z),V_w) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_w),V_z)) # label(fact_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 104 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_a)))) # label(fact_abs__ge__minus__self) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 105 (all V_n all V_m c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n),c_Nat_OSuc(V_m))) # label(fact_diff__less__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 106 (all V_n_2 all V_m_2 all V_k_2 (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_m_2 = V_n_2))) # label(fact_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 107 (all V_y all V_x (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_x) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_y) -> 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]. 2.14/2.47 108 (all V_a all V_r c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Complex_Orcis(V_r,V_a)) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_r)) # label(fact_complex__mod__rcis) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 109 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) | c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y))) # label(fact_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 110 (all V_c all V_b all V_e all V_a all T_a (class_Rings_Osemiring(T_a) -> c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)),V_e),V_c) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_e),c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_e),V_c)))) # label(fact_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 111 (all V_x all V_r all V_ra all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_r,c_RealVector_OscaleR__class_OscaleR(T_a,V_ra,V_x)) = c_RealVector_OscaleR__class_OscaleR(T_a,V_ra,c_RealVector_OscaleR__class_OscaleR(T_a,V_r,V_x)))) # label(fact_scaleR__right_OscaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 112 (all V_m all V_j all V_k (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_j),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_m)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k)),V_m))) # label(fact_diff__Suc__diff__eq2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 113 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b))))) # label(fact_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 114 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Groups_Oabs__class_Oabs(T_a,V_a) = c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a)))) # label(fact_abs__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 115 (all V_n 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]. 2.14/2.47 116 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a)) # label(fact_power__one__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 117 (all V_x (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]. 2.14/2.47 118 (all V_n_2 all V_m_2 all V_k_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_n_2) <-> V_n_2 = V_m_2)) # label(fact_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 119 (all V_y all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)),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]. 2.14/2.47 120 (all V_m all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Oone__class_Oone(T_a))),V_m) = 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]. 2.14/2.47 121 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_m_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_n_2) -> (V_m_2 = V_n_2 <-> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n_2,V_k_2) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m_2,V_k_2))))) # label(fact_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 122 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> c_Groups_Oabs__class_Oabs(T_a,V_a) = V_a))) # label(fact_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 123 (all V_m all V_n c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m),V_n) = V_m) # label(fact_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 124 (all V_y all V_x c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,V_x,V_y) = c_Complex_Ocomplex_OComplex(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Complex_ORe(V_x),c_Complex_ORe(V_y)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Complex_OIm(V_x),c_Complex_OIm(V_y)))) # label(fact_complex__add__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 125 (all V_n_2 all V_m_2 all V_u_2 all V_j_2 all V_i_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2,V_j_2) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_i_2)),V_u_2),V_n_2)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2))))) # label(fact_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 126 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> -c_Orderings_Oord__class_Oless(T_a,V_x,V_y)))) # label(fact_leD) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 127 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,V_c,V_d))),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_c)),c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_d)))))) # label(fact_abs__diff__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 128 (all V_w all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,V_z)),V_w) = c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_w))) # label(fact_zmult__zminus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 129 (all V_i all V_j all V_k (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),V_j) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k)))) # label(fact_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 130 (all V_n all V_k all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_add__leD1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 131 (all V_y all V_z all V_x (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_z) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,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]. 2.14/2.47 132 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.47 133 (all V_r all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (exists B_K all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_r,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_scaleR__right_Obounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 134 (all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) = c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n))) # label(fact_add__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.47 135 (all V_y all V_x all T_a (class_Rings_Ocomm__ring__1(T_a) -> c_Groups_Ominus__class_Ominus(T_a,V_x,V_y) = c_Groups_Oplus__class_Oplus(T_a,V_x,c_Groups_Ouminus__class_Ouminus(T_a,V_y)))) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 136 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> (V_n != c_Nat_OSuc(V_m) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n)))) # label(fact_Suc__lessI) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 137 (all V_x_2 all V_y_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2) -> (-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]. 2.14/2.48 138 (all V_w c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_w,V_w)) # label(fact_zle__refl) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 139 (all V_x_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x_2) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),V_x_2) <-> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Transcendental_Oln(V_x_2))))) # label(fact_ln__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 140 (all V_y_2 all V_x_2 all T_a (class_Lattices_Oboolean__algebra(T_a) -> (c_Groups_Ouminus__class_Ouminus(T_a,V_x_2) = c_Groups_Ouminus__class_Ouminus(T_a,V_y_2) <-> V_x_2 = V_y_2))) # label(fact_compl__eq__compl__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 141 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n))) # label(fact_Suc__leI) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 142 (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]. 2.14/2.48 143 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)) = 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))) # label(fact_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 144 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> 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]. 2.14/2.48 145 (all V_b all V_a all V_r c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,V_r,c_Complex_Ocomplex_OComplex(V_a,V_b)) = c_Complex_Ocomplex_OComplex(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),V_b))) # label(fact_complex__scaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 146 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) | V_m_2 = V_n_2)) # label(fact_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 147 (all V_x c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x)))) # label(fact_abs__add__one__gt__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 148 (all V_a_2 all V_b_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b_2),c_Groups_Ouminus__class_Ouminus(T_a,V_a_2))))) # label(fact_neg__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 149 (all V_i all V_j all V_k (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_k) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k)))) # label(fact_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 150 (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]. 2.14/2.48 151 (all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2),V_b_2) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b_2),V_a_2)))) # label(fact_minus__less__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 152 (all V_n (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) -> V_n = c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))))) # label(fact_Suc__pred) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 153 (all V_m all V_i c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_i)))) # label(fact_less__add__Suc2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 154 (all V_i all V_j all V_k (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j) -> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k),V_i) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,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]. 2.14/2.48 155 (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]. 2.14/2.48 156 (all V_n_2 all V_m_2 all V_u_2 all V_i_2 all V_j_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j_2,V_i_2) -> (c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2) <-> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i_2,V_j_2)),V_u_2),V_m_2) = V_n_2))) # label(fact_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 157 (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),c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)),V_ya) = 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)))) # label(fact_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 158 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_n_2),V_m_2) <-> -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]. 2.14/2.48 159 (all V_m all V_n (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]. 2.14/2.48 160 (all V_z (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_z) -> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),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]. 2.14/2.48 161 (all V_m_2 all V_n_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) <-> 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)))) # label(fact_real__of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 162 (all V_y_2 all V_x_2 (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x_2,V_y_2) <-> V_y_2 = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_x_2))) # label(fact_real__add__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 163 (all T_a (class_Rings_Olinordered__semidom(T_a) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),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]. 2.14/2.48 164 (all V_y all V_x all T_a (class_RealVector_Ometric__space(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(T_a,V_x,V_y)))) # label(fact_zero__le__dist) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 165 (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)),c_Nat_OSuc(V_n)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 166 (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]. 2.14/2.48 167 (all V_z all V_y all V_x all T_a (class_RealVector_Ometric__space(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(T_a,V_x,V_y),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(T_a,V_x,V_z),c_RealVector_Odist__class_Odist(T_a,V_y,V_z))))) # label(fact_dist__triangle2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 168 (all V_a all V_y all V_x all T_a (class_RealVector_Ometric__space(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(T_a,V_x,V_y),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(T_a,V_a,V_x),c_RealVector_Odist__class_Odist(T_a,V_a,V_y))))) # label(fact_dist__triangle3) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 169 (all V_x all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x)) = c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_a),V_x))) # label(fact_scaleR__minus__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 170 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> V_a = c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a)))) # label(fact_minus__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 171 (all V_b_2 all V_r_2 all V_a_2 all V_prod_2 all T_b all T_c all T_a (class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_c) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> c_RealVector_OscaleR__class_OscaleR(T_c,V_r_2,hAPP(hAPP(V_prod_2,V_a_2),V_b_2)) = hAPP(hAPP(V_prod_2,V_a_2),c_RealVector_OscaleR__class_OscaleR(T_b,V_r_2,V_b_2))))) # label(fact_bounded__bilinear_OscaleR__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 172 (all V_a ((c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_a,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) -> c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_a) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_a)) & (-c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_a,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) -> V_a = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_a)))) # label(fact_abs__real__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 173 (all V_a all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))) # label(fact_times_Oidem) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 174 (all V_z V_z = c_Complex_Ocomplex_OComplex(c_Complex_ORe(V_z),c_Complex_OIm(V_z))) # label(fact_complex__surj) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 175 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2) -> (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)) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)))) # label(fact_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 176 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_d) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)))))) # label(fact_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 177 (all V_p all V_m ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),V_m)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_m -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Oone__class_Oone(tc_Nat_Onat)))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),V_m)))) # label(fact_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 178 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ouminus__class_Ouminus(T_a,V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ouminus__class_Ouminus(T_a,V_a)),V_b))) # label(fact_minus__mult__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 179 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_n_2)))) # label(fact_Suc__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 180 (all V_y all V_x c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,V_y)) = c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,V_x,V_y)) # label(fact_dist__real__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 181 (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),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),V_x) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Nat_OSuc(V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 182 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oring(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2) = 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) <-> V_d_2 = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2)),V_e_2),V_c_2)))) # label(fact_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 183 (all V_k all V_j all V_u all V_i c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j)),V_u),V_k) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u),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]. 2.14/2.48 184 (all V_a all T_a (class_Rings_Olinordered__ring(T_a) -> 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]. 2.14/2.48 185 (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_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2) <-> V_n_2 = V_m_2 | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2)) # label(fact_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 186 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> 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]. 2.14/2.48 187 (all V_m all V_n (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) -> c_RealDef_Oreal(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)) = c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_m),c_RealDef_Oreal(tc_Nat_Onat,V_n)))) # label(fact_real__of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 188 (all V_n all V_a all T_a (class_Groups_Omonoid__mult(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_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a))) # label(fact_power__commutes) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 189 (all V_d all V_c all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,V_c,V_d))),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_c)),c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_d)))))) # label(fact_norm__diff__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 190 (all V_x_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_RComplete_Onatfloor(V_x_2)) <-> 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]. 2.14/2.48 191 (all V_a_2 all V_b_2 all T_a (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_a) <-> V_b_2 = c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_a_2)))) # label(fact_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 192 (all V_n hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_power__Suc__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 193 (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) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_u) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_v) -> (c_Groups_Oplus__class_Oplus(T_a,V_u,V_v) = c_Groups_Oone__class_Oone(T_a) -> c_Orderings_Oord__class_Oless(T_a,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]. 2.14/2.48 194 (all V_n_2 all V_m_2 all V_k_2 (V_n_2 = V_m_2 <-> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2))) # label(fact_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 195 (all V_y_2 all V_x_2 (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_y_2,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_x_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x_2,V_y_2),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))) # label(fact_real__add__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 196 (all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_abs__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 197 (all V_m_2 all V_n_2 (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) -> (V_n_2 = V_m_2 <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,c_Nat_OSuc(V_m_2))))) # label(fact_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 198 (all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K all B_a all B_b 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))))) # label(fact_mult_Obounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 199 (all V_y all V_x (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_x,V_y) | V_y = V_x | c_Orderings_Oord__class_Oless(tc_Int_Oint,V_y,V_x))) # label(fact_zless__linear) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 200 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b)),c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b))))) # label(fact_norm__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 201 (all V_z2 all V_z1 all V_w hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_z1,V_z2)) = c_Groups_Ominus__class_Ominus(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_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 202 (all V_n all V_m all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_i) -> (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)) -> 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]. 2.14/2.48 203 (all V_y_2 all V_x_2 all V_za_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_za_2) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x_2),V_za_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_y_2),V_za_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2,V_y_2)))) # label(fact_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 204 (all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_Onorm__class_Onorm(T_a,V_a) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a)))) # label(fact_abs__norm__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 205 (all V_n all V_x all T_a (class_RealVector_Oreal__normed__div__algebra(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,V_x)),V_n) = c_RealVector_Onorm__class_Onorm(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)))) # label(fact_norm__power) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 206 (all V_b_2 all V_a_2 all V_c_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2)) <-> c_Orderings_Oord__class_Oless(T_a,V_b_2,V_a_2))))) # label(fact_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 207 (all V_m all V_n (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) -> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)) = V_m)) # label(fact_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 208 (all V_r_H all V_q_H all V_b_H (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),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_Orderings_Oord__class_Oless(tc_Int_Oint,V_r_H,V_b_H) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b_H) -> 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]. 2.14/2.48 209 (all V_b_2 all V_a_2 all V_c_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2)) <-> 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]. 2.14/2.48 210 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 211 (all V_y all V_x all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x),c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_y)) = c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)))) # label(fact_scaleR__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 212 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)))) # label(fact_abs__le__D1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 213 (all V_y_2 all V_x_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x_2) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_y_2) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Transcendental_Oln(V_x_2),c_Transcendental_Oln(V_y_2)) <-> 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]. 2.14/2.48 214 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)))))) # label(fact_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 215 (all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> V_x = V_y)))) # label(fact_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 216 (all V_xa all V_y all V_x all T_a (class_RealVector_Oreal__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,V_y),V_xa) = c_Groups_Oplus__class_Oplus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_x,V_xa),c_RealVector_OscaleR__class_OscaleR(T_a,V_y,V_xa)))) # label(fact_scaleR__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 217 (all V_c all V_a all V_b all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,V_c,V_a) = c_Groups_Oplus__class_Oplus(T_a,V_b,V_a) -> V_c = V_b))) # label(fact_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 218 (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]. 2.14/2.48 219 (all V_b_2 all V_a_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (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)) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,c_Groups_Ozero__class_Ozero(T_a)) & c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) | c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 220 (all V_x 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]. 2.14/2.48 221 (all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_z) = V_z) # label(fact_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 222 (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]. 2.14/2.48 223 (all V_a_2 all V_x_2 (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x_2) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2,c_RealDef_Oreal(tc_Nat_Onat,V_a_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_RComplete_Onatceiling(V_x_2),V_a_2)))) # label(fact_natceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 224 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,c_Groups_Oone__class_Oone(T_a)) -> 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]. 2.14/2.48 225 (all V_x all T_a (class_RealVector_Ometric__space(T_a) -> c_RealVector_Odist__class_Odist(T_a,V_x,V_x) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))) # label(fact_dist__self) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 226 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ouminus__class_Ouminus(T_a,V_a)),V_b))) # label(fact_mult_Ominus__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 227 (all V_n c_Nat_OSuc(V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 228 (all V_y all V_x c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_y,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_x))) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_y)))) # label(fact_abs__minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 229 (all V_m all V_n (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) -> V_m = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n),V_n))) # label(fact_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 230 (all V_n_2 all V_m_2 all V_u_2 all V_i_2 all V_j_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j_2,V_i_2) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i_2,V_j_2)),V_u_2),V_m_2),V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2))))) # label(fact_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 231 (all V_b_2 all V_prod_2 all T_c all T_a all T_b (class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_c) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(V_prod_2,c_Groups_Ozero__class_Ozero(T_a)),V_b_2)))) # label(fact_bounded__bilinear_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 232 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> c_Groups_Ouminus__class_Ouminus(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),c_Groups_Ouminus__class_Ouminus(T_a,V_b)))) # label(fact_mult_Ominus__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 233 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k) -> 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]. 2.14/2.48 234 (all V_y all V_x all T_a (class_RealVector_Ometric__space(T_a) -> c_RealVector_Odist__class_Odist(T_a,V_x,V_y) = c_RealVector_Odist__class_Odist(T_a,V_y,V_x))) # label(fact_dist__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 235 (all V_y all V_x (V_x != V_y -> (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_y,V_x)))) # label(fact_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 236 (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]. 2.14/2.48 237 (all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2),c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2)))) # label(fact_neg__le__0__iff__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 238 (all V_a_2 all T_a (class_Groups_Ogroup__add(T_a) -> (c_Groups_Ouminus__class_Ouminus(T_a,V_a_2) = c_Groups_Ozero__class_Ozero(T_a) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_neg__0__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 239 (all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_scaleR__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 240 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x) | c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y))) # label(fact_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 241 (all T_a all V_g_2 all V_f_2 all T_b (class_Orderings_Oord(T_b) -> ((all B_x c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x))) -> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)))) # label(fact_le__funI) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 242 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_a) -> (V_c = V_b -> c_Orderings_Oord__class_Oless(T_a,V_c,V_a))))) # label(fact_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 243 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> 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]. 2.14/2.48 244 (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),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_p,V_q)) = 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)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 245 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2) | V_y_2 = V_x_2))) # label(fact_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 246 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_Suc__leD) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 247 (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),c_Nat_OSuc(V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 248 (all V_w all V_z c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_z,V_w) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,V_w))) # label(fact_diff__int__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 249 (all V_n_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) <-> 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]. 2.14/2.48 250 (all V_c all V_b all V_a (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_c,V_a) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_a,V_c),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_b,V_c))))) # label(fact_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 251 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_abs__minus__le__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 252 (all V_n all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) -> 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]. 2.14/2.48 253 (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_x_2 & c_Groups_Ozero__class_Ozero(T_a) = V_y_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,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]. 2.14/2.48 254 (all V_a all V_b c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,V_b,V_a)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_b)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_a))) # label(fact_complex__mod__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 255 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)))) # label(fact_abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 256 (all V_n all V_a all V_r hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Complex_Orcis(V_r,V_a)),V_n) = c_Complex_Orcis(hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_r),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n)),V_a))) # label(fact_DeMoivre2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 257 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) = c_Groups_Oplus__class_Oplus(T_a,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]. 2.14/2.48 258 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_a = V_b -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_c))))) # label(fact_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 259 (all V_x_2 all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) -> 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]. 2.14/2.48 260 (all V_b all V_a hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Complex_Ocomplex_OComplex(V_a,V_b)),c_Complex_Oii) = c_Complex_Ocomplex_OComplex(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_b),V_a)) # label(fact_Complex__mult__i) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 261 (all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2),c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2)))) # label(fact_neg__less__0__iff__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 262 (all V_z V_z = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),c_Groups_Oone__class_Oone(tc_Int_Oint))) # label(fact_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.48 263 (all V_i all V_j -c_Orderings_Oord__class_Oless(tc_Nat_Onat,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]. 2.14/2.48 264 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (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]. 2.14/2.48 265 (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]. 2.14/2.48 266 (all V_x all T_a (class_RealVector_Oreal__vector(T_a) -> V_x = c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),V_x))) # label(fact_real__vector_Oscale__one) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 267 (all V_y all V_x V_y = c_Complex_OIm(c_Complex_Ocomplex_OComplex(V_x,V_y))) # label(fact_Im) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 268 (all V_ya all V_y all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> 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)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_y)),V_ya))) # label(fact_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 269 (all V_a_2 all V_b_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b_2),c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)))) # label(fact_neg__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 270 (all V_z3 all V_z2 all V_z1 c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z1,V_z2),V_z3) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z1,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z2,V_z3))) # label(fact_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 271 (all V_b_2 all V_a_2 all T_a (class_Groups_Ogroup__add(T_a) -> (V_a_2 = V_b_2 <-> c_Groups_Ouminus__class_Ouminus(T_a,V_b_2) = c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)))) # label(fact_neg__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 272 (all V_r_H all V_q_H all V_b_H all V_r all V_q all V_b (c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q),V_r) = 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_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),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_Orderings_Oord__class_Oless(tc_Int_Oint,V_r_H,V_b_H) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_r) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b_H) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_b_H,V_b) -> 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]. 2.14/2.49 273 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_N) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a),V_a) -> 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]. 2.14/2.49 274 (all V_n all V_m all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a) -> (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)) -> 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]. 2.14/2.49 275 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) = V_b)) # label(fact_minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 276 (all V_b_2 all V_a_2 all V_r_2 all V_prod_2 all T_b all T_c all T_a (class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_c) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> hAPP(hAPP(V_prod_2,c_RealVector_OscaleR__class_OscaleR(T_a,V_r_2,V_a_2)),V_b_2) = c_RealVector_OscaleR__class_OscaleR(T_c,V_r_2,hAPP(hAPP(V_prod_2,V_a_2),V_b_2))))) # label(fact_bounded__bilinear_OscaleR__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 277 (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 c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,hAPP(V_X_2,B_n)),B_K)) & 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]. 2.14/2.49 278 (all V_m_2 all V_n_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,V_m_2) <-> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_n_2),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]. 2.14/2.49 279 (all V_b all V_a c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Complex_Ocomplex_OComplex(V_a,V_b)) = c_Complex_Ocomplex_OComplex(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_a),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_b))) # label(fact_complex__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 280 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) <-> V_x_2 = V_y_2)))) # label(fact_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 281 (all V_n c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_n)) # label(fact_le__refl) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 282 (all V_b_2 all V_a_2 all V_P_2 (hBOOL(hAPP(V_P_2,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_a_2,V_b_2))) <-> -((exists B_d (-hBOOL(hAPP(V_P_2,B_d)) & V_a_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_b_2,B_d))) | -hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) & c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_a_2,V_b_2)))) # label(fact_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 283 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b)),c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_a))))) # label(fact_abs__triangle__ineq2__sym) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 284 (all V_y_2 all V_x_2 all V_b_2 all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_b_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x_2,V_y_2) <-> 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)))))) # label(fact_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 285 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_m)),V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n))) # label(fact_mult__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 286 (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]. 2.14/2.49 287 (all T_2 all T_1 (class_Groups_Ouminus(T_1) -> class_Groups_Ouminus(tc_fun(T_2,T_1)))) # label(arity_fun__Groups_Ouminus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 288 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) & c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) | c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) & c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.49 289 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) <-> (all B_x 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]. 2.14/2.49 290 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 291 (all V_m all V_n (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_m) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Nat_OSuc(V_m)) -> V_n = V_m))) # label(fact_less__antisym) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 292 (all V_a_2 all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a_2),c_Groups_Ozero__class_Ozero(T_a)) <-> c_Groups_Ozero__class_Ozero(T_a) = V_a_2))) # label(fact_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 293 (all V_x all V_z all V_d (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_d) -> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_z,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_x,V_z)),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_d))))) # label(fact_incr__lemma) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 294 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_z))))) # label(fact_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 295 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)))) # label(fact_le__imp__less__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 296 (all V_d all V_b all V_c all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),V_c) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oabs__class_Oabs(T_a,V_b),V_d) -> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Groups_Oabs__class_Oabs(T_a,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_d)))))) # label(fact_abs__mult__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 297 (all V_m c_Nat_OSuc(V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 298 (all V_y all V_x (c_Complex_ORe(V_y) = c_Complex_ORe(V_x) -> (c_Complex_OIm(V_y) = c_Complex_OIm(V_x) -> V_x = V_y))) # label(fact_complex__equality) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 299 (all V_n all V_m (c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) = V_m -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n)) # label(fact_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 300 (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]. 2.14/2.49 301 (all V_c all V_a all V_b all T_a (class_Rings_Oordered__ring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.49 302 (all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 303 (all V_y all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,V_y) -> 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]. 2.14/2.49 304 (all V_y_2 all V_x_2 (V_x_2 != V_y_2 & c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x_2,V_y_2))) # label(fact_real__less__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 305 (all V_m_2 all V_n_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n_2,V_m_2)))) # label(fact_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 306 (all V_y all V_x all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_Groups_Ominus__class_Ominus(T_a,V_x,V_y)) = c_Groups_Ominus__class_Ominus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x),c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_y)))) # label(fact_scaleR__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 307 (all V_n all V_x (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> c_Transcendental_Oln(hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealDef_Oreal(tc_Nat_Onat,V_n)),c_Transcendental_Oln(V_x)))) # label(fact_ln__realpow) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 308 (all V_z all V_y all V_x c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_y,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_x,V_z)) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_x,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_y,V_z))) # label(fact_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 309 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> 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]. 2.14/2.49 310 (all V_y_2 all V_x_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x_2,V_y_2),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) <-> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_y_2,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_x_2)))) # label(fact_real__add__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 311 (all V_y all V_x all T_a (class_Rings_Olinordered__ring(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),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]. 2.14/2.49 312 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_d) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.49 313 (all V_b all V_x all V_a (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_a,V_x) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,V_b) -> (exists B_d ((all B_y (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,B_y)),B_d) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B_y,V_b) & c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_a,B_y))) & c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_d)))))) # label(fact_lemma__interval) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 314 (all V_a all V_b all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (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)) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> 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]. 2.14/2.49 315 (all V_y all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K ((all B_x 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))) & c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K))))) # label(fact_mult__left_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 316 (all V_xa all V_y all V_x all T_a (class_RealVector_Oreal__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,V_y),V_xa) = c_Groups_Ominus__class_Ominus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_x,V_xa),c_RealVector_OscaleR__class_OscaleR(T_a,V_y,V_xa)))) # label(fact_scaleR__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 317 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_m_2),V_n_2))) # label(fact_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 318 (all V_y all V_x c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Complex_ORe(V_x),c_Complex_ORe(V_y)) = c_Complex_ORe(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,V_x,V_y))) # label(fact_Re_Odiff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 319 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (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)) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> 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]. 2.14/2.49 320 (all V_z all V_y all V_x hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),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]. 2.14/2.49 321 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_a),c_RealVector_Onorm__class_Onorm(T_a,V_b)),c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b))))) # label(fact_norm__diff__ineq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 322 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i,V_j) -> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_k,V_i),c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_k,V_j)))) # label(fact_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 323 (all V_y all V_a all V_x all T_a (class_RealVector_Oreal__algebra(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_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),c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_y)))) # label(fact_mult__scaleR__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 324 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2),V_a_2)))) # label(fact_minus__le__self__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 325 (all V_y_2 all V_x_2 (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x_2,V_y_2),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2,V_y_2))) # label(fact_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 326 (all V_x 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]. 2.14/2.49 327 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_z,V_x))))) # label(fact_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 328 (all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) = c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n))) # label(fact_add__Suc__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 329 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),V_n)))) # label(fact_zero__le__power__abs) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 330 (all V_n all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),V_n) = c_Groups_Oone__class_Oone(T_a))) # label(fact_power__one) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 331 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_z,V_x))))) # label(fact_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 332 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,c_Nat_OSuc(V_m_2)) <-> -c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_not__less__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 333 (all V_b all V_a all V_y all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_a)),c_Groups_Ominus__class_Ominus(T_a,V_y,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_a)),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ominus__class_Ominus(T_a,V_y,V_b))) = c_Groups_Ominus__class_Ominus(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_a),V_b)))) # label(fact_mult_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 334 (all V_k_2 all V_f_2 all V_n_2 ((all B_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_i,V_n_2) -> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,hAPP(V_f_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B_i,c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(V_f_2,B_i))),c_Groups_Oone__class_Oone(tc_Int_Oint)))) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_k_2,hAPP(V_f_2,V_n_2)) -> (exists B_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_i,V_n_2) & hAPP(V_f_2,B_i) = V_k_2)))))) # label(fact_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 335 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless(T_a,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]. 2.14/2.49 336 (all V_l_2 all V_k_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_l_2) <-> (exists B_n V_l_2 = 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]. 2.14/2.49 337 (all V_b_2 all V_a_2 all V_c_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2),c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_b_2)) <-> 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]. 2.14/2.49 338 (all V_y all V_x (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_x) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_y) -> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),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]. 2.14/2.49 339 (all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (exists B_K all B_a all B_b c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(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(tc_RealDef_Oreal,B_a)),c_RealVector_Onorm__class_Onorm(T_a,B_b))),B_K))))) # label(fact_scaleR_Obounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 340 (exists B_K (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Complex_ORe(B_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B_x)),B_K))))) # label(fact_Re_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 341 (all T_a (class_Rings_Olinordered__idom(T_a) -> c_Groups_Oone__class_Oone(T_a) = c_Groups_Oabs__class_Oabs(T_a,c_Groups_Oone__class_Oone(T_a)))) # label(fact_abs__one) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 342 (all V_y_2 all V_x_2 all T_a (class_Lattices_Oboolean__algebra(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_x_2),c_Groups_Ouminus__class_Ouminus(T_a,V_y_2))))) # label(fact_compl__le__compl__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 343 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 344 (all V_n -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]. 2.14/2.49 345 (all V_nat_H_1 c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != c_Nat_OSuc(V_nat_H_1)) # label(fact_nat_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 346 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a != V_b -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> c_Orderings_Oord__class_Oless(T_a,V_b,V_a))))) # label(fact_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 347 (all V_a_2 all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Groups_Oabs__class_Oabs(T_a,V_a_2) = c_Groups_Ozero__class_Ozero(T_a) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 348 (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),c_Nat_OSuc(V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> V_b = V_a))))) # label(fact_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 349 (all V_b all V_a all V_y all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,V_a),c_Groups_Ominus__class_Ominus(T_a,V_y,V_b)),c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,V_a),V_b)),c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_Groups_Ominus__class_Ominus(T_a,V_y,V_b))) = c_Groups_Ominus__class_Ominus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_x,V_y),c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b)))) # label(fact_scaleR_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 350 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> (V_c = V_b -> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a))))) # label(fact_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 351 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a != V_b -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 352 (all V_a c_Complex_Orcis(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_a) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)) # label(fact_rcis__zero__mod) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 353 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) = 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))) # label(fact_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 354 (all V_y all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_x 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_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 355 (all V_n all V_m all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a) -> (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)) -> 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]. 2.14/2.49 356 (all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Ouminus__class_Ouminus(T_a,V_a_2))))) # label(fact_neg__0__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 357 (all V_n_2 all V_m_2 (c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> V_n_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Nat_OSuc(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__1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 358 (all V_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Complex_OIm(V_x)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_x))) # label(fact_abs__Im__le__cmod) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 359 (all V_b all V_a all V_r all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_RealVector_OscaleR__class_OscaleR(T_a,V_r,V_a)),V_b) = c_RealVector_OscaleR__class_OscaleR(T_a,V_r,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_mult_OscaleR__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 360 (all V_m all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_m)))) # label(fact_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 361 (all V_n all V_b all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_n) = 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)))) # label(fact_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 362 (all V_w all V_z2 all V_z1 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),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]. 2.14/2.49 363 (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 | c_Groups_Ozero__class_Ozero(T_a) != V_x_2 <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),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]. 2.14/2.49 364 (all V_z all V_y all V_x c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,V_z))) # label(fact_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 365 (all V_c all V_b all V_a all T_a (class_Groups_Oab__semigroup__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) = c_Groups_Oplus__class_Oplus(T_a,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]. 2.14/2.49 366 (all V_z all V_x all V_d (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_d) -> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_x,V_z)),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_d)),V_z))) # label(fact_decr__lemma) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 367 (all V_x c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Complex_OIm(V_x)) = c_Complex_OIm(c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,V_x))) # label(fact_Im_Ominus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 368 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Groups_Oabs__class_Oabs(T_a,V_a) = c_Groups_Oabs__class_Oabs(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a)))) # label(fact_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 369 (all V_y_2 all V_x_2 all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x_2) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y_2) -> (c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Oplus__class_Oplus(T_a,V_x_2,V_y_2) <-> 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]. 2.14/2.49 370 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (V_a != V_b -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 371 (all V_i all V_j all V_k (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j) -> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k),V_i) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,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]. 2.14/2.49 372 (all V_x_2 all T_a (class_Groups_Oone(T_a) -> (c_Groups_Oone__class_Oone(T_a) = V_x_2 <-> c_Groups_Oone__class_Oone(T_a) = V_x_2))) # label(fact_one__reorient) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 373 (all V_n_2 all V_m_2 all V_k_2 ((c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)) <-> 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]. 2.14/2.49 374 (all V_x_2 all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) -> 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]. 2.14/2.49 375 (all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_a all B_b 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))))))) # label(fact_mult_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 376 (all V_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Complex_ORe(V_x),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_x))) # label(fact_complex__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 377 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) | V_m = V_n -> 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]. 2.14/2.49 378 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> c_Groups_Ominus__class_Ominus(T_a,V_b,V_a) = c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)))) # label(fact_minus__diff__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 379 (all V_n -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_n),V_n)) # label(fact_Suc__n__not__le__n) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 380 (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]. 2.14/2.49 381 (all V_m all V_j all V_k (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k))) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),c_Nat_OSuc(V_j)))) # label(fact_diff__Suc__diff__eq1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 382 (all T_a (class_Rings_Olinordered__semidom(T_a) -> 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]. 2.14/2.49 383 (all V_r all V_q all V_r_H all V_q_H all V_b (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,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),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q),V_r)) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_r_H) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_r_H,V_b) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_r,V_b) -> 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]. 2.14/2.49 384 (all V_b_2 all V_c_2 all V_a_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2),c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_c_2)) <-> c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)))) # label(fact_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 385 (all V_n all V_m all V_k (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_m) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_k),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,V_k)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)))) # label(fact_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 386 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Ozero__class_Ozero(T_a) != c_Groups_Oone__class_Oone(T_a))) # label(fact_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 387 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i,V_j) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_j,V_k) -> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_i,V_k)))) # label(fact_zle__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 388 (all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> 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]. 2.14/2.49 389 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m),c_Nat_OSuc(V_n)))) # label(fact_Suc__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 390 (all V_nat_H_2 all V_nat_2 (V_nat_2 = V_nat_H_2 <-> c_Nat_OSuc(V_nat_2) = c_Nat_OSuc(V_nat_H_2))) # label(fact_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 391 (all V_b_2 all V_a_2 all V_c_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_a_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2)))))) # label(fact_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 2.14/2.49 392 (all V_x (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),V_x) -> 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]. 2.14/2.50 393 (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]. 2.14/2.50 394 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,c_Nat_OSuc(V_n_2)))) # label(fact_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 395 (all V_z all V_y all V_x (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,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]. 2.14/2.50 396 (all V_b all V_c all V_a all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) -> 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]. 2.14/2.50 397 (all V_za_2 all V_wa_2 (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_wa_2,V_za_2) <-> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_wa_2,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_za_2,c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 398 (all V_y all V_x all V_xa all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> 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)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),c_Groups_Ominus__class_Ominus(T_a,V_x,V_y)))) # label(fact_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 399 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_i,V_j) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_k) -> 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]. 2.14/2.50 400 (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]. 2.14/2.50 401 (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) -> V_b = c_Groups_Ozero__class_Ozero(T_a) | c_Groups_Ozero__class_Ozero(T_a) = V_a))) # label(fact_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 402 (all V_b_2 all V_a_2 all V_c_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) & c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2) | c_Orderings_Oord__class_Oless(T_a,V_b_2,V_a_2) & c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2))))) # label(fact_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 403 (all V_n c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),V_n) = c_Nat_OSuc(V_n)) # label(fact_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 404 (all V_b_2 all V_a_2 all V_c_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2),c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_b_2))))) # label(fact_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 405 (all V_prod_2 all T_a all T_b all T_c (class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_c) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> (exists B_K (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_a all B_b c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_c,hAPP(hAPP(V_prod_2,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_b,B_b))),B_K)))))))) # label(fact_bounded__bilinear_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 406 (all V_r all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (exists B_K ((all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_r,B_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,B_x)),B_K))) & c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K))))) # label(fact_scaleR__right_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 407 (all V_a_2 all V_x_2 (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_a_2,c_RComplete_Onatfloor(V_x_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_a_2),V_x_2)))) # label(fact_le__natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 408 (all V_za_2 all V_wa_2 (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_wa_2,c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_za_2,c_Groups_Oone__class_Oone(tc_Int_Oint))) <-> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_wa_2,V_za_2))) # label(fact_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 409 (all V_w c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_w,V_w)) # label(fact_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 410 (all V_i_2 all V_j_2 all V_k_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_j_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_k_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i_2,V_k_2),V_j_2)))) # label(fact_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 411 (all V_n all V_m (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n) -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,V_m) -> V_n = V_m))) # label(fact_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 412 (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_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))) = 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_I14_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 413 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x))))) # label(fact_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 414 (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),c_Nat_OSuc(V_n_2))) # label(fact_power_Opower_Opower__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 415 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ominus__class_Ominus(T_a,V_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b)) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_b))) # label(fact_diff__minus__eq__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 416 (all V_n 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]. 2.14/2.50 417 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> 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]. 2.14/2.50 418 (all V_x all V_b all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_a,V_b),V_x) = c_Groups_Oplus__class_Oplus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x),c_RealVector_OscaleR__class_OscaleR(T_a,V_b,V_x)))) # label(fact_real__vector_Oscale__left__distrib) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 419 (all V_a all T_a (class_Rings_Omult__zero(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_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 420 (all V_X_2 all V_K_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_K_2) -> ((all B_n 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]. 2.14/2.50 421 (all V_n all V_k all V_m c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_k)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)) # label(fact_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 422 (all V_d all V_c all V_b all V_a c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Complex_Ocomplex_OComplex(V_a,V_b),c_Complex_Ocomplex_OComplex(V_c,V_d)) = c_Complex_Ocomplex_OComplex(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_a,V_c),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_b,V_d))) # label(fact_complex__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 423 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b),V_b) = V_a)) # label(fact_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 424 (all V_a all T_a (class_Groups_Oab__group__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_ab__left__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 425 (all V_x all V_a (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_a),V_x) -> c_RComplete_Onatfloor(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,c_RealDef_Oreal(tc_Nat_Onat,V_a))) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_RComplete_Onatfloor(V_x),V_a))) # label(fact_natfloor__subtract) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 426 (all V_z all V_y all V_x (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,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]. 2.14/2.50 427 (all V_n all V_x (V_x = c_RealDef_Oreal(tc_Nat_Onat,c_RComplete_Onatfloor(V_x)) -> c_RComplete_Onatfloor(hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),V_x),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_RComplete_Onatfloor(V_x)),V_n))) # label(fact_natfloor__power) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 428 (all V_x (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Transcendental_Oln(V_x)) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> 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]. 2.14/2.50 429 (all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K ((all B_a all B_b 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))) & 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]. 2.14/2.50 430 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)))))) # label(fact_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 431 (all V_real2_H_2 all V_real1_H_2 all V_real2_2 all V_real1_2 (c_Complex_Ocomplex_OComplex(V_real1_2,V_real2_2) = c_Complex_Ocomplex_OComplex(V_real1_H_2,V_real2_H_2) <-> V_real1_2 = V_real1_H_2 & V_real2_H_2 = V_real2_2)) # label(fact_complex_Oinject) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 432 (all V_m all V_n c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m))) # label(fact_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 433 (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]. 2.14/2.50 434 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c))))) # label(fact_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 435 (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]. 2.14/2.50 436 (all V_n all V_m c_Groups_Oplus__class_Oplus(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,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n))) # label(fact_real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 437 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) -> 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]. 2.14/2.50 438 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),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]. 2.14/2.50 439 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_left__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 440 (all V_k all V_j all V_i c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,V_k),V_j) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,V_j),V_k)) # label(fact_diff__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 441 (all V_r_H all V_q_H all V_b_H (c_Orderings_Oord__class_Oless(tc_Int_Oint,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)) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_r_H) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b_H) -> 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]. 2.14/2.50 442 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (V_b = V_c -> c_Orderings_Oord__class_Oless(T_a,V_a,V_c))))) # label(fact_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 443 (all V_real2_2 all V_real1_2 all V_f1_2 all T_a c_Complex_Ocomplex_Ocomplex__rec(T_a,V_f1_2,c_Complex_Ocomplex_OComplex(V_real1_2,V_real2_2)) = hAPP(hAPP(V_f1_2,V_real1_2),V_real2_2)) # label(fact_complex_Orecs) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 444 (all V_r all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (exists B_K ((all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_r,B_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_a,B_x)),B_K))) & c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K))))) # label(fact_scaleR__right_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 445 (all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (exists B_K all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,B_x,V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B_x)),B_K))))) # label(fact_scaleR__left_Obounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 446 (all V_x_2 (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = V_x_2 <-> -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]. 2.14/2.50 447 (all V_w all V_z c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_z,V_w) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,V_w))) # label(fact_diff__int__def__symmetric) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 448 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.50 449 (all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 450 (all V_a_2 all V_prod_2 all T_c all T_a all T_b (class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_c) & class_RealVector_Oreal__normed__vector(T_b) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(V_prod_2,V_a_2),c_Groups_Ozero__class_Ozero(T_b))))) # label(fact_bounded__bilinear_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 451 (all V_b_2 all V_a_2 all T_a (class_Rings_Oring__no__zero__divisors(T_a) -> (V_b_2 = c_Groups_Ozero__class_Ozero(T_a) | V_a_2 = c_Groups_Ozero__class_Ozero(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_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 452 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 453 (all V_n all V_x all T_a (class_RealVector_Oreal__normed__algebra__1(T_a) -> 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]. 2.14/2.50 454 (all V_n all V_m c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n),V_n) = V_m) # label(fact_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 455 (all V_m all V_n (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) = c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)))) # label(fact_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 456 (all V_b_2 all V_a_2 all V_P_2 ((all B_d (V_a_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_b_2,B_d) -> hBOOL(hAPP(V_P_2,B_d)))) & (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_a_2,V_b_2) -> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) <-> hBOOL(hAPP(V_P_2,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_a_2,V_b_2))))) # label(fact_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 457 (all V_y all V_x c_Complex_ORe(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),V_x),V_y)) = c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Complex_ORe(V_x)),c_Complex_ORe(V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Complex_OIm(V_x)),c_Complex_OIm(V_y)))) # label(fact_complex__Re__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 458 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> c_Orderings_Oord__class_Oless__eq(T_a,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]. 2.14/2.50 459 (all V_b_2 all V_a_2 all V_c_2 (V_c_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_a_2),V_c_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_b_2),V_c_2)))) # label(fact_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 460 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> -c_Orderings_Oord__class_Oless(T_a,V_y,V_x)))) # label(fact_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 461 (all V_y all V_x c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Complex_OIm(V_x),c_Complex_OIm(V_y)) = c_Complex_OIm(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,V_x,V_y))) # label(fact_Im_Oadd) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 462 (all V_y all V_x c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,V_y) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_y))) # label(fact_minus__real__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 463 (all V_c 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),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)),V_c) = 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)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 464 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (V_x_2 = V_y_2 <-> c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) & 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]. 2.14/2.50 465 (all V_b all V_a (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_a) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_b) -> 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]. 2.14/2.50 466 (all V_x_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_RealVector_Onorm__class_Onorm(T_a,V_x_2) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) <-> V_x_2 = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_norm__eq__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 467 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,c_Nat_OSuc(V_n_2)) <-> V_m_2 = c_Nat_OSuc(V_n_2) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2))) # label(fact_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 468 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | 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]. 2.14/2.50 469 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.50 470 (all V_X_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> ((exists B_k ((exists B_N all B_n c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(V_X_2,B_n),c_Groups_Ouminus__class_Ouminus(T_a,hAPP(V_X_2,B_N)))),B_k)) & c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_k))) <-> c_SEQ_OBseq(T_a,V_X_2)))) # label(fact_Bseq__iff3) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 471 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a = V_b -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_b) -> c_Orderings_Oord__class_Oless(T_a,V_c,V_a))))) # label(fact_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 472 (all V_b_2 all V_x_2 all V_a_2 all T_a (class_RealVector_Oreal__vector(T_a) -> (V_a_2 = V_b_2 | c_Groups_Ozero__class_Ozero(T_a) = V_x_2 <-> c_RealVector_OscaleR__class_OscaleR(T_a,V_a_2,V_x_2) = c_RealVector_OscaleR__class_OscaleR(T_a,V_b_2,V_x_2)))) # label(fact_scaleR__cancel__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 473 (exists B_K all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Complex_ORe(B_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B_x)),B_K))) # label(fact_Re_Obounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 474 (all V_n all V_x 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),c_Groups_Oabs__class_Oabs(tc_Int_Oint,V_x)),V_n))) # label(fact_zero__le__zpower__abs) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 475 (all V_t all V_s (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_s,V_t) -> V_t != V_s)) # label(fact_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 476 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2),V_a_2) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2)))) # label(fact_neg__less__nonneg) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 477 (all V_xa all V_x all T_a (class_RealVector_Oreal__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_x),V_xa) = c_Groups_Ouminus__class_Ouminus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_x,V_xa)))) # label(fact_scaleR__left_Ominus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 478 (all V_b all V_x all V_a (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_a,V_x) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,V_b) -> (exists B_d (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_d) & (all B_y (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,B_y)),B_d) -> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B_y,V_b) & c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_a,B_y)))))))) # label(fact_lemma__interval__lt) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 479 (all V_real2 all V_real1 c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Complex_Ocomplex_Ocomplex__size(c_Complex_Ocomplex_OComplex(V_real1,V_real2))) # label(fact_complex_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 480 (all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (exists B_K (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,B_x,V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B_x)),B_K))))))) # label(fact_scaleR__left_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 481 (all V_x 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]. 2.14/2.50 482 (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_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 483 (all V_y all V_x all V_r all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_RealVector_OscaleR__class_OscaleR(T_a,V_r,V_x)),V_y) = c_RealVector_OscaleR__class_OscaleR(T_a,V_r,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)))) # label(fact_mult__left_OscaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 484 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b),c_Groups_Ouminus__class_Ouminus(T_a,V_a))))) # label(fact_le__imp__neg__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 485 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_m_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k_2,V_n_2) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m_2,V_k_2),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n_2,V_k_2)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2))))) # label(fact_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 486 (all V_m all V_n c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n))) # label(fact_le__add2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 487 (all V_b_H all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_b_H)) = c_Groups_Ominus__class_Ominus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b),c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b_H)))) # label(fact_scaleR_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 488 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a) -> 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]. 2.14/2.50 489 (all V_m_2 all V_n_2 (c_RealDef_Oreal(tc_Nat_Onat,V_n_2) = c_RealDef_Oreal(tc_Nat_Onat,V_m_2) <-> V_m_2 = V_n_2)) # label(fact_real__of__nat__inject) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 490 (all V_xa all V_x all V_r all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,V_r,V_x),V_xa) = c_RealVector_OscaleR__class_OscaleR(T_a,V_r,c_RealVector_OscaleR__class_OscaleR(T_a,V_x,V_xa)))) # label(fact_scaleR__left_OscaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 491 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_le__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 492 (all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Transcendental_Oln(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]. 2.14/2.50 493 (all V_b_2 all V_a_2 all T_a (class_Groups_Ogroup__add(T_a) -> (c_Groups_Ouminus__class_Ouminus(T_a,V_b_2) = V_a_2 <-> c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_b_2) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_eq__neg__iff__add__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 494 (all V_n c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 495 (all V_prod_2 all T_a all T_b all T_c (class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_c) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> (exists B_K all B_a all B_b c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_c,hAPP(hAPP(V_prod_2,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_b,B_b))),B_K)))))) # label(fact_bounded__bilinear_Obounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 496 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Nat_OSuc(V_k_2)),V_n_2)))) # label(fact_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 497 (all V_a_2 all V_x_2 (c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x_2,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_a_2)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) <-> V_x_2 = V_a_2)) # label(fact_real__add__minus__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 498 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (V_y != V_x -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x))))) # label(fact_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 499 (all V_s all V_y all V_r all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_x),V_r) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_y),V_s) -> 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]. 2.14/2.50 500 (all V_b_2 all V_a_2 all T_a (class_Groups_Ogroup__add(T_a) -> (V_a_2 = c_Groups_Ouminus__class_Ouminus(T_a,V_b_2) <-> c_Groups_Ouminus__class_Ouminus(T_a,V_a_2) = V_b_2))) # label(fact_minus__equation__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 501 (all V_x all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_a)),c_RealVector_Onorm__class_Onorm(T_a,V_x)) = c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x)))) # label(fact_norm__scaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 502 (all V_y all V_x c_Complex_OIm(c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,V_x,V_y)) = c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Complex_OIm(V_x),c_Complex_OIm(V_y))) # label(fact_Im_Odiff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 503 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> V_a = c_Groups_Oabs__class_Oabs(T_a,V_a)))) # label(fact_abs__of__nonneg) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 504 (all V_n all V_m all V_l all V_k (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_k,V_l) -> (c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_l) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)))) # label(fact_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.50 505 (all V_x all V_n (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_n),V_x) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,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) = 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]. 2.14/2.51 506 (all V_y_2 all V_x_2 (V_y_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) & c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = V_x_2 <-> c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = 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)))) # label(fact_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 507 (all V_n V_n = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 508 (all V_x_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x_2) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x_2,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) <-> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Transcendental_Oln(V_x_2),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))) # label(fact_ln__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 509 (all V_x_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x_2) -> (c_Transcendental_Oln(V_x_2) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) <-> c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = V_x_2))) # label(fact_ln__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 510 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 511 (all V_x c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),c_RealDef_Oreal(tc_Nat_Onat,c_RComplete_Onatfloor(V_x)))) # label(fact_real__natfloor__gt__diff__one) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 512 (all V_a_2 all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a_2)) <-> V_a_2 != c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 513 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) <-> -c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)))) # label(fact_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 514 (all V_w all V_z (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_z,V_w) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_w,V_z) -> V_z = V_w))) # label(fact_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 515 (all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_Onorm__class_Onorm(T_a,V_x) = c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_x)))) # label(fact_norm__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 516 (all V_n all V_m ((V_m = 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),V_n)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_m -> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)))) # label(fact_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 517 (all V_b all V_a all V_x all T_a (class_RealVector_Oreal__vector(T_a) -> (V_x != c_Groups_Ozero__class_Ozero(T_a) -> (c_RealVector_OscaleR__class_OscaleR(T_a,V_b,V_x) = c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x) -> V_a = V_b)))) # label(fact_scaleR__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 518 (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),c_Groups_Ouminus__class_Ouminus(T_a,V_x)),V_y) = c_Groups_Ouminus__class_Ouminus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)))) # label(fact_mult__left_Ominus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 519 (all V_b_2 all V_a_2 all V_c_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2) -> (c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c_2),V_b_2)) <-> 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]. 2.14/2.51 520 (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]. 2.14/2.51 521 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_j,V_k) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_i),V_k)))) # label(fact_less__trans__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 522 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) <-> (exists B_k V_n_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k)))) # label(fact_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 523 (all V_m all V_n (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_m) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n),V_m)))) # label(fact_diff__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 524 (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),c_Groups_Ozero__class_Ozero(T_a)),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 525 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ominus__class_Ominus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 526 (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]. 2.14/2.51 527 (all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.51 528 (all V_l all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_k,V_l) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_l))))) # label(fact_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 529 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.51 530 (all V_m all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_j)))) # label(fact_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 531 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ominus__class_Ominus(T_a,V_a,V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_diff__self) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 532 (all V_x all V_b all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_b,V_x)) = c_RealVector_OscaleR__class_OscaleR(T_a,V_b,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x)))) # label(fact_scaleR__left__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 533 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),V_b))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 534 (all V_n_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 535 (all V_n all V_m (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = V_m -> V_n = c_Groups_Oone__class_Oone(tc_Nat_Onat) | V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 536 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != c_Nat_OSuc(V_m)) # label(fact_Zero__not__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 537 (all V_d all V_b all V_c all V_a all T_a (class_Groups_Oab__group__add(T_a) -> c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b),c_Groups_Ominus__class_Ominus(T_a,V_c,V_d)))) # label(fact_Deriv_Oadd__diff__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 538 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)))))) # label(fact_add__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 539 (all V_m_2 all V_n_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,V_m_2) <-> 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]. 2.14/2.51 540 (all V_c all V_b all V_a all T_a (class_Groups_Ocancel__ab__semigroup__add(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,V_a,V_b) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c) -> V_c = V_b))) # label(fact_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 541 (all V_b_2 all V_a_2 all T_a (class_Groups_Oab__group__add(T_a) -> (c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) = c_Groups_Ozero__class_Ozero(T_a) <-> V_b_2 = V_a_2))) # label(fact_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 542 (all V_y all V_x all T_a (class_RealVector_Odist__norm(T_a) -> c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_x,V_y)) = c_RealVector_Odist__class_Odist(T_a,V_x,V_y))) # label(fact_dist__norm) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 543 (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]. 2.14/2.51 544 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n))) # label(fact_Suc__le__lessD) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 545 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> 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]. 2.14/2.51 546 (all V_w all V_z c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z,V_w) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_w,V_z)) # label(fact_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 547 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_b = V_a -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c))))) # label(fact_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 548 (all V_r all V_q all V_r_H all V_q_H all V_b (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,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),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q),V_r)) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_r,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_b,V_r) -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_b,V_r_H) -> 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]. 2.14/2.51 549 (all V_real2 all V_real1 c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Nat_Osize__class_Osize(tc_Complex_Ocomplex,c_Complex_Ocomplex_OComplex(V_real1,V_real2))) # label(fact_complex_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 550 (all V_d all V_c all V_b all V_a c_Complex_Ocomplex_OComplex(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_b),V_d)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_a),V_d),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_b),V_c))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Complex_Ocomplex_OComplex(V_a,V_b)),c_Complex_Ocomplex_OComplex(V_c,V_d))) # label(fact_complex__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 551 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_m_2) -> (c_Groups_Oone__class_Oone(tc_Int_Oint) = V_n_2 & c_Groups_Oone__class_Oone(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)))) # label(fact_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 552 (all V_x_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (V_x_2 != c_Groups_Ozero__class_Ozero(T_a) <-> 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))))) # label(fact_zero__less__norm__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 553 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (V_y != V_x -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x))))) # label(fact_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 554 (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]. 2.14/2.51 555 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2) -> (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)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)))) # label(fact_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 556 (all V_n all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,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),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]. 2.14/2.51 557 (all V_y_2 all V_x_2 (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_x_2),V_y_2) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x_2,V_y_2)))) # label(fact_real__0__le__add__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 558 (all V_n all V_x (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_x) -> 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]. 2.14/2.51 559 (all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K all B_x 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_Obounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 560 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ouminus__class_Ouminus(T_a,V_a)),V_b))) # label(fact_minus__mult__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 561 (all V_n_2 all V_m_2 (c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2) <-> 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 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_one__is__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 562 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2) | 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]. 2.14/2.51 563 (all V_n all T_a (class_Power_Opower(T_a) & class_Rings_Osemiring__0(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n) = c_Groups_Ozero__class_Ozero(T_a)) & (V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n) = c_Groups_Oone__class_Oone(T_a)))) # label(fact_power__0__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 564 (all V_m 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]. 2.14/2.51 565 (all V_n_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))) # label(fact_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 566 (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]. 2.14/2.51 567 (all V_b all V_a all V_c all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (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)) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 568 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ouminus__class_Ouminus(T_a,V_a)),c_Groups_Ouminus__class_Ouminus(T_a,V_b)))) # label(fact_minus__mult__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 569 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_N) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Oone__class_Oone(T_a)) -> 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]. 2.14/2.51 570 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> V_x = V_y | c_Orderings_Oord__class_Oless(T_a,V_y,V_x) | c_Orderings_Oord__class_Oless(T_a,V_x,V_y))) # label(fact_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 571 (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) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_u) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_v) -> (c_Groups_Oplus__class_Oplus(T_a,V_u,V_v) = c_Groups_Oone__class_Oone(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,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]. 2.14/2.51 572 (all V_b all V_a all V_c all T_a (class_Rings_Olinordered__semiring(T_a) -> (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)) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 573 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m_2),c_Nat_OSuc(V_n_2)))) # label(fact_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 574 (all V_b_2 all V_a_2 (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = V_b_2 & c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = V_a_2 <-> c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = c_Complex_Ocomplex_OComplex(V_a_2,V_b_2))) # label(fact_Complex__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 575 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (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)) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 576 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (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]. 2.14/2.51 577 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> -c_Orderings_Oord__class_Oless(T_a,V_y,V_x)))) # label(fact_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 578 (all V_b all V_a all T_a (class_Lattices_Oab__semigroup__idem__mult(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_mult__left__idem) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 579 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),V_n) = c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__abs) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 580 (all V_x c_RealDef_Oreal(tc_Nat_Onat,V_x) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_x))) # label(fact_abs__real__of__nat__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 581 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_i,V_j) -> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_j,V_k)))) # label(fact_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 582 (all V_x_2 all T_a (class_Groups_Ozero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_x_2 <-> V_x_2 = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 583 (all V_X_2 all T_a (class_Orderings_Oorder(T_a) -> ((all B_m all B_n (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m,B_n) -> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(V_X_2,B_m),hAPP(V_X_2,B_n)))) <-> c_SEQ_Oincseq(T_a,V_X_2)))) # label(fact_incseq__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 584 (all V_z c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = V_z) # label(fact_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 585 (all V_b_2 all V_a_H_2 all V_a_2 all V_prod_2 all T_b all T_c all T_a (class_RealVector_Oreal__normed__vector(T_c) & class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_a) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> c_Groups_Oplus__class_Oplus(T_c,hAPP(hAPP(V_prod_2,V_a_2),V_b_2),hAPP(hAPP(V_prod_2,V_a_H_2),V_b_2)) = hAPP(hAPP(V_prod_2,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_H_2)),V_b_2)))) # label(fact_bounded__bilinear_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 586 (all V_x -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),V_x)) # label(fact_abs__add__one__not__less__self) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 587 (all V_n V_n != c_Nat_OSuc(V_n)) # label(fact_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 588 (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]. 2.14/2.51 589 (all V_m all V_l all V_y all V_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,V_y),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_l),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_m)))),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_l))),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_y,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_m)))))) # label(fact_abs__sum__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 590 (all V_k all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_k)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n),V_k)) # label(fact_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 591 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> -c_Orderings_Oord__class_Oless(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_abs__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 592 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Nat_OSuc(V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),c_Nat_OSuc(V_n))) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> 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]. 2.14/2.51 593 (all V_y all V_x all T_a (class_Rings_Olinordered__ring(T_a) -> -c_Orderings_Oord__class_Oless(T_a,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]. 2.14/2.51 594 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Groups_Oabs__class_Oabs(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b))) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b)))) # label(fact_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 595 (all V_b all V_a c_Complex_Oexpi(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,V_a,V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Complex_Oexpi(V_a)),c_Complex_Oexpi(V_b))) # label(fact_expi__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 596 (all V_n_2 (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)) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_real__of__nat__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 597 (exists B_K (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Complex_ORe(B_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B_x)),B_K))))) # label(fact_Re_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 598 (all V_n all V_m all V_k c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_m),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)) # label(fact_diff__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 599 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring(T_a) -> 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),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]. 2.14/2.51 600 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.51 601 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> -c_Orderings_Oord__class_Oless(T_a,V_y,V_x)))) # label(fact_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 602 (all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a_2),V_b_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2),V_b_2) & c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)))) # label(fact_abs__le__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 603 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_d) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.51 604 (all V_c all V_b all V_a all T_a (class_Rings_Oordered__comm__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.51 605 (all V_z2 all V_z1 all V_w 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)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z1,V_z2))) # label(fact_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 606 (all V_c all V_a all V_b all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.51 607 (all V_q all V_r all V_a (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_a) -> (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 -> (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_r,V_a) -> 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]. 2.14/2.51 608 (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]. 2.14/2.51 609 (all V_n V_n = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_diff__Suc__1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 610 (all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_minus__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 611 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oring(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2) = 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) <-> V_c_2 = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_b_2,V_a_2)),V_e_2),V_d_2)))) # label(fact_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 612 (all V_y all V_x all T_a (class_RealVector_Ometric__space(T_a) -> -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(T_a,V_x,V_y),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))) # label(fact_dist__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 613 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a),V_b)))) # label(fact_abs__le__D2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 614 (all V_b all V_a all V_y all V_x all T_a (class_Rings_Oring(T_a) -> c_Groups_Ominus__class_Ominus(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_a),V_b)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),c_Groups_Ominus__class_Ominus(T_a,V_y,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_x,V_a)),V_b)))) # label(fact_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 615 (all V_n -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]. 2.14/2.51 616 (all V_n_2 all V_a_2 all T_a (class_Power_Opower(T_a) & class_Rings_Ozero__neq__one(T_a) & class_Rings_Ono__zero__divisors(T_a) & class_Rings_Omult__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_a_2 & V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),V_n_2) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 617 (all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Transcendental_Oln(V_x)) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> 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]. 2.14/2.51 618 (all V_n all V_m all V_u all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_i)),V_u),V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u),V_m),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u),V_n)))) # label(fact_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 619 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)))) # label(fact_less__SucI) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 620 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 <-> c_Groups_Oplus__class_Oplus(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]. 2.14/2.51 621 (all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (exists B_K (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_a all B_b c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(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(tc_RealDef_Oreal,B_a)),c_RealVector_Onorm__class_Onorm(T_a,B_b))),B_K))))))) # label(fact_scaleR_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 622 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,c_Nat_OSuc(V_m_2)) <-> V_n_2 = V_m_2))) # label(fact_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 623 (all V_b all V_r2 all V_a all V_r1 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Complex_Orcis(V_r1,V_a)),c_Complex_Orcis(V_r2,V_b)) = c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r1),V_r2),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_a,V_b))) # label(fact_rcis__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 624 (all V_n all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_n) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,V_i)) = V_i)) # label(fact_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 625 (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),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]. 2.14/2.51 626 (all V_i_2 all V_k_2 all V_j_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_k_2),V_i_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j_2,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]. 2.14/2.51 627 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)),V_k) = 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))) # label(fact_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 628 (all V_n_2 all V_m_2 all V_u_2 all V_j_2 all V_i_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2,V_j_2) -> (c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2) <-> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_i_2)),V_u_2),V_n_2) = V_m_2))) # label(fact_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 629 (exists B_K all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Complex_OIm(B_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B_x)),B_K))) # label(fact_Im_Obounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 630 (all V_a 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_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.51 631 (all V_a all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,c_RealDef_Oreal(tc_Nat_Onat,V_a)) -> 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]. 2.14/2.51 632 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oordered__ring(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2),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)) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2)),V_e_2),V_c_2),V_d_2)))) # label(fact_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 633 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,V_a) = c_Groups_Ominus__class_Ominus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a))) # label(fact_diff__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 634 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k)))) # label(fact_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 635 (all V_x all V_a hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_a),V_x) = c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,V_a,V_x)) # label(fact_real__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 636 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),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]. 2.14/2.52 637 (all V_y_2 all V_x_2 (V_y_2 = V_x_2 <-> c_Complex_ORe(V_x_2) = c_Complex_ORe(V_y_2) & c_Complex_OIm(V_y_2) = c_Complex_OIm(V_x_2))) # label(fact_expand__complex__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 638 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_z))))) # label(fact_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 639 (all V_b all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Groups_Oabs__class_Oabs(T_a,V_b)))) # label(fact_abs__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 640 (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]. 2.14/2.52 641 (all V_n all V_k all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n) -> -(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n)))) # label(fact_add__leE) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 642 (all V_n_2 all V_m_2 all V_a_2 all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a_2) -> (V_n_2 = V_m_2 <-> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),V_n_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a_2),V_m_2))))) # label(fact_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 643 (all V_d all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),V_d) = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 644 (all V_z all V_y all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_z,V_x),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_z,V_y)))) # label(fact_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 645 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> 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]. 2.14/2.52 646 (all V_n -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]. 2.14/2.52 647 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_i,V_j) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_j,V_k) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_i,V_k)))) # label(fact_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 648 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) | V_n_2 = V_m_2 <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,c_Nat_OSuc(V_n_2)))) # label(fact_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 649 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a),V_a) -> 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]. 2.14/2.52 650 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)) = c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_a)))) # label(fact_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 651 (all V_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Complex_ORe(V_x)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_x))) # label(fact_abs__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 652 (all V_b_2 all V_a_2 (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = V_b_2 & c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = V_a_2 <-> c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = c_Complex_Ocomplex_OComplex(V_a_2,V_b_2))) # label(fact_Complex__eq__1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 653 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> V_x = V_y | c_Orderings_Oord__class_Oless(T_a,V_x,V_y)))) # label(fact_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 654 (all V_w all V_z2 all V_z1 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)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_z1,V_z2)),V_w)) # label(fact_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 655 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_N) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a) -> 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]. 2.14/2.52 656 (all V_b all V_c all V_a all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b)))) # label(fact_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 657 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> c_Orderings_Oord__class_Oless(T_a,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]. 2.14/2.52 658 (all V_m_2 all V_n_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,V_m_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_n_2),c_Nat_OSuc(V_m_2)))) # label(fact_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 659 (all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> c_Orderings_Oord__class_Oless(T_a,V_a,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]. 2.14/2.52 660 (all V_n all V_m c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n)) # label(fact_diff__Suc__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 661 (all V_m V_m = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 662 (all V_y all V_x all V_c all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_c,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) -> (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]. 2.14/2.52 663 (all V_x all V_a (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealDef_Oreal(tc_Nat_Onat,V_a),V_x) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_RComplete_Onatceiling(V_x),V_a) = c_RComplete_Onatceiling(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_x,c_RealDef_Oreal(tc_Nat_Onat,V_a))))) # label(fact_natceiling__subtract) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 664 (all V_y all V_x c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,V_x,V_y)) = c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,V_x,V_y)) # label(fact_dist__complex__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 665 (all V_w all V_z2 all V_z1 c_Groups_Ominus__class_Ominus(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),c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_z1,V_z2)),V_w)) # label(fact_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 666 (all V_n_2 all V_x_2 (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)) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_x_2))) # label(fact_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 667 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b)) = c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_a),V_b))) # label(fact_scaleR_Ominus__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 668 (all V_b_2 all V_a_2 all T_a (class_Groups_Ogroup__add(T_a) -> (c_Groups_Ouminus__class_Ouminus(T_a,V_b_2) = V_a_2 <-> V_b_2 = c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)))) # label(fact_equation__minus__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 669 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2) <-> -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]. 2.14/2.52 670 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n)) # label(fact_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 671 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oordered__ring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2),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)) <-> c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_b_2,V_a_2)),V_e_2),V_d_2))))) # label(fact_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 672 (all V_m all V_i c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_m)))) # label(fact_less__add__Suc1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 673 (all V_g_2 all V_f_2 ((all B_x hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x)) -> V_f_2 = V_g_2)) # label(fact_ext) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 674 (all V_n_2 all V_m_2 (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)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_m_2) & 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]. 2.14/2.52 675 (all V_za_2 (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_za_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) <-> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_za_2),V_za_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 676 (all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> (exists B_K ((all B_x 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))) & 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]. 2.14/2.52 677 (all V_b_2 all V_a_2 all V_prod_2 all T_b all T_c all T_a (class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_c) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> c_Groups_Ouminus__class_Ouminus(T_c,hAPP(hAPP(V_prod_2,V_a_2),V_b_2)) = hAPP(hAPP(V_prod_2,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)),V_b_2)))) # label(fact_bounded__bilinear_Ominus__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 678 (all V_za_2 all V_x_2 all V_y_2 all V_wa_2 all T_a (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_wa_2),V_y_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x_2),V_za_2)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_wa_2),V_za_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x_2),V_y_2)) <-> V_wa_2 = V_x_2 | V_y_2 = V_za_2))) # label(fact_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 679 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b)) = c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b)))) # label(fact_scaleR_Ominus__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 680 (all V_n 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]. 2.14/2.52 681 (all V_y_2 all V_x_2 all T_a (class_Orderings_Olinorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) -> (V_y_2 = V_x_2 <-> -c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2))))) # label(fact_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 682 (all V_y_2 all V_x_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x_2) -> (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]. 2.14/2.52 683 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.52 684 (all V_X_2 all T_b (class_RealVector_Oreal__normed__vector(T_b) -> ((exists B_N all B_n 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,c_Nat_OSuc(B_N)))) <-> (exists B_K (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_n c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_b,hAPP(V_X_2,B_n)),B_K))))))) # label(fact_lemma__NBseq__def2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 685 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),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]. 2.14/2.52 686 (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 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_k,V_n_2) & hBOOL(hAPP(V_P_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B_k,c_Groups_Oone__class_Oone(tc_Nat_Onat)))) & (all B_i (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]. 2.14/2.52 687 (all V_a_2 all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 688 (all V_prod_2 all T_a all T_b all T_c (class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_c) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> (exists B_K ((all B_a all B_b c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_c,hAPP(hAPP(V_prod_2,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_b,B_b))),B_K))) & c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K)))))) # label(fact_bounded__bilinear_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 689 (all V_y all V_x all V_c (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_c,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_c),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_y))) -> V_x = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))) # label(fact_rabs__ratiotest__lemma) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 690 (all V_b_H_2 all V_b_2 all V_a_2 all V_prod_2 all T_b all T_c all T_a (class_RealVector_Oreal__normed__vector(T_c) & class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_a) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> c_Groups_Ominus__class_Ominus(T_c,hAPP(hAPP(V_prod_2,V_a_2),V_b_2),hAPP(hAPP(V_prod_2,V_a_2),V_b_H_2)) = hAPP(hAPP(V_prod_2,V_a_2),c_Groups_Ominus__class_Ominus(T_b,V_b_2,V_b_H_2))))) # label(fact_bounded__bilinear_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 691 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> (V_a != V_b -> c_Orderings_Oord__class_Oless(T_a,V_b,V_a))))) # label(fact_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 692 (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]. 2.14/2.52 693 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)))) # label(fact_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 694 (all V_k all V_n all V_m c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n),V_k) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n),c_Nat_OSuc(V_k))) # label(fact_Suc__diff__diff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 695 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)))) # label(fact_le__SucI) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 696 (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),c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)) = 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)))) # label(fact_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 697 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)) = c_Groups_Ominus__class_Ominus(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a),c_Groups_Ouminus__class_Ouminus(T_a,V_b)))) # label(fact_Limits_Ominus__diff__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 698 (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),c_Groups_Oplus__class_Oplus(T_a,V_a,V_a_H)),V_b) = 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)))) # label(fact_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 699 (all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n)) # label(fact_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 700 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_y)))) # label(fact_not__leE) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 701 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_b) -> c_Groups_Ouminus__class_Ouminus(T_a,V_a) = V_b))) # label(fact_minus__unique) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 702 (all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_b,V_a)) = c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)))) # label(fact_norm__minus__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 703 (all V_n_2 all V_m_2 (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) <-> c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m_2 & V_n_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 704 (all V_j all V_i -c_Orderings_Oord__class_Oless(tc_Nat_Onat,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]. 2.14/2.52 705 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_add__increasing) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 706 (all V_m c_Nat_OSuc(V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Suc__not__Zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 707 (all V_z V_z = c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,V_z))) # label(fact_zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 708 (all V_b_2 all V_a_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (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)) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) & c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b_2) | c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) & c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 709 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_y)),V_x) = c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_x))))) # label(fact_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 710 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> c_Groups_Oabs__class_Oabs(T_a,V_a) = c_Groups_Ouminus__class_Ouminus(T_a,V_a)))) # label(fact_abs__of__neg) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 711 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oordered__ring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2)),V_e_2),V_c_2),V_d_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2),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]. 2.14/2.52 712 (all V_n all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> 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]. 2.14/2.52 713 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (c_Orderings_Oord__class_Oless(T_a,V_y,V_z) -> c_Orderings_Oord__class_Oless(T_a,V_x,V_z))))) # label(fact_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 714 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != c_Nat_OSuc(V_m)) # label(fact_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 715 (all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),V_x) -> 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]. 2.14/2.52 716 (all V_n_2 all V_m_2 all V_k_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 | 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_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 717 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) -> V_n = V_m))) # label(fact_le__antisym) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 718 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_d) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.52 719 (all V_n_2 all V_k_2 all V_m_2 (c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,V_k_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_k_2) <-> V_m_2 = V_n_2)) # label(fact_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 720 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)))) # label(fact_leI) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 721 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a) -> (V_c_2 != V_d_2 & V_a_2 != V_b_2 <-> 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_c_2)) != c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_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]. 2.14/2.52 722 (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 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_k) & (exists B_x all B_n c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(V_X_2,B_n),c_Groups_Ouminus__class_Ouminus(T_a,B_x))),B_k))))))) # label(fact_Bseq__iff2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 723 (all V_b all V_a all V_c all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b)) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b)))) # label(fact_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 724 (all V_y_2 all V_x_2 (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = V_x_2 & V_y_2 = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) <-> c_Complex_Oii = c_Complex_Ocomplex_OComplex(V_x_2,V_y_2))) # label(fact_Complex__eq__i) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 725 (all V_b all V_a_H all V_a all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> 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)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_a,V_a_H)),V_b))) # label(fact_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 726 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b))))) # label(fact_abs__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 727 (all V_x c_Complex_Ocomplex_OComplex(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Complex_ORe(V_x)),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Complex_OIm(V_x))) = c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,V_x)) # label(fact_complex__minus__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 728 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ominus__class_Ominus(T_a,V_a,V_b) = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b)))) # label(fact_diff__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 729 (all V_x all V_y ((-c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_y,V_x) -> c_Nat__Transfer_Otsub(V_x,V_y) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) & (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_y,V_x) -> c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_x,V_y) = c_Nat__Transfer_Otsub(V_x,V_y)))) # label(fact_tsub__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 730 (all V_x_2 all T_a (class_Rings_Oring__1__no__zero__divisors(T_a) -> (c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x_2),V_x_2) <-> V_x_2 = c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Oone__class_Oone(T_a)) | V_x_2 = c_Groups_Oone__class_Oone(T_a)))) # label(fact_square__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 731 (all V_x all T_a (class_RealVector_Oreal__vector(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x))) # label(fact_scaleR__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 732 (all V_x (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,c_Nat_OSuc(c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_x | c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_x)) # label(fact_nat__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 733 (all V_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_x)),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,V_x))) # label(fact_complex__mod__minus__le__complex__mod) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 734 (all V_b_2 all V_c_2 all V_a_2 all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Ozero__class_Ozero(T_a)) & c_Orderings_Oord__class_Oless(T_a,V_b_2,V_a_2) | c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c_2) & c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_c_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_c_2))))) # label(fact_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 735 (all V_n c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Nat_OSuc(V_n))) # label(fact_zero__less__Suc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 736 (all V_z all V_z_H all V_w all V_w_H (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_w_H,V_w) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_z_H,V_z) -> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_w_H,V_z_H),c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_w,V_z))))) # label(fact_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 737 (all V_i all V_j all V_k (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_j) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_k) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j,V_k)))) # label(fact_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 738 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_a (class_Rings_Oordered__ring(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_e_2),V_c_2),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)) <-> c_Orderings_Oord__class_Oless(T_a,V_c_2,c_Groups_Oplus__class_Oplus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ominus__class_Ominus(T_a,V_b_2,V_a_2)),V_e_2),V_d_2))))) # label(fact_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 739 (all V_g_2 all V_f_2 all T_a all T_b (class_Orderings_Oord(T_b) -> (c_Orderings_Oord__class_Oless(tc_fun(T_a,T_b),V_f_2,V_g_2) <-> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2) & -c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_g_2,V_f_2)))) # label(fact_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 740 (all V_l all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_l) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_l,V_n),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_l,V_m))))) # label(fact_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 741 (all V_n all V_k all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n))) # label(fact_add__leD2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 742 (all V_x hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Complex_Oii),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Complex_Oii),V_x)) = c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,V_x)) # label(fact_complex__i__mult__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 743 (all V_n_2 all V_k_2 all V_m_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_m_2),V_k_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))) # label(fact_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 744 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) -> (-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]. 2.14/2.52 745 (all V_X_2 all T_a (class_Orderings_Oorder(T_a) -> (c_SEQ_Odecseq(T_a,V_X_2) <-> (all B_m all B_n (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m,B_n) -> c_Orderings_Oord__class_Oless__eq(T_a,hAPP(V_X_2,B_n),hAPP(V_X_2,B_m))))))) # label(fact_decseq__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 746 (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]. 2.14/2.52 747 (all V_x all V_r hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),c_Complex_ORe(V_x)) = c_Complex_ORe(c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,V_r,V_x))) # label(fact_complex__Re__scaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 748 (all V_b_H all V_b all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b),c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b_H)) = c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_b,V_b_H)))) # label(fact_scaleR_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 749 (all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_Onorm__class_Onorm(T_a,V_x) = c_RealVector_Odist__class_Odist(T_a,V_x,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_norm__conv__dist) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 750 (all V_y_2 all V_x_2 all V_za_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_za_2) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x_2,V_y_2) <-> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x_2),V_za_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_y_2),V_za_2))))) # label(fact_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 751 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless__eq(T_a,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]. 2.14/2.52 752 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (V_c = V_b -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c))))) # label(fact_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 753 (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]. 2.14/2.52 754 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Nat_OSuc(V_n)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n))) # label(fact_mult__Suc__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 755 (all T_2 all T_1 (class_Groups_Ominus(T_1) -> class_Groups_Ominus(tc_fun(T_2,T_1)))) # label(arity_fun__Groups_Ominus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 756 (all V_y all V_z all V_x all T_a (class_RealVector_Ometric__space(T_a) -> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(T_a,V_x,V_z),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(T_a,V_x,V_y),c_RealVector_Odist__class_Odist(T_a,V_y,V_z))))) # label(fact_dist__triangle) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 757 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m),V_n) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n))) # label(fact_Suc__lessD) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 758 (all V_x all V_b all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> c_Groups_Ominus__class_Ominus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x),c_RealVector_OscaleR__class_OscaleR(T_a,V_b,V_x)) = c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_a,V_b),V_x))) # label(fact_scaleR__left__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 759 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)))))) # label(fact_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 2.14/2.52 760 (all V_X_2 all V_M_2 all T_a all T_b (class_Orderings_Oord(T_a) & class_RealVector_Oreal__normed__vector(T_b) -> ((all B_n (c_Orderings_Oord__class_Oless__eq(T_a,V_M_2,B_n) -> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_b,c_Groups_Ominus__class_Ominus(T_b,hAPP(V_X_2,V_M_2),hAPP(V_X_2,B_n))),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))) -> (all B_n (c_Orderings_Oord__class_Oless__eq(T_a,V_M_2,B_n) -> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_b,hAPP(V_X_2,B_n)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(T_b,hAPP(V_X_2,V_M_2))))))))) # label(fact_lemmaCauchy) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 761 (all V_l all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_l) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_l))))) # label(fact_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 762 (all V_k_2 all V_j_2 all V_i_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2,V_j_2) -> (V_j_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_i_2) <-> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_i_2) = V_k_2))) # label(fact_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 763 (all V_n_2 all V_m_2 all V_u_2 all V_i_2 all V_j_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j_2,V_i_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i_2,V_j_2)),V_u_2),V_m_2),V_n_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2))))) # label(fact_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 764 (all V_m all V_n all T_a (class_Power_Opower(T_a) -> (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_m),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_m),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_m),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,c_Groups_Oone__class_Oone(tc_Nat_Onat))))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_m),V_n)))) # label(fact_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 765 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n))) # label(fact_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 766 (all V_x c_Complex_ORe(c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,V_x)) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Complex_ORe(V_x))) # label(fact_Re_Ominus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 767 (all V_y_2 all V_x_2 all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) & V_y_2 != V_x_2 <-> c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)))) # label(fact_order__less__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 768 (all V_y_2 all V_x_2 all T_a (class_Groups_Ogroup__add(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,V_x_2,V_y_2) = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ouminus__class_Ouminus(T_a,V_x_2) = V_y_2))) # label(fact_add__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 769 (all V_z c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,V_z),V_z)) # label(fact_zadd__zminus__inverse2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 770 (all V_b_2 all V_a_2 all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a_2),V_b_2) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2),V_b_2) & c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)))) # label(fact_abs__less__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 771 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 772 (all V_n all V_z (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_RComplete_Onatfloor(V_z),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) -> 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]. 2.14/2.53 773 (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]. 2.14/2.53 774 (all V_b all V_a all T_a (class_Rings_Oordered__ring__abs(T_a) -> ((c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) | c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))) & (c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) | c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Groups_Oabs__class_Oabs(T_a,V_b)) = c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))) # label(fact_abs__eq__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 775 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 | (exists B_j (V_m_2 = c_Nat_OSuc(B_j) & c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_j,V_n_2))) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,c_Nat_OSuc(V_n_2)))) # label(fact_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 776 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Oone__class_Oone(T_a)) -> 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]. 2.14/2.53 777 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,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)) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2) & c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2))) # label(fact_one__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 778 (all V_x all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x)) = c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_Groups_Ouminus__class_Ouminus(T_a,V_x)))) # label(fact_scaleR__minus__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 779 (all V_m all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_m)))) # label(fact_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 780 (all V_r c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,V_r) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_r)) # label(fact_real__norm__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 781 (all V_d all V_c all V_b all V_a c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,c_Complex_Ocomplex_OComplex(V_a,V_b),c_Complex_Ocomplex_OComplex(V_c,V_d)) = c_Complex_Ocomplex_OComplex(c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_a,V_c),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_b,V_d))) # label(fact_complex__diff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 782 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_a) <-> V_a_2 = c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)))) # label(fact_neg__equal__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 783 (all V_n c_Nat_OSuc(V_n) != V_n) # label(fact_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 784 (all V_n -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]. 2.14/2.53 785 (all V_y all V_x all V_a all T_a (class_RealVector_Oreal__algebra(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x)),V_y))) # label(fact_mult__scaleR__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 786 (all V_b_2 all V_a_2 all T_a (class_Groups_Ogroup__add(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) <-> V_b_2 = V_a_2))) # label(fact_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 787 (all V_n (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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]. 2.14/2.53 788 (all V_n -c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_n)) # label(fact_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 789 (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]. 2.14/2.53 790 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a) -> 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),c_Nat_OSuc(V_n)))))) # label(fact_power__gt1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 791 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ouminus__class_Ouminus(T_a,V_a)),V_n)) = c_Groups_Oabs__class_Oabs(T_a,hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_abs__power__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 792 (all V_k_2 all V_j_2 all V_i_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i_2,V_k_2),V_j_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i_2,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]. 2.14/2.53 793 (all V_n all V_m (c_Groups_Oone__class_Oone(tc_Int_Oint) = c_Groups_Oabs__class_Oabs(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m),V_n)) -> c_Groups_Oone__class_Oone(tc_Int_Oint) = c_Groups_Oabs__class_Oabs(tc_Int_Oint,V_m))) # label(fact_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 794 (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]. 2.14/2.53 795 (all V_n all V_m (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Nat_OSuc(V_m),c_Nat_OSuc(V_n)) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n))) # label(fact_Suc__less__SucD) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 796 (all T_2 all T_1 (class_Lattices_Oboolean__algebra(T_1) -> class_Lattices_Oboolean__algebra(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Oboolean__algebra) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 797 (all V_m all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_j)))) # label(fact_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 798 (all V_x all V_y (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_y,V_x) -> c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_x,V_y) = c_Nat__Transfer_Otsub(V_x,V_y))) # label(fact_tsub__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 799 (all V_y_2 all V_x_2 all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2) & -c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) <-> 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]. 2.14/2.53 800 (all V_n all V_m c_RealDef_Oreal(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = 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))) # label(fact_real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 801 (all V_m all V_n c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m))) # label(fact_le__add1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 802 (all V_l all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_l) -> 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]. 2.14/2.53 803 (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_n_2 & c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_m_2)) # label(fact_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 804 (all V_n_2 all V_m_2 all V_u_2 all V_j_2 all V_i_2 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i_2,V_j_2) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2),V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2),V_n_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_j_2,V_i_2)),V_u_2),V_n_2))))) # label(fact_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 805 (exists B_K (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Complex_OIm(B_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,B_x)),B_K))))) # label(fact_Im_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 806 (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),c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oone__class_Oone(T_a))),V_m) = 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]. 2.14/2.53 807 (all V_y all V_x c_Complex_OIm(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),V_x),V_y)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Complex_ORe(V_x)),c_Complex_OIm(V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Complex_OIm(V_x)),c_Complex_ORe(V_y)))) # label(fact_complex__Im__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 808 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ominus__class_Ominus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b)),c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b))))) # label(fact_abs__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 809 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k) -> 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]. 2.14/2.53 810 (all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_c,V_a) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 811 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> V_a = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a))) # label(fact_add__0__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 812 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> (exists B_k V_n_2 = c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k))))) # label(fact_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 813 (all V_s all V_r c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_r,V_s) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_r,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_s))) # label(fact_real__diff__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 814 (all V_x_2 all V_B_2 all V_A_2 all T_b all T_a (class_Groups_Ominus(T_a) -> hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,T_a),V_A_2,V_B_2),V_x_2) = c_Groups_Ominus__class_Ominus(T_a,hAPP(V_A_2,V_x_2),hAPP(V_B_2,V_x_2)))) # label(fact_minus__apply) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 815 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_y),c_Groups_Ouminus__class_Ouminus(T_a,V_x))))) # label(fact_compl__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 816 (all V_m_2 all V_x_2 (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) <-> V_x_2 = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2)) # label(fact_nat__power__eq__Suc__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 817 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.53 818 (all V_b all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_b) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_scaleR_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 819 (all V_n all V_m all V_u all V_i all V_j (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j,V_i) -> c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u),V_m),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u),V_n)) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,V_j)),V_u),V_m),V_n))) # label(fact_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 820 (all V_x all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_x,c_Groups_Oone__class_Oone(T_a))),c_Groups_Ominus__class_Ominus(T_a,V_x,c_Groups_Oone__class_Oone(T_a))) = 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)))) # label(fact_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 821 (all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) <-> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Ouminus__class_Ouminus(T_a,V_a_2))))) # label(fact_neg__0__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 822 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) -> (c_Orderings_Oord__class_Oless(T_a,V_z,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_z,V_x))))) # label(fact_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 823 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (V_x = V_y -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)))) # label(fact_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 824 (all V_y all V_x all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != V_a -> (c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_y) = c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_x) -> V_y = V_x)))) # label(fact_scaleR__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 825 (all V_nat_H 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]. 2.14/2.53 826 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,c_Nat_OSuc(V_n)) -> (-c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> V_m = c_Nat_OSuc(V_n)))) # label(fact_le__SucE) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 827 (all V_y all V_x V_x = c_Complex_ORe(c_Complex_Ocomplex_OComplex(V_x,V_y))) # label(fact_Re) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 828 (all V_y all V_x c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,V_x,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,V_y)) = c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,V_x,V_y)) # label(fact_complex__diff__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 829 (all V_x (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]. 2.14/2.53 830 (all V_b_2 all V_a_H_2 all V_a_2 all V_prod_2 all T_b all T_c all T_a (class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_c) & class_RealVector_Oreal__normed__vector(T_b) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> c_Groups_Ominus__class_Ominus(T_c,hAPP(hAPP(V_prod_2,V_a_2),V_b_2),hAPP(hAPP(V_prod_2,V_a_H_2),V_b_2)) = hAPP(hAPP(V_prod_2,c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_a_H_2)),V_b_2)))) # label(fact_bounded__bilinear_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 831 (all V_y all V_x (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_x) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_y) -> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Nat__Transfer_Otsub(V_x,V_y))))) # label(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 832 (all V_a all V_b all V_c all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 833 (all V_b all V_a all T_a (class_Rings_Oordered__ring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.53 834 (all V_y all V_x all V_z (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_z) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,V_y) -> 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]. 2.14/2.53 835 (all V_b all V_r 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_RealVector_OscaleR__class_OscaleR(T_a,V_r,V_b)) = c_RealVector_OscaleR__class_OscaleR(T_a,V_r,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_mult_OscaleR__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 836 (all V_l all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_l,V_n),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_l,V_m)))) # label(fact_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 837 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> -c_Orderings_Oord__class_Oless(T_a,V_x,V_x))) # label(fact_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 838 (all V_b_H_2 all V_b_2 all V_a_2 all V_prod_2 all T_b all T_c all T_a (class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_c) & class_RealVector_Oreal__normed__vector(T_a) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> hAPP(hAPP(V_prod_2,V_a_2),c_Groups_Oplus__class_Oplus(T_b,V_b_2,V_b_H_2)) = c_Groups_Oplus__class_Oplus(T_c,hAPP(hAPP(V_prod_2,V_a_2),V_b_2),hAPP(hAPP(V_prod_2,V_a_2),V_b_H_2))))) # label(fact_bounded__bilinear_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 839 (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]. 2.14/2.53 840 (all V_a all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_RComplete_Onatfloor(V_x),V_a) = c_RComplete_Onatfloor(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,c_RealDef_Oreal(tc_Nat_Onat,V_a))))) # label(fact_natfloor__add) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 841 (all V_y all V_x all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> 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]. 2.14/2.53 842 (all V_n_2 all V_x_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_x_2 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) <-> 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),c_Groups_Oabs__class_Oabs(tc_Int_Oint,V_x_2)),V_n_2)))) # label(fact_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 843 (all V_za_2 (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,V_za_2),c_Groups_Oone__class_Oone(tc_Int_Oint)) <-> V_za_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 844 (all V_c all V_a all V_b all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_c,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.53 845 (all V_b_H all V_b 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_Ominus__class_Ominus(T_a,V_b,V_b_H)) = 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)))) # label(fact_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 846 (all V_n all V_m (V_n = V_m -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n))) # label(fact_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 847 (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]. 2.14/2.53 848 (all V_y all V_x c_Complex_ORe(c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,V_x,V_y)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Complex_ORe(V_x),c_Complex_ORe(V_y))) # label(fact_Re_Oadd) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 849 (all V_r ((c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_r,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) -> c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_r) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_r)) & (-c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_r,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) -> c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_r) = V_r))) # label(fact_real__abs__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 850 (all V_n all V_m all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_m) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_n) -> 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]. 2.14/2.53 851 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_k) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_k))) # label(fact_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 852 (all V_y all V_z all V_x (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,V_z) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,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]. 2.14/2.53 853 (all V_n_2 all V_k_2 all V_m_2 (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)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) & c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2))) # label(fact_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 854 (all V_za_2 all V_wa_2 (V_wa_2 = V_za_2 | c_Orderings_Oord__class_Oless(tc_Int_Oint,V_wa_2,V_za_2) <-> c_Orderings_Oord__class_Oless(tc_Int_Oint,V_wa_2,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_za_2,c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 855 (all V_x all V_r hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_r),c_Complex_OIm(V_x)) = c_Complex_OIm(c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,V_r,V_x))) # label(fact_complex__Im__scaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 856 (all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> (V_m != V_n -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)))) # label(fact_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 857 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oabs__class_Oabs(T_a,V_a)),c_Groups_Oabs__class_Oabs(T_a,V_a)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))) # label(fact_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 858 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)))) # label(fact_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 859 (all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 860 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)),V_k) = 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]. 2.14/2.53 861 (all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> c_RComplete_Onatceiling(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))) = 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]. 2.14/2.53 862 (all V_n_2 all V_m_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_m_2) | 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]. 2.14/2.53 863 (all V_m_2 all V_n_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,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)))) # label(fact_nat__less__real__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 864 (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]. 2.14/2.53 865 (all V_n all V_m c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m)) # label(fact_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 866 (all V_y all V_x (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_y) -> c_Transcendental_Oln(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_x),V_y)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Transcendental_Oln(V_x),c_Transcendental_Oln(V_y))))) # label(fact_ln__mult) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 867 (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]. 2.14/2.53 868 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_N) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Oone__class_Oone(T_a)) -> 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]. 2.14/2.53 869 (all V_x all V_r all V_xa all T_a (class_RealVector_Oreal__normed__algebra(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_r,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_x)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),c_RealVector_OscaleR__class_OscaleR(T_a,V_r,V_x)))) # label(fact_mult__right_OscaleR) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 870 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a)))) # label(fact_right__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 871 (all V_n_2 (c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = c_RealDef_Oreal(tc_Nat_Onat,V_n_2) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_real__of__nat__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 872 (all V_b all V_a all T_a (class_Rings_Oordered__ring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) & c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) | c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) & c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.53 873 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a)) # label(fact_add__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 874 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> 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]. 2.14/2.53 875 (all V_b_2 all V_c_2 all V_a_2 all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2),c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_c_2))))) # label(fact_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 876 (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]. 2.14/2.53 877 (all V_b all V_c all V_a all T_a (class_Rings_Olinordered__semiring(T_a) -> (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)) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> c_Orderings_Oord__class_Oless(T_a,V_a,V_b))))) # label(fact_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 878 (all V_x_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x_2) -> (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Transcendental_Oln(V_x_2)) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),V_x_2)))) # label(fact_ln__ge__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 879 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2) = c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) -> (c_Orderings_Oord__class_Oless(T_a,V_c_2,V_d_2) <-> c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2))))) # label(fact_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 880 (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),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]. 2.14/2.53 881 (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]. 2.14/2.53 882 (all V_b_2 all V_a_2 all T_a (class_Rings_Oidom(T_a) -> (c_Groups_Ouminus__class_Ouminus(T_a,V_b_2) = V_a_2 | V_a_2 = V_b_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b_2),V_b_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_2),V_a_2)))) # label(fact_square__eq__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 883 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_x)) = V_x)) # label(fact_double__compl) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 884 (all V_z c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_z) = V_z) # label(fact_zadd__0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 885 (all V_k all V_j all V_i c_Groups_Ominus__class_Ominus(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,V_j),V_k) = c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k))) # label(fact_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 886 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,V_c) -> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)))))) # label(fact_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 887 (all V_k_2 all V_P_2 all V_d_2 (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_d_2) -> ((all B_x (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(V_P_2,c_Groups_Ominus__class_Ominus(tc_Int_Oint,B_x,V_d_2))))) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_k_2) -> (all B_x (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(V_P_2,c_Groups_Ominus__class_Ominus(tc_Int_Oint,B_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),V_d_2)))))))))) # label(fact_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 888 (all V_w all V_z (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_w,V_z) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_z,V_w))) # label(fact_zle__linear) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 889 (all V_real2_2 all V_real1_2 all V_f1_2 all T_a c_Complex_Ocomplex_Ocomplex__case(T_a,V_f1_2,c_Complex_Ocomplex_OComplex(V_real1_2,V_real2_2)) = hAPP(hAPP(V_f1_2,V_real1_2),V_real2_2)) # label(fact_complex_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.53 890 (all V_a_2 all T_a (class_Groups_Ogroup__add(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_a_2 <-> c_Groups_Ouminus__class_Ouminus(T_a,V_a_2) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_neg__equal__0__iff__equal) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 891 (all V_m 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]. 2.14/2.54 892 (all V_y_2 all V_x_2 (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2,V_y_2) <-> V_x_2 = V_y_2 | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x_2,V_y_2))) # label(fact_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 893 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j) -> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k)))) # label(fact_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 894 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> (c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> c_Orderings_Oord__class_Oless(T_a,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]. 2.14/2.54 895 (all V_m_2 all V_n_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Nat_OSuc(V_n_2),V_m_2))) # label(fact_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 896 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_a) -> 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]. 2.14/2.54 897 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2)))) # label(fact_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 898 (all V_za_2 all V_wa_2 (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_wa_2,V_za_2) <-> c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_wa_2,c_Groups_Oone__class_Oone(tc_Int_Oint)),V_za_2))) # label(fact_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 899 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.54 900 (all V_n all V_m (-c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) -> V_m = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_n)))) # label(fact_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 901 (all V_l_2 all V_k_2 (c_Orderings_Oord__class_Oless(tc_Int_Oint,V_k_2,V_l_2) <-> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,V_k_2,V_l_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 902 (all V_b all V_a_H all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_Groups_Ominus__class_Ominus(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b),c_RealVector_OscaleR__class_OscaleR(T_a,V_a_H,V_b)) = c_RealVector_OscaleR__class_OscaleR(T_a,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,V_a,V_a_H),V_b))) # label(fact_scaleR_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 903 (all V_n_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) # label(fact_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 904 (all V_y_2 all V_x_2 all V_b_2 all T_a (class_Rings_Olinordered__semidom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),V_b_2) -> (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)) <-> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x_2,V_y_2))))) # label(fact_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 905 (all V_x all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> V_x = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x))) # label(fact_mult__idem) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 906 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_a (class_Groups_Oab__group__add(T_a) -> (c_Groups_Ominus__class_Ominus(T_a,V_c_2,V_d_2) = c_Groups_Ominus__class_Ominus(T_a,V_a_2,V_b_2) -> (V_d_2 = V_c_2 <-> V_b_2 = V_a_2)))) # label(fact_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 907 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> 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]. 2.14/2.54 908 (all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.54 909 (all V_r_2 all V_x_2 (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,V_x_2),V_r_2) <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,V_x_2,V_r_2) & c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_r_2),V_x_2))) # label(fact_abs__le__interval__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 910 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c))))) # label(fact_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 911 (all V_y_2 all V_x_2 all T_a (class_RealVector_Ometric__space(T_a) -> (V_x_2 = V_y_2 <-> c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(T_a,V_x_2,V_y_2),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))) # label(fact_dist__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 912 (all V_n c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Nat_OSuc(V_n))) # label(fact_lessI) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 913 (all V_b all V_a c_Complex_Ocomplex_OComplex(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_b),V_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Complex_Oii),c_Complex_Ocomplex_OComplex(V_a,V_b))) # label(fact_i__mult__Complex) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 914 (all V_a all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_scaleR_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 915 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_a) <-> V_a_2 = c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)))) # label(fact_equal__neg__zero) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 916 (all V_x_2 all V_a_2 all T_a (class_RealVector_Oreal__vector(T_a) -> (c_RealVector_OscaleR__class_OscaleR(T_a,V_a_2,V_x_2) = c_Groups_Ozero__class_Ozero(T_a) <-> c_Groups_Ozero__class_Ozero(T_a) = V_x_2 | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = V_a_2))) # label(fact_scaleR__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 917 (all V_b_2 all V_a_2 all V_c_2 (V_c_2 != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_c_2),V_b_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_c_2),V_a_2) <-> V_a_2 = V_b_2))) # label(fact_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 918 (all V_q all V_r all V_a (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_a) -> (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 -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_r) -> 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]. 2.14/2.54 919 (all V_w all V_z (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,V_z,V_w) -> (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]. 2.14/2.54 920 (all V_z c_Groups_Oplus__class_Oplus(tc_Int_Oint,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]. 2.14/2.54 921 (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) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) -> V_a = V_b)))))) # label(fact_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 922 (all V_b_2 all V_a_2 all V_prod_2 all T_b all T_c all T_a (class_RealVector_Oreal__normed__vector(T_a) & class_RealVector_Oreal__normed__vector(T_c) & class_RealVector_Oreal__normed__vector(T_b) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> c_Groups_Ouminus__class_Ouminus(T_c,hAPP(hAPP(V_prod_2,V_a_2),V_b_2)) = hAPP(hAPP(V_prod_2,V_a_2),c_Groups_Ouminus__class_Ouminus(T_b,V_b_2))))) # label(fact_bounded__bilinear_Ominus__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 923 (all V_k all V_j all V_i (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j) -> 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]. 2.14/2.54 924 (all V_l all V_n all V_m (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) -> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_m,V_l),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,V_n,V_l)))) # label(fact_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 925 (all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> -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]. 2.14/2.54 926 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)))))) # label(fact_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 927 (all V_a all T_a (class_Rings_Olinordered__ring(T_a) -> -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]. 2.14/2.54 928 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> c_Groups_Ouminus__class_Ouminus(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),c_Groups_Ouminus__class_Ouminus(T_a,V_b)))) # label(fact_minus__mult__right) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 929 (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 (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_N_2,B_n) -> 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]. 2.14/2.54 930 (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_d != V_c & V_b = V_a -> c_Groups_Oplus__class_Oplus(T_a,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_c)) != 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]. 2.14/2.54 931 (all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b_2),V_a_2) <-> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2),V_b_2)))) # label(fact_minus__le__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 932 (all V_y_2 all V_x_2 all T_a (class_RealVector_Ometric__space(T_a) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(T_a,V_x_2,V_y_2)) <-> V_y_2 != V_x_2))) # label(fact_zero__less__dist__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 933 (all V_n -c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_n)) # label(fact_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 934 (all V_a all T_a (class_Groups_Oabs__if(T_a) -> (-c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> c_Groups_Oabs__class_Oabs(T_a,V_a) = V_a) & (c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> c_Groups_Oabs__class_Oabs(T_a,V_a) = c_Groups_Ouminus__class_Ouminus(T_a,V_a)))) # label(fact_abs__if) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 935 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Oabs__class_Oabs(T_a,V_a)))) # label(fact_abs__ge__self) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 936 (all V_b_2 all V_a_2 all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) <-> c_Orderings_Oord__class_Oless(T_a,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]. 2.14/2.54 937 (all V_y all V_x (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_y) -> 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]. 2.14/2.54 938 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) -> c_Groups_Ouminus__class_Ouminus(T_a,V_a) = c_Groups_Oabs__class_Oabs(T_a,V_a)))) # label(fact_abs__of__nonpos) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 939 (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]. 2.14/2.54 940 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 941 (all V_x_2 all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_x_2 <-> 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]. 2.14/2.54 942 (all V_y_2 all V_x_2 (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,V_x_2),V_y_2) <-> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_x_2,V_y_2)))) # label(fact_real__0__less__add__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 943 (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),c_Groups_Ouminus__class_Ouminus(T_a,V_x)) = c_Groups_Ouminus__class_Ouminus(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_xa),V_x)))) # label(fact_mult__right_Ominus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 944 (all V_y all V_x (c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y) -> 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]. 2.14/2.54 945 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ouminus__class_Ouminus(T_a,V_b)) = c_Groups_Ominus__class_Ominus(T_a,V_a,V_b))) # label(fact_ab__diff__minus) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 946 (all V_k_2 all V_P_2 all V_d_2 (c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_d_2) -> ((all B_x (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(V_P_2,c_Groups_Oplus__class_Oplus(tc_Int_Oint,B_x,V_d_2))))) -> (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_k_2) -> (all B_x (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(V_P_2,c_Groups_Oplus__class_Oplus(tc_Int_Oint,B_x,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),V_d_2)))))))))) # label(fact_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 947 (all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (exists B_K (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_K) & (all B_x c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_RealVector_OscaleR__class_OscaleR(T_a,B_x,V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,B_x)),B_K))))))) # label(fact_scaleR__left_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 948 (all V_P_2 all V_n_2 all V_m_2 ((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))) -> ((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]. 2.14/2.54 949 (all V_b all V_a all V_c all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (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)) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c) -> 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]. 2.14/2.54 950 (all V_x all T_a (class_Rings_Ocomm__ring__1(T_a) -> c_Groups_Ouminus__class_Ouminus(T_a,V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ouminus__class_Ouminus(T_a,c_Groups_Oone__class_Oone(T_a))),V_x))) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 951 (all V_w all V_z c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,V_z,V_w)) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,V_z),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,V_w))) # label(fact_zminus__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 952 (all V_a all V_n (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_a) -> (exists B_x (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_x) & (all B_y (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B_y) & V_a = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B_y),V_n) -> B_y = B_x)) & V_a = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B_x),V_n)))))) # label(fact_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 953 (all V_x all V_b all V_a all T_a (class_RealVector_Oreal__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),V_a),V_b),V_x) = c_RealVector_OscaleR__class_OscaleR(T_a,V_a,c_RealVector_OscaleR__class_OscaleR(T_a,V_b,V_x)))) # label(fact_real__vector_Oscale__scale) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 954 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (V_x != V_y -> (-c_Orderings_Oord__class_Oless(T_a,V_x,V_y) -> c_Orderings_Oord__class_Oless(T_a,V_y,V_x))))) # label(fact_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 955 (all V_y_2 all V_x_2 all V_a_2 all T_a (class_RealVector_Oreal__vector(T_a) -> (c_RealVector_OscaleR__class_OscaleR(T_a,V_a_2,V_x_2) = c_RealVector_OscaleR__class_OscaleR(T_a,V_a_2,V_y_2) <-> V_a_2 = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | V_x_2 = V_y_2))) # label(fact_scaleR__cancel__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 956 (all V_x_2 all V_A_2 all T_b all T_a (class_Groups_Ouminus(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,T_a),V_A_2),V_x_2) = c_Groups_Ouminus__class_Ouminus(T_a,hAPP(V_A_2,V_x_2)))) # label(fact_uminus__apply) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 957 (all V_b_2 all V_a_2 all V_y_2 all V_x_2 all V_prod_2 all T_b all T_c all T_a (class_RealVector_Oreal__normed__vector(T_b) & class_RealVector_Oreal__normed__vector(T_c) & class_RealVector_Oreal__normed__vector(T_a) -> (c_RealVector_Obounded__bilinear(T_a,T_b,T_c,V_prod_2) -> c_Groups_Ominus__class_Ominus(T_c,hAPP(hAPP(V_prod_2,V_x_2),V_y_2),hAPP(hAPP(V_prod_2,V_a_2),V_b_2)) = c_Groups_Oplus__class_Oplus(T_c,c_Groups_Oplus__class_Oplus(T_c,hAPP(hAPP(V_prod_2,c_Groups_Ominus__class_Ominus(T_a,V_x_2,V_a_2)),c_Groups_Ominus__class_Ominus(T_b,V_y_2,V_b_2)),hAPP(hAPP(V_prod_2,c_Groups_Ominus__class_Ominus(T_a,V_x_2,V_a_2)),V_b_2)),hAPP(hAPP(V_prod_2,V_a_2),c_Groups_Ominus__class_Ominus(T_b,V_y_2,V_b_2)))))) # label(fact_bounded__bilinear_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 958 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x) -> (c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_y,c_Groups_Oone__class_Oone(T_a)) -> 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]. 2.14/2.54 959 (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),c_Nat_OSuc(V_q)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 960 (all T_a (class_Rings_Olinordered__semidom(T_a) -> -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]. 2.14/2.54 961 (all V_s all V_y all V_r all V_x all T_a (class_RealVector_Oreal__normed__vector(T_a) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_x),V_r) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,V_y),V_s) -> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(T_a,c_Groups_Oplus__class_Oplus(T_a,V_x,V_y)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,V_r,V_s)))))) # label(fact_norm__add__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 962 (all V_a_2 all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 963 (all V_n_2 all V_m_2 all V_k_2 (c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2) & c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) <-> 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_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 964 (all V_c all V_b all V_a all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (c_Groups_Oplus__class_Oplus(T_a,V_a,V_b) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c) -> V_c = V_b))) # label(fact_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 965 (all T_a (class_Rings_Olinordered__semidom(T_a) -> -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]. 2.14/2.54 966 (all V_a all V_m all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oplus__class_Oplus(T_a,V_m,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oone__class_Oone(T_a))),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 967 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) -> (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]. 2.14/2.54 968 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a,V_b) -> (c_Orderings_Oord__class_Oless(T_a,V_c,V_d) -> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)))))) # label(fact_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 2.14/2.54 969 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) -> (c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a)) -> 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]. 2.14/2.54 970 (all V_a_2 all T_a (class_Rings_Olinordered__idom(T_a) -> (c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ouminus__class_Ouminus(T_a,V_a_2)) <-> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_less__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 2.14/2.56 971 (all V_b all V_a all T_a (class_Rings_Ono__zero__divisors(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_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]. 2.14/2.56 972 (all V_za_2 (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_za_2) <-> c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_za_2))) # label(fact_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 2.14/2.56 973 (all V_n all V_x (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),V_x) -> (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,V_x,c_RealDef_Oreal(tc_Nat_Onat,V_n)) -> 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]. 2.14/2.56 974 (all V_b all V_a all V_r all T_a (class_RealVector_Oreal__normed__vector(T_a) -> c_RealVector_OscaleR__class_OscaleR(T_a,V_r,c_RealVector_OscaleR__class_OscaleR(T_a,V_a,V_b)) = c_RealVector_OscaleR__class_OscaleR(T_a,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,V_r,V_a),V_b))) # label(fact_scaleR_OscaleR__left) # label(axiom) # label(non_clause). [assumption]. 2.14/2.56 975 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oabs__class_Oabs(T_a,c_Groups_Ominus__class_Ominus(T_a,V_a,V_b)),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oabs__class_Oabs(T_a,V_a),c_Groups_Oabs__class_Oabs(T_a,V_b))))) # label(fact_abs__triangle__ineq4) # label(axiom) # label(non_clause). [assumption]. 2.14/2.56 2.14/2.56 ============================== end of process non-clausal formulas === 2.14/2.56 2.14/2.56 ============================== PROCESS INITIAL CLAUSES =============== 2.14/2.56 2.14/2.56 ============================== PREDICATE ELIMINATION ================= 2.14/2.56 976 class_Rings_Olinordered__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.14/2.56 977 -class_Rings_Olinordered__semiring__strict(A) | -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)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless(A,B,D) # label(fact_mult__less__imp__less__right) # label(axiom). [clausify(2)]. 2.14/2.56 978 -class_Rings_Olinordered__semiring__strict(A) | -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)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless__eq(A,B,D) # label(fact_mult__right__le__imp__le) # label(axiom). [clausify(95)]. 2.14/2.56 979 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,C,c_Groups_Ozero__class_Ozero(A)) | 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(132)]. 2.14/2.56 980 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) | 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(144)]. 2.14/2.56 981 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | 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(312)]. 2.14/2.59 982 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) # label(fact_zero__less__mult__pos2) # label(axiom). [clausify(314)]. 2.14/2.59 983 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) # label(fact_zero__less__mult__pos) # label(axiom). [clausify(319)]. 2.14/2.59 Derived: -c_Orderings_Oord__class_Oless__eq(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)) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,C). [resolve(976,a,978,a)]. 2.14/2.59 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(976,a,981,a)]. 2.14/2.59 984 -class_Rings_Olinordered__semiring__strict(A) | -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)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless(A,C,D) # label(fact_mult__less__imp__less__left) # label(axiom). [clausify(567)]. 2.14/2.59 985 class_Rings_Olinordered__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.14/2.59 Derived: -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)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,A,C). [resolve(985,a,977,a)]. 2.14/2.59 Derived: -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)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,C). [resolve(985,a,978,a)]. 2.14/2.59 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,B,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | 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(985,a,979,a)]. 2.14/2.59 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | 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(985,a,980,a)]. 2.14/2.59 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,D) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(985,a,981,a)]. 2.14/2.59 Derived: -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)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A). [resolve(985,a,982,a)]. 2.14/2.61 Derived: -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)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B). [resolve(985,a,983,a)]. 2.14/2.61 Derived: -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)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,C). [resolve(985,a,984,a)]. 2.14/2.61 986 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | 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(603)]. 2.14/2.61 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(986,a,976,a)]. 2.14/2.61 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,D) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(986,a,985,a)]. 2.14/2.61 987 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),D) | 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(683)]. 2.14/2.61 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(987,a,985,a)]. 2.14/2.61 988 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | 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(718)]. 2.14/2.61 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(988,a,976,a)]. 2.14/2.61 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(988,a,985,a)]. 2.14/2.66 989 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,E) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),D) | 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(817)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(989,a,976,a)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(989,a,985,a)]. 2.14/2.66 990 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),D) | 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(899)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(990,a,985,a)]. 2.14/2.66 991 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) | 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(907)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | 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),A),B)). [resolve(991,a,976,a)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | 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(991,a,985,a)]. 2.14/2.66 992 class_Rings_Olinordered__semiring__strict(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.14/2.66 Derived: -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)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,C). [resolve(992,a,977,a)]. 2.14/2.66 Derived: -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)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,C). [resolve(992,a,978,a)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | 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(992,a,979,a)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | 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(992,a,980,a)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,D) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(992,a,981,a)]. 2.14/2.66 Derived: -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)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A). [resolve(992,a,982,a)]. 2.14/2.66 Derived: -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)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B). [resolve(992,a,983,a)]. 2.14/2.66 Derived: -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)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,C). [resolve(992,a,984,a)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,D) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(992,a,986,a)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(992,a,987,a)]. 2.14/2.66 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(992,a,988,a)]. 2.14/2.70 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(992,a,989,a)]. 2.14/2.70 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(992,a,990,a)]. 2.14/2.70 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | 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(992,a,991,a)]. 2.14/2.70 993 -class_Rings_Olinordered__semiring__strict(A) | -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)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_mult__left__le__imp__le) # label(axiom). [clausify(949)]. 2.14/2.70 Derived: -c_Orderings_Oord__class_Oless__eq(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_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C). [resolve(993,a,976,a)]. 2.14/2.70 Derived: -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)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,C). [resolve(993,a,985,a)]. 2.14/2.70 Derived: -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)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,C). [resolve(993,a,992,a)]. 2.14/2.70 994 -class_Rings_Olinordered__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,C,c_Groups_Ozero__class_Ozero(A)) | 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(969)]. 2.14/2.70 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(994,a,976,a)]. 2.14/2.70 995 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Oplus__class_Oplus(A,C,B) = C # label(fact_add__0__iff) # label(axiom). [clausify(191)]. 2.14/2.70 996 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]. 2.14/2.70 997 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]. 2.48/2.74 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,A) = B. [resolve(995,a,996,a)]. 2.48/2.74 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,A) = B. [resolve(995,a,997,a)]. 2.48/2.74 998 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Oplus__class_Oplus(A,C,B) != C # label(fact_add__0__iff) # label(axiom). [clausify(191)]. 2.48/2.74 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,A) != B. [resolve(998,a,996,a)]. 2.48/2.74 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,A) != B. [resolve(998,a,997,a)]. 2.48/2.74 999 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]. 2.48/2.74 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,A) = B. [resolve(999,a,995,a)]. 2.48/2.74 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,A) != B. [resolve(999,a,998,a)]. 2.48/2.74 1000 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | 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),E)) != c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) | B = D | C = E # label(fact_crossproduct__eq) # label(axiom). [clausify(678)]. 2.48/2.74 Derived: 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),D)) != 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),C),B)) | A = C | B = D. [resolve(1000,a,996,a)]. 2.48/2.74 Derived: 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),D)) != 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),C),B)) | A = C | B = D. [resolve(1000,a,997,a)]. 2.48/2.74 Derived: 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),D)) != 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),C),B)) | A = C | B = D. [resolve(1000,a,999,a)]. 2.48/2.74 1001 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | 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),E)) = c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) | B != D # label(fact_crossproduct__eq) # label(axiom). [clausify(678)]. 2.48/2.74 Derived: 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),D)) = 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),C),B)) | A != C. [resolve(1001,a,996,a)]. 2.48/2.74 Derived: 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),D)) = 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),C),B)) | A != C. [resolve(1001,a,997,a)]. 2.52/2.77 Derived: 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),D)) = 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),C),B)) | A != C. [resolve(1001,a,999,a)]. 2.52/2.77 1002 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | 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),E)) = c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) | C != E # label(fact_crossproduct__eq) # label(axiom). [clausify(678)]. 2.52/2.77 Derived: 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),D)) = 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),C),B)) | B != D. [resolve(1002,a,996,a)]. 2.52/2.77 Derived: 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),D)) = 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),C),B)) | B != D. [resolve(1002,a,997,a)]. 2.52/2.77 Derived: 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),D)) = 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),C),B)) | B != D. [resolve(1002,a,999,a)]. 2.52/2.77 1003 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B = C | D = E | 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)) != 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(721)]. 2.52/2.77 1004 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | 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)) = 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(721)]. 2.52/2.77 1005 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | 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)) = 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(721)]. 2.52/2.77 1006 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]. 2.52/2.77 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,A) = B. [resolve(1006,a,995,a)]. 2.52/2.77 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,A) != B. [resolve(1006,a,998,a)]. 2.52/2.77 Derived: c_Groups_Oplus__class_Oplus(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),D)) != 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),C),B)) | A = C | B = D. [resolve(1006,a,1000,a)]. 2.52/2.83 Derived: c_Groups_Oplus__class_Oplus(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),D)) = 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),C),B)) | A != C. [resolve(1006,a,1001,a)]. 2.52/2.83 Derived: c_Groups_Oplus__class_Oplus(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),D)) = 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),C),B)) | B != D. [resolve(1006,a,1002,a)]. 2.52/2.83 1007 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Ozero__class_Ozero(A) = B | C = D | E != F | c_Groups_Oplus__class_Oplus(A,E,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) != c_Groups_Oplus__class_Oplus(A,F,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) # label(fact_add__scale__eq__noteq) # label(axiom). [clausify(930)]. 2.52/2.83 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | B = C | D != E | c_Groups_Oplus__class_Oplus(tc_Int_Oint,D,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)) != c_Groups_Oplus__class_Oplus(tc_Int_Oint,E,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)). [resolve(1007,a,996,a)]. 2.52/2.83 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | B = C | D != E | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,D,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),C)) != c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,E,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B)). [resolve(1007,a,997,a)]. 2.52/2.83 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | B = C | D != E | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,D,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)) != c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,E,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B)). [resolve(1007,a,999,a)]. 2.52/2.83 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | B = C | D != E | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,D,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)) != c_Groups_Oplus__class_Oplus(tc_Nat_Onat,E,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)). [resolve(1007,a,1006,a)]. 2.52/2.83 1008 class_Groups_Oordered__ab__group__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 2.52/2.83 1009 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ouminus__class_Ouminus(A,C)) | c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_le__minus__iff) # label(axiom). [clausify(10)]. 2.52/2.83 1010 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ouminus__class_Ouminus(A,C)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_le__minus__iff) # label(axiom). [clausify(10)]. 2.52/2.83 1011 -class_Groups_Oordered__ab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) != c_Groups_Ominus__class_Ominus(A,D,E) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,D,E) # label(fact_diff__eq__diff__less__eq) # label(axiom). [clausify(28)]. 2.52/2.83 1012 -class_Groups_Oordered__ab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) != c_Groups_Ominus__class_Ominus(A,D,E) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,E) # label(fact_diff__eq__diff__less__eq) # label(axiom). [clausify(28)]. 2.52/2.83 1013 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ouminus__class_Ouminus(A,C)) | c_Orderings_Oord__class_Oless(A,C,c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_less__minus__iff) # label(axiom). [clausify(72)]. 2.52/2.84 1014 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ouminus__class_Ouminus(A,C)) | -c_Orderings_Oord__class_Oless(A,C,c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_less__minus__iff) # label(axiom). [clausify(72)]. 2.52/2.84 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)). [resolve(1008,a,1009,a)]. 2.52/2.84 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) != c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,D). [resolve(1008,a,1011,a)]. 2.52/2.84 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) != c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,C,D) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,D). [resolve(1008,a,1012,a)]. 2.52/2.84 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)). [resolve(1008,a,1013,a)]. 2.52/2.84 1015 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,C),c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_neg__less__iff__less) # label(axiom). [clausify(148)]. 2.52/2.84 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)). [resolve(1015,a,1008,a)]. 2.52/2.84 1016 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,C),c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_neg__less__iff__less) # label(axiom). [clausify(148)]. 2.52/2.84 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)). [resolve(1016,a,1008,a)]. 2.52/2.84 1017 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,B),C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,C),B) # label(fact_minus__less__iff) # label(axiom). [clausify(151)]. 2.52/2.84 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B),A). [resolve(1017,a,1008,a)]. 2.52/2.84 1018 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,B),C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,C),B) # label(fact_minus__less__iff) # label(axiom). [clausify(151)]. 2.52/2.84 1019 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) # label(fact_neg__le__0__iff__le) # label(axiom). [clausify(237)]. 2.52/2.84 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A). [resolve(1019,a,1008,a)]. 2.52/2.84 1020 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) # label(fact_neg__le__0__iff__le) # label(axiom). [clausify(237)]. 2.52/2.84 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A). [resolve(1020,a,1008,a)]. 2.52/2.85 1021 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(261)]. 2.52/2.85 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A). [resolve(1021,a,1008,a)]. 2.52/2.85 1022 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(261)]. 2.52/2.85 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A). [resolve(1022,a,1008,a)]. 2.52/2.85 1023 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Ouminus__class_Ouminus(A,C)) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_neg__le__iff__le) # label(axiom). [clausify(269)]. 2.52/2.85 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,A). [resolve(1023,a,1008,a)]. 2.52/2.85 1024 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Ouminus__class_Ouminus(A,C)) | -c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_neg__le__iff__le) # label(axiom). [clausify(269)]. 2.52/2.85 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,A). [resolve(1024,a,1008,a)]. 2.52/2.85 1025 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_neg__0__le__iff__le) # label(axiom). [clausify(356)]. 2.52/2.85 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)). [resolve(1025,a,1008,a)]. 2.52/2.85 1026 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_neg__0__le__iff__le) # label(axiom). [clausify(356)]. 2.52/2.85 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)). [resolve(1026,a,1008,a)]. 2.52/2.85 1027 class_Groups_Oordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 2.52/2.85 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)). [resolve(1027,a,1009,a)]. 2.52/2.85 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) != c_Groups_Ominus__class_Ominus(tc_Int_Oint,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,D). [resolve(1027,a,1011,a)]. 2.62/2.86 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) != c_Groups_Ominus__class_Ominus(tc_Int_Oint,C,D) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,D). [resolve(1027,a,1012,a)]. 2.62/2.86 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)). [resolve(1027,a,1013,a)]. 2.62/2.86 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)). [resolve(1027,a,1015,a)]. 2.62/2.86 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)). [resolve(1027,a,1016,a)]. 2.62/2.86 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B),A). [resolve(1027,a,1017,a)]. 2.62/2.86 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A). [resolve(1027,a,1019,a)]. 2.62/2.86 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A). [resolve(1027,a,1020,a)]. 2.62/2.86 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A). [resolve(1027,a,1021,a)]. 2.62/2.86 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A). [resolve(1027,a,1022,a)]. 2.62/2.86 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,A). [resolve(1027,a,1023,a)]. 2.62/2.86 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,A). [resolve(1027,a,1024,a)]. 2.62/2.86 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)). [resolve(1027,a,1025,a)]. 2.62/2.86 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)). [resolve(1027,a,1026,a)]. 2.62/2.86 1028 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ominus__class_Ominus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(449)]. 2.62/2.86 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1028,a,1008,a)]. 2.62/2.86 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1028,a,1027,a)]. 2.62/2.86 1029 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ominus__class_Ominus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(449)]. 2.67/2.93 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1029,a,1008,a)]. 2.67/2.93 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1029,a,1027,a)]. 2.67/2.93 1030 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,C),c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_le__imp__neg__le) # label(axiom). [clausify(484)]. 2.67/2.93 1031 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_neg__0__less__iff__less) # label(axiom). [clausify(821)]. 2.67/2.93 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)). [resolve(1031,a,1008,a)]. 2.67/2.93 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)). [resolve(1031,a,1027,a)]. 2.67/2.93 1032 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_neg__0__less__iff__less) # label(axiom). [clausify(821)]. 2.67/2.93 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)). [resolve(1032,a,1008,a)]. 2.67/2.93 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)). [resolve(1032,a,1027,a)]. 2.67/2.93 1033 -class_Groups_Oordered__ab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) != c_Groups_Ominus__class_Ominus(A,D,E) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,D,E) # label(fact_diff__eq__diff__less) # label(axiom). [clausify(879)]. 2.67/2.93 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) != c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,C,D) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,D). [resolve(1033,a,1008,a)]. 2.67/2.93 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) != c_Groups_Ominus__class_Ominus(tc_Int_Oint,C,D) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,C,D). [resolve(1033,a,1027,a)]. 2.67/2.93 1034 -class_Groups_Oordered__ab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) != c_Groups_Ominus__class_Ominus(A,D,E) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) # label(fact_diff__eq__diff__less) # label(axiom). [clausify(879)]. 2.67/2.93 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) != c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,C,D) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,D). [resolve(1034,a,1008,a)]. 2.67/2.93 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) != c_Groups_Ominus__class_Ominus(tc_Int_Oint,C,D) | c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,D). [resolve(1034,a,1027,a)]. 2.67/2.93 1035 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,C),B) # label(fact_minus__le__iff) # label(axiom). [clausify(931)]. 2.82/3.10 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B),A). [resolve(1035,a,1008,a)]. 2.82/3.10 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B),A). [resolve(1035,a,1027,a)]. 2.82/3.10 1036 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,C),B) # label(fact_minus__le__iff) # label(axiom). [clausify(931)]. 2.82/3.10 1037 -class_Groups_Oordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ominus__class_Ominus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(936)]. 2.82/3.10 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1037,a,1008,a)]. 2.82/3.10 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1037,a,1027,a)]. 2.82/3.10 1038 -class_Groups_Oordered__ab__group__add(A) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ominus__class_Ominus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(936)]. 2.82/3.10 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1038,a,1008,a)]. 2.82/3.10 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1038,a,1027,a)]. 2.82/3.10 1039 -class_Groups_Omonoid__add(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ozero__class_Ozero(A)) = B # label(fact_add__0__right) # label(axiom). [clausify(93)]. 2.82/3.10 1040 class_Groups_Omonoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Omonoid__add) # label(axiom). [assumption]. 2.82/3.10 1041 class_Groups_Omonoid__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Omonoid__add) # label(axiom). [assumption]. 2.82/3.10 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)) = A. [resolve(1041,a,1039,a)]. 2.82/3.10 1042 class_Groups_Omonoid__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Omonoid__add) # label(axiom). [assumption]. 2.82/3.10 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = A. [resolve(1042,a,1039,a)]. 2.82/3.10 1043 class_Groups_Omonoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__add) # label(axiom). [assumption]. 2.82/3.10 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(1043,a,1039,a)]. 2.82/3.10 1044 -class_Groups_Omonoid__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ozero__class_Ozero(A),B) = B # label(fact_add__0__left) # label(axiom). [clausify(811)]. 2.82/3.10 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex),A) = A. [resolve(1044,a,1041,a)]. 2.82/3.10 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) = A. [resolve(1044,a,1042,a)]. 2.82/3.10 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) = A. [resolve(1044,a,1043,a)]. 2.82/3.10 1045 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__linear) # label(axiom). [clausify(109)]. 2.82/3.11 1046 class_Orderings_Olinorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Olinorder) # label(axiom). [assumption]. 2.82/3.11 1047 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_leD) # label(axiom). [clausify(126)]. 2.82/3.11 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(1047,a,1046,a)]. 2.82/3.11 1048 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) | B = C # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(137)]. 2.82/3.11 1049 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) | B != C # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(137)]. 2.82/3.11 1050 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__le__less__linear) # label(axiom). [clausify(240)]. 2.82/3.11 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(1050,a,1046,a)]. 2.82/3.11 1051 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) | C = B # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(245)]. 2.82/3.11 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A) | B = A. [resolve(1051,a,1046,a)]. 2.82/3.11 1052 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(245)]. 2.82/3.11 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,A). [resolve(1052,a,1046,a)]. 2.82/3.11 1053 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | C != B # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(245)]. 2.82/3.11 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | B != A. [resolve(1053,a,1046,a)]. 2.82/3.11 1054 class_Orderings_Olinorder(tc_Int_Oint) # label(arity_Int__Oint__Orderings_Olinorder) # label(axiom). [assumption]. 2.82/3.11 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,B,A). [resolve(1054,a,1047,a)]. 2.82/3.11 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,A) | A = B. [resolve(1054,a,1048,a)]. 2.82/3.11 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,B,A) | A != B. [resolve(1054,a,1049,a)]. 2.82/3.11 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,A). [resolve(1054,a,1050,a)]. 2.82/3.11 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,B,A). [resolve(1054,a,1052,a)]. 2.82/3.11 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | B != A. [resolve(1054,a,1053,a)]. 2.82/3.11 1055 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | C = B # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(280)]. 2.82/3.11 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | B = A. [resolve(1055,a,1046,a)]. 2.82/3.11 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | B = A. [resolve(1055,a,1054,a)]. 2.82/3.11 1056 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,B,C) | C != B # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(280)]. 2.82/3.11 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | B != A. [resolve(1056,a,1046,a)]. 2.82/3.11 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | B != A. [resolve(1056,a,1054,a)]. 2.93/3.17 1057 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | C = B | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__cases) # label(axiom). [clausify(498)]. 2.93/3.17 1058 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__not__less) # label(axiom). [clausify(513)]. 2.93/3.17 1059 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__not__less) # label(axiom). [clausify(513)]. 2.93/3.17 1060 -class_Orderings_Olinorder(A) | B = C | c_Orderings_Oord__class_Oless(A,C,B) | c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__neqE) # label(axiom). [clausify(553)]. 2.93/3.17 1061 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | C != B # label(fact_linorder__neq__iff) # label(axiom). [clausify(562)]. 2.93/3.17 1062 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | B != C # label(fact_linorder__neq__iff) # label(axiom). [clausify(562)]. 2.93/3.17 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | A != B. [resolve(1062,a,1046,a)]. 2.93/3.17 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | A != B. [resolve(1062,a,1054,a)]. 2.93/3.17 1063 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) | C = B # label(fact_linorder__neq__iff) # label(axiom). [clausify(562)]. 2.93/3.17 1064 -class_Orderings_Olinorder(A) | B = C | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_linorder__less__linear) # label(axiom). [clausify(570)]. 2.93/3.17 1065 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__not__le) # label(axiom). [clausify(669)]. 2.93/3.17 1066 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__not__le) # label(axiom). [clausify(669)]. 2.93/3.17 1067 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | C != B | -c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(681)]. 2.93/3.17 1068 -class_Orderings_Olinorder(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | C = B | c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(681)]. 2.93/3.17 1069 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_linorder__le__cases) # label(axiom). [clausify(693)]. 2.93/3.17 Derived: c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,A). [resolve(1069,a,1046,a)]. 2.93/3.17 1070 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) # label(fact_not__leE) # label(axiom). [clausify(700)]. 2.93/3.17 1071 -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,C,B) # label(fact_leI) # label(axiom). [clausify(720)]. 2.93/3.17 1072 class_Orderings_Olinorder(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Orderings_Olinorder) # label(axiom). [assumption]. 2.93/3.17 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,A). [resolve(1072,a,1045,a)]. 2.93/3.17 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,A). [resolve(1072,a,1047,a)]. 2.93/3.17 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,A) | A = B. [resolve(1072,a,1048,a)]. 2.93/3.17 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,A) | A != B. [resolve(1072,a,1049,a)]. 2.93/3.17 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,A). [resolve(1072,a,1050,a)]. 2.93/3.17 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,A). [resolve(1072,a,1052,a)]. 3.04/3.29 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | B != A. [resolve(1072,a,1053,a)]. 3.04/3.29 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | A != B. [resolve(1072,a,1062,a)]. 3.04/3.29 1073 -class_Rings_Ono__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) | c_Groups_Ozero__class_Ozero(A) = C | c_Groups_Ozero__class_Ozero(A) = B # label(fact_divisors__zero) # label(axiom). [clausify(401)]. 3.04/3.29 1074 class_Rings_Ono__zero__divisors(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.04/3.29 1075 class_Rings_Ono__zero__divisors(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.04/3.29 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A. [resolve(1073,a,1074,a)]. 3.04/3.29 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1073,a,1075,a)]. 3.04/3.29 1076 -class_Power_Opower(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Omult__zero(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = C | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) # label(fact_power__eq__0__iff) # label(axiom). [clausify(616)]. 3.04/3.29 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | -class_Rings_Ozero__neq__one(tc_Complex_Ocomplex) | -class_Rings_Omult__zero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B). [resolve(1076,c,1074,a)]. 3.04/3.29 Derived: -class_Power_Opower(tc_RealDef_Oreal) | -class_Rings_Ozero__neq__one(tc_RealDef_Oreal) | -class_Rings_Omult__zero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B). [resolve(1076,c,1075,a)]. 3.04/3.29 1077 -class_Power_Opower(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Omult__zero(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) # label(fact_power__eq__0__iff) # label(axiom). [clausify(616)]. 3.04/3.29 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | -class_Rings_Ozero__neq__one(tc_Complex_Ocomplex) | -class_Rings_Omult__zero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B). [resolve(1077,c,1074,a)]. 3.04/3.29 Derived: -class_Power_Opower(tc_RealDef_Oreal) | -class_Rings_Ozero__neq__one(tc_RealDef_Oreal) | -class_Rings_Omult__zero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B). [resolve(1077,c,1075,a)]. 3.04/3.29 1078 -class_Power_Opower(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Omult__zero(A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Power_Opower__class_Opower(A),C),B) # label(fact_power__eq__0__iff) # label(axiom). [clausify(616)]. 3.04/3.29 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | -class_Rings_Ozero__neq__one(tc_Complex_Ocomplex) | -class_Rings_Omult__zero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),B),A). [resolve(1078,c,1074,a)]. 3.15/3.39 Derived: -class_Power_Opower(tc_RealDef_Oreal) | -class_Rings_Ozero__neq__one(tc_RealDef_Oreal) | -class_Rings_Omult__zero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B),A). [resolve(1078,c,1075,a)]. 3.15/3.39 1079 class_Rings_Ono__zero__divisors(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.15/3.39 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(1079,a,1073,a)]. 3.15/3.39 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B). [resolve(1079,a,1076,c)]. 3.15/3.39 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B). [resolve(1079,a,1077,c)]. 3.15/3.39 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),A). [resolve(1079,a,1078,c)]. 3.15/3.39 1080 class_Rings_Ono__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.15/3.39 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(1080,a,1073,a)]. 3.15/3.39 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B). [resolve(1080,a,1076,c)]. 3.15/3.39 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B). [resolve(1080,a,1077,c)]. 3.15/3.39 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),A). [resolve(1080,a,1078,c)]. 3.15/3.39 1081 -class_Rings_Ono__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) # label(fact_no__zero__divisors) # label(axiom). [clausify(971)]. 3.15/3.39 1082 class_Rings_Ocomm__semiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 3.15/3.39 1083 -class_Rings_Ocomm__semiring__1(A) | 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),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(15)]. 3.15/3.39 1084 -class_Rings_Ocomm__semiring__1(A) | 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),c_Groups_Oplus__class_Oplus(A,B,D)),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(32)]. 3.15/3.40 1085 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) = c_Groups_Oplus__class_Oplus(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(83)]. 3.15/3.40 1086 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,B,B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oplus__class_Oplus(A,c_Groups_Oone__class_Oone(A),c_Groups_Oone__class_Oone(A))),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(120)]. 3.15/3.40 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),A). [resolve(1082,a,1086,a)]. 3.15/3.40 1087 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom). [clausify(155)]. 3.15/3.40 1088 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),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_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom). [clausify(181)]. 3.15/3.40 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),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(1088,a,1082,a)]. 3.15/3.40 1089 class_Rings_Ocomm__semiring__1(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 3.15/3.40 Derived: 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),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C)). [resolve(1089,a,1083,a)]. 3.15/3.40 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C)). [resolve(1089,a,1085,a)]. 3.15/3.40 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),A). [resolve(1089,a,1086,a)]. 3.15/3.40 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A). [resolve(1089,a,1087,a)]. 3.15/3.40 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),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(1089,a,1088,a)]. 3.15/3.40 1090 -class_Rings_Ocomm__semiring__1(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_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom). [clausify(218)]. 3.15/3.40 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(1090,a,1089,a)]. 3.15/3.40 1091 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ozero__class_Ozero(A)) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(243)]. 3.15/3.40 1092 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),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_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom). [clausify(244)]. 3.15/3.41 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),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(1092,a,1089,a)]. 3.15/3.41 1093 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Nat_OSuc(C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom). [clausify(247)]. 3.15/3.41 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Nat_OSuc(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(1093,a,1082,a)]. 3.15/3.41 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),c_Nat_OSuc(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(1093,a,1089,a)]. 3.15/3.41 1094 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),D) = c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(257)]. 3.15/3.41 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),C) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C)). [resolve(1094,a,1089,a)]. 3.15/3.41 1095 class_Rings_Ocomm__semiring__1(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 3.15/3.41 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat))),A). [resolve(1095,a,1086,a)]. 3.15/3.41 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),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(1095,a,1088,a)]. 3.15/3.41 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),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(1095,a,1092,a)]. 3.15/3.41 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Nat_OSuc(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(1095,a,1093,a)]. 3.15/3.41 1096 -class_Rings_Ocomm__semiring__1(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_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom). [clausify(400)]. 3.15/3.41 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(1096,a,1082,a)]. 3.15/3.41 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(1096,a,1089,a)]. 3.15/3.41 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(1096,a,1095,a)]. 3.15/3.42 1097 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom). [clausify(412)]. 3.15/3.42 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D))). [resolve(1097,a,1082,a)]. 3.15/3.42 1098 -class_Rings_Ocomm__semiring__1(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom). [clausify(433)]. 3.15/3.42 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),B). [resolve(1098,a,1082,a)]. 3.15/3.42 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)),B). [resolve(1098,a,1089,a)]. 3.15/3.42 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),B). [resolve(1098,a,1095,a)]. 3.15/3.42 1099 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom). [clausify(435)]. 3.15/3.42 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(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(1099,a,1082,a)]. 3.15/3.42 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(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(1099,a,1089,a)]. 3.15/3.42 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(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(1099,a,1095,a)]. 3.15/3.42 1100 class_Rings_Ocomm__semiring__1(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 3.15/3.42 Derived: 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),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,C)). [resolve(1100,a,1083,a)]. 3.15/3.42 Derived: 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),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,C)),B). [resolve(1100,a,1084,a)]. 3.15/3.42 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,C)) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,C)). [resolve(1100,a,1085,a)]. 3.15/3.42 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))),A). [resolve(1100,a,1086,a)]. 3.15/3.42 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),A). [resolve(1100,a,1087,a)]. 3.15/3.42 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),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(1100,a,1088,a)]. 3.15/3.42 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(1100,a,1090,a)]. 3.15/3.42 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),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(1100,a,1092,a)]. 3.15/3.42 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),c_Nat_OSuc(B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B)). [resolve(1100,a,1093,a)]. 3.15/3.42 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B),C) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,C)). [resolve(1100,a,1094,a)]. 3.15/3.42 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(1100,a,1096,a)]. 3.15/3.42 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),C)),B). [resolve(1100,a,1098,a)]. 3.15/3.42 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(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)). [resolve(1100,a,1099,a)]. 3.15/3.42 1101 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,E)) = c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,E)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom). [clausify(438)]. 3.15/3.44 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,D)) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,D)). [resolve(1101,a,1082,a)]. 3.15/3.44 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,D)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,D)). [resolve(1101,a,1089,a)]. 3.15/3.44 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,D)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,D)). [resolve(1101,a,1095,a)]. 3.15/3.44 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,C,D)) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,C),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,D)). [resolve(1101,a,1100,a)]. 3.15/3.44 1102 -class_Rings_Ocomm__semiring__1(A) | 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),c_Groups_Oplus__class_Oplus(A,B,D)),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(463)]. 3.15/3.44 Derived: 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),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C)),B). [resolve(1102,a,1082,a)]. 3.15/3.44 1103 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom). [clausify(482)]. 3.15/3.44 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)). [resolve(1103,a,1082,a)]. 3.15/3.44 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C)). [resolve(1103,a,1089,a)]. 3.15/3.44 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)). [resolve(1103,a,1095,a)]. 3.15/3.44 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),C)). [resolve(1103,a,1100,a)]. 3.15/3.44 1104 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Oone__class_Oone(A)) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(510)]. 3.15/3.44 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A. [resolve(1104,a,1082,a)]. 3.15/3.44 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) = A. [resolve(1104,a,1089,a)]. 3.15/3.44 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1104,a,1095,a)]. 3.23/3.46 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)) = A. [resolve(1104,a,1100,a)]. 3.23/3.46 1105 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(524)]. 3.23/3.46 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(1105,a,1082,a)]. 3.23/3.46 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A). [resolve(1105,a,1089,a)]. 3.23/3.46 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)),A). [resolve(1105,a,1100,a)]. 3.23/3.46 1106 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),D) = c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,D),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(533)]. 3.23/3.46 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),C) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C),B). [resolve(1106,a,1082,a)]. 3.23/3.46 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),C) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C),B). [resolve(1106,a,1089,a)]. 3.23/3.46 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),C) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),B). [resolve(1106,a,1095,a)]. 3.23/3.46 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B),C) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,C),B). [resolve(1106,a,1100,a)]. 3.23/3.46 1107 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom). [clausify(543)]. 3.23/3.46 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1107,a,1082,a)]. 3.23/3.46 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1107,a,1089,a)]. 3.23/3.46 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1107,a,1095,a)]. 3.23/3.46 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(1107,a,1100,a)]. 3.23/3.46 1108 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom). [clausify(566)]. 3.23/3.46 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(1108,a,1082,a)]. 3.23/3.46 Derived: c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(1108,a,1089,a)]. 3.23/3.46 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(1108,a,1095,a)]. 3.23/3.46 Derived: c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(1108,a,1100,a)]. 3.23/3.46 1109 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),D) = c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom). [clausify(643)]. 3.26/3.49 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),C) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,C)). [resolve(1109,a,1082,a)]. 3.26/3.49 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),C) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C)). [resolve(1109,a,1095,a)]. 3.26/3.49 1110 -class_Rings_Ocomm__semiring__1(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_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom). [clausify(753)]. 3.26/3.49 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)). [resolve(1110,a,1082,a)]. 3.26/3.49 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(1110,a,1095,a)]. 3.26/3.49 1111 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ozero__class_Ozero(A),B) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(771)]. 3.26/3.49 1112 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oone__class_Oone(A))),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(806)]. 3.26/3.49 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,c_Groups_Oone__class_Oone(tc_Int_Oint))),B). [resolve(1112,a,1082,a)]. 3.26/3.49 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),B). [resolve(1112,a,1089,a)]. 3.26/3.49 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat))),B). [resolve(1112,a,1095,a)]. 3.26/3.49 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))),B). [resolve(1112,a,1100,a)]. 3.26/3.49 1113 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,B,C) = c_Groups_Oplus__class_Oplus(A,C,B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom). [clausify(810)]. 3.26/3.49 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,A). [resolve(1113,a,1082,a)]. 3.26/3.49 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,A). [resolve(1113,a,1089,a)]. 3.26/3.49 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,A). [resolve(1113,a,1100,a)]. 3.26/3.49 1114 -class_Rings_Ocomm__semiring__1(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_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom). [clausify(847)]. 3.26/3.52 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(1114,a,1082,a)]. 3.26/3.52 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(1114,a,1089,a)]. 3.26/3.52 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(1114,a,1095,a)]. 3.26/3.52 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(1114,a,1100,a)]. 3.26/3.52 1115 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ozero__class_Ozero(A)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(876)]. 3.26/3.52 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1115,a,1082,a)]. 3.26/3.52 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1115,a,1089,a)]. 3.26/3.52 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(1115,a,1095,a)]. 3.26/3.52 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)). [resolve(1115,a,1100,a)]. 3.26/3.52 1116 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),E)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom). [clausify(881)]. 3.26/3.52 1117 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oone__class_Oone(A)),B) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(940)]. 3.26/3.52 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A) = A. [resolve(1117,a,1082,a)]. 3.26/3.52 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oone__class_Oone(tc_RealDef_Oreal)),A) = A. [resolve(1117,a,1089,a)]. 3.26/3.52 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A) = A. [resolve(1117,a,1095,a)]. 3.26/3.52 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)),A) = A. [resolve(1117,a,1100,a)]. 3.26/3.52 1118 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Nat_OSuc(C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom). [clausify(959)]. 3.26/3.52 1119 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oplus__class_Oplus(A,B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oplus__class_Oplus(A,C,c_Groups_Oone__class_Oone(A))),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(966)]. 3.32/3.60 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,c_Groups_Oone__class_Oone(tc_Int_Oint))),A). [resolve(1119,a,1082,a)]. 3.32/3.60 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),A). [resolve(1119,a,1089,a)]. 3.32/3.60 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,c_Groups_Oone__class_Oone(tc_Nat_Onat))),A). [resolve(1119,a,1095,a)]. 3.32/3.60 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))),A). [resolve(1119,a,1100,a)]. 3.32/3.60 1120 class_RealVector_Oreal__normed__algebra(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra) # label(axiom). [assumption]. 3.32/3.60 1121 -class_RealVector_Oreal__normed__algebra(A) | 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),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_mult_Oadd__right) # label(axiom). [clausify(17)]. 3.32/3.60 1122 -class_RealVector_Oreal__normed__algebra(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f5(B,A)) # label(fact_mult__right_Ononneg__bounded) # label(axiom). [clausify(57)]. 3.32/3.60 1123 -class_RealVector_Oreal__normed__algebra(A) | 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)),f5(B,A))) # label(fact_mult__right_Ononneg__bounded) # label(axiom). [clausify(57)]. 3.32/3.60 1124 -class_RealVector_Oreal__normed__algebra(A) | 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)),f6(C,A))) # label(fact_mult__left_Obounded) # label(axiom). [clausify(63)]. 3.32/3.60 1125 -class_RealVector_Oreal__normed__algebra(A) | 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),c_Groups_Oplus__class_Oplus(A,B,D)),C) # label(fact_mult__left_Oadd) # label(axiom). [clausify(157)]. 3.32/3.60 1126 -class_RealVector_Oreal__normed__algebra(A) | 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))),f11(A))) # label(fact_mult_Obounded) # label(axiom). [clausify(198)]. 3.32/3.60 1127 -class_RealVector_Oreal__normed__algebra(A) | c_Groups_Ouminus__class_Ouminus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ouminus__class_Ouminus(A,B)),C) # label(fact_mult_Ominus__left) # label(axiom). [clausify(226)]. 3.32/3.60 1128 -class_RealVector_Oreal__normed__algebra(A) | c_Groups_Ouminus__class_Ouminus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ouminus__class_Ouminus(A,C)) # label(fact_mult_Ominus__right) # label(axiom). [clausify(232)]. 3.32/3.60 1129 -class_RealVector_Oreal__normed__algebra(A) | 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),c_Groups_Ominus__class_Ominus(A,B,D)),C) # label(fact_mult__left_Odiff) # label(axiom). [clausify(268)]. 3.32/3.60 1130 -class_RealVector_Oreal__normed__algebra(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ozero__class_Ozero(A)) # label(fact_mult__right_Ozero) # label(axiom). [clausify(302)]. 3.32/3.60 1131 -class_RealVector_Oreal__normed__algebra(A) | 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)),f18(C,A))) # label(fact_mult__left_Ononneg__bounded) # label(axiom). [clausify(315)]. 3.32/3.60 1132 -class_RealVector_Oreal__normed__algebra(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f18(B,A)) # label(fact_mult__left_Ononneg__bounded) # label(axiom). [clausify(315)]. 3.32/3.60 1133 -class_RealVector_Oreal__normed__algebra(A) | 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),E)) = c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,B,D)),c_Groups_Ominus__class_Ominus(A,C,E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,B,D)),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),c_Groups_Ominus__class_Ominus(A,C,E))) # label(fact_mult_Oprod__diff__prod) # label(axiom). [clausify(333)]. 3.32/3.60 1134 -class_RealVector_Oreal__normed__algebra(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f23(B,A)) # label(fact_mult__left_Opos__bounded) # label(axiom). [clausify(354)]. 3.32/3.60 1135 -class_RealVector_Oreal__normed__algebra(A) | 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)),f23(C,A))) # label(fact_mult__left_Opos__bounded) # label(axiom). [clausify(354)]. 3.32/3.60 1136 -class_RealVector_Oreal__normed__algebra(A) | c_RealVector_OscaleR__class_OscaleR(A,B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_RealVector_OscaleR__class_OscaleR(A,B,C)),D) # label(fact_mult_OscaleR__left) # label(axiom). [clausify(359)]. 3.32/3.60 1137 -class_RealVector_Oreal__normed__algebra(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f24(A)) # label(fact_mult_Ononneg__bounded) # label(axiom). [clausify(375)]. 3.32/3.60 1138 -class_RealVector_Oreal__normed__algebra(A) | 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))),f24(A))) # label(fact_mult_Ononneg__bounded) # label(axiom). [clausify(375)]. 3.32/3.60 1139 -class_RealVector_Oreal__normed__algebra(A) | 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),c_Groups_Ominus__class_Ominus(A,C,D)) # label(fact_mult__right_Odiff) # label(axiom). [clausify(398)]. 3.32/3.60 1140 -class_RealVector_Oreal__normed__algebra(A) | 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))),f28(A))) # label(fact_mult_Opos__bounded) # label(axiom). [clausify(429)]. 3.32/3.61 1141 -class_RealVector_Oreal__normed__algebra(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f28(A)) # label(fact_mult_Opos__bounded) # label(axiom). [clausify(429)]. 3.32/3.61 1142 -class_RealVector_Oreal__normed__algebra(A) | c_RealVector_OscaleR__class_OscaleR(A,B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_RealVector_OscaleR__class_OscaleR(A,B,C)),D) # label(fact_mult__left_OscaleR) # label(axiom). [clausify(483)]. 3.32/3.61 1143 -class_RealVector_Oreal__normed__algebra(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,B),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,D),E) | 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(499)]. 3.32/3.61 1144 -class_RealVector_Oreal__normed__algebra(A) | c_Groups_Ouminus__class_Ouminus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ouminus__class_Ouminus(A,B)),C) # label(fact_mult__left_Ominus) # label(axiom). [clausify(518)]. 3.32/3.61 1145 -class_RealVector_Oreal__normed__algebra(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_mult__left_Ozero) # label(axiom). [clausify(554)]. 3.32/3.61 1146 -class_RealVector_Oreal__normed__algebra(A) | 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)),f39(B,A))) # label(fact_mult__right_Obounded) # label(axiom). [clausify(559)]. 3.32/3.61 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f5(A,tc_Complex_Ocomplex)). [resolve(1120,a,1122,a)]. 3.32/3.61 Derived: 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)),f5(A,tc_Complex_Ocomplex))). [resolve(1120,a,1123,a)]. 3.32/3.61 Derived: 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)),f6(B,tc_Complex_Ocomplex))). [resolve(1120,a,1124,a)]. 3.32/3.61 Derived: 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))),f11(tc_Complex_Ocomplex))). [resolve(1120,a,1126,a)]. 3.32/3.61 Derived: c_Groups_Ouminus__class_Ouminus(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_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A)),B). [resolve(1120,a,1127,a)]. 3.32/3.61 Derived: c_Groups_Ouminus__class_Ouminus(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_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B)). [resolve(1120,a,1128,a)]. 3.32/3.61 Derived: 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),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,C)),B). [resolve(1120,a,1129,a)]. 3.32/3.61 Derived: 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)),f18(B,tc_Complex_Ocomplex))). [resolve(1120,a,1131,a)]. 3.32/3.61 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f18(A,tc_Complex_Ocomplex)). [resolve(1120,a,1132,a)]. 3.32/3.61 Derived: 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),D)) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,C)),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,B,D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,C)),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),C),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,B,D))). [resolve(1120,a,1133,a)]. 3.32/3.61 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f23(A,tc_Complex_Ocomplex)). [resolve(1120,a,1134,a)]. 3.32/3.61 Derived: 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)),f23(B,tc_Complex_Ocomplex))). [resolve(1120,a,1135,a)]. 3.32/3.61 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B)),C). [resolve(1120,a,1136,a)]. 3.32/3.61 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f24(tc_Complex_Ocomplex)). [resolve(1120,a,1137,a)]. 3.32/3.61 Derived: 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))),f24(tc_Complex_Ocomplex))). [resolve(1120,a,1138,a)]. 3.32/3.61 Derived: 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),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,B,C)). [resolve(1120,a,1139,a)]. 3.32/3.61 Derived: 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))),f28(tc_Complex_Ocomplex))). [resolve(1120,a,1140,a)]. 3.32/3.61 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f28(tc_Complex_Ocomplex)). [resolve(1120,a,1141,a)]. 3.32/3.61 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,A),B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,C),D) | 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(1120,a,1143,a)]. 3.39/3.64 Derived: 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)),f39(A,tc_Complex_Ocomplex))). [resolve(1120,a,1146,a)]. 3.39/3.64 1147 -class_RealVector_Oreal__normed__algebra(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ozero__class_Ozero(A)) # label(fact_mult_Ozero__right) # label(axiom). [clausify(630)]. 3.39/3.64 1148 -class_RealVector_Oreal__normed__algebra(A) | 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)),f44(B,A))) # label(fact_mult__right_Opos__bounded) # label(axiom). [clausify(676)]. 3.39/3.64 Derived: 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)),f44(A,tc_Complex_Ocomplex))). [resolve(1148,a,1120,a)]. 3.39/3.64 1149 -class_RealVector_Oreal__normed__algebra(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f44(B,A)) # label(fact_mult__right_Opos__bounded) # label(axiom). [clausify(676)]. 3.39/3.64 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f44(A,tc_Complex_Ocomplex)). [resolve(1149,a,1120,a)]. 3.39/3.64 1150 -class_RealVector_Oreal__normed__algebra(A) | 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),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_mult__right_Oadd) # label(axiom). [clausify(696)]. 3.39/3.64 1151 -class_RealVector_Oreal__normed__algebra(A) | 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),c_Groups_Oplus__class_Oplus(A,B,D)),C) # label(fact_mult_Oadd__left) # label(axiom). [clausify(698)]. 3.39/3.64 1152 -class_RealVector_Oreal__normed__algebra(A) | 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),c_Groups_Ominus__class_Ominus(A,B,D)),C) # label(fact_mult_Odiff__left) # label(axiom). [clausify(725)]. 3.39/3.64 1153 -class_RealVector_Oreal__normed__algebra(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_mult_Ozero__left) # label(axiom). [clausify(773)]. 3.39/3.64 1154 -class_RealVector_Oreal__normed__algebra(A) | c_RealVector_OscaleR__class_OscaleR(A,B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),c_RealVector_OscaleR__class_OscaleR(A,B,D)) # label(fact_mult_OscaleR__right) # label(axiom). [clausify(835)]. 3.39/3.64 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C)). [resolve(1154,a,1120,a)]. 3.39/3.64 1155 class_RealVector_Oreal__normed__algebra(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Oreal__normed__algebra) # label(axiom). [assumption]. 3.39/3.64 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f5(A,tc_RealDef_Oreal)). [resolve(1155,a,1122,a)]. 3.39/3.64 Derived: 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)),f5(A,tc_RealDef_Oreal))). [resolve(1155,a,1123,a)]. 3.39/3.65 Derived: 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)),f6(B,tc_RealDef_Oreal))). [resolve(1155,a,1124,a)]. 3.39/3.65 Derived: 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),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C)),B). [resolve(1155,a,1125,a)]. 3.39/3.65 Derived: 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))),f11(tc_RealDef_Oreal))). [resolve(1155,a,1126,a)]. 3.39/3.65 Derived: c_Groups_Ouminus__class_Ouminus(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_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)),B). [resolve(1155,a,1127,a)]. 3.39/3.65 Derived: c_Groups_Ouminus__class_Ouminus(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_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)). [resolve(1155,a,1128,a)]. 3.39/3.65 Derived: 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),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,C)),B). [resolve(1155,a,1129,a)]. 3.39/3.65 Derived: 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)),f18(B,tc_RealDef_Oreal))). [resolve(1155,a,1131,a)]. 3.39/3.65 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f18(A,tc_RealDef_Oreal)). [resolve(1155,a,1132,a)]. 3.39/3.65 Derived: 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),D)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,C)),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,C)),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),C),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,D))). [resolve(1155,a,1133,a)]. 3.39/3.65 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f23(A,tc_RealDef_Oreal)). [resolve(1155,a,1134,a)]. 3.39/3.65 Derived: 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)),f23(B,tc_RealDef_Oreal))). [resolve(1155,a,1135,a)]. 3.39/3.65 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B)),C). [resolve(1155,a,1136,a)]. 3.39/3.65 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f24(tc_RealDef_Oreal)). [resolve(1155,a,1137,a)]. 3.39/3.65 Derived: 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))),f24(tc_RealDef_Oreal))). [resolve(1155,a,1138,a)]. 3.39/3.65 Derived: 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),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,C)). [resolve(1155,a,1139,a)]. 3.39/3.65 Derived: 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))),f28(tc_RealDef_Oreal))). [resolve(1155,a,1140,a)]. 3.39/3.65 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f28(tc_RealDef_Oreal)). [resolve(1155,a,1141,a)]. 3.39/3.65 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,A),B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,C),D) | 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(1155,a,1143,a)]. 3.39/3.65 Derived: 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)),f39(A,tc_RealDef_Oreal))). [resolve(1155,a,1146,a)]. 3.39/3.65 Derived: 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)),f44(A,tc_RealDef_Oreal))). [resolve(1155,a,1148,a)]. 3.39/3.65 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f44(A,tc_RealDef_Oreal)). [resolve(1155,a,1149,a)]. 3.39/3.65 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C)). [resolve(1155,a,1154,a)]. 3.39/3.65 1156 -class_RealVector_Oreal__normed__algebra(A) | 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(841)]. 3.39/3.65 Derived: 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(1156,a,1120,a)]. 3.39/3.65 Derived: 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(1156,a,1155,a)]. 3.45/3.74 1157 -class_RealVector_Oreal__normed__algebra(A) | 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),c_Groups_Ominus__class_Ominus(A,C,D)) # label(fact_mult_Odiff__right) # label(axiom). [clausify(845)]. 3.45/3.74 1158 -class_RealVector_Oreal__normed__algebra(A) | c_RealVector_OscaleR__class_OscaleR(A,B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),c_RealVector_OscaleR__class_OscaleR(A,B,D)) # label(fact_mult__right_OscaleR) # label(axiom). [clausify(869)]. 3.45/3.74 1159 -class_RealVector_Oreal__normed__algebra(A) | c_Groups_Ouminus__class_Ouminus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ouminus__class_Ouminus(A,C)) # label(fact_mult__right_Ominus) # label(axiom). [clausify(943)]. 3.45/3.74 1160 class_Groups_Olinordered__ab__group__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 3.45/3.74 1161 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Oplus__class_Oplus(A,B,B) = c_Groups_Ozero__class_Ozero(A) # label(fact_double__eq__0__iff) # label(axiom). [clausify(20)]. 3.45/3.74 1162 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Oplus__class_Oplus(A,B,B) != c_Groups_Ozero__class_Ozero(A) # label(fact_double__eq__0__iff) # label(axiom). [clausify(20)]. 3.45/3.74 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1160,a,1161,a)]. 3.45/3.74 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1160,a,1162,a)]. 3.45/3.74 1163 -class_Groups_Olinordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),B) # label(fact_minus__le__self__iff) # label(axiom). [clausify(324)]. 3.45/3.74 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),A). [resolve(1163,a,1160,a)]. 3.45/3.74 1164 -class_Groups_Olinordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),B) # label(fact_minus__le__self__iff) # label(axiom). [clausify(324)]. 3.45/3.74 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),A). [resolve(1164,a,1160,a)]. 3.45/3.74 1165 -class_Groups_Olinordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,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(335)]. 3.45/3.74 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1165,a,1160,a)]. 3.45/3.74 1166 -class_Groups_Olinordered__ab__group__add(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,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(335)]. 3.45/3.74 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1166,a,1160,a)]. 3.53/3.77 1167 -class_Groups_Olinordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,B),B) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) # label(fact_neg__less__nonneg) # label(axiom). [clausify(476)]. 3.53/3.77 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A). [resolve(1167,a,1160,a)]. 3.53/3.77 1168 -class_Groups_Olinordered__ab__group__add(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,B),B) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) # label(fact_neg__less__nonneg) # label(axiom). [clausify(476)]. 3.53/3.77 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A). [resolve(1168,a,1160,a)]. 3.53/3.77 1169 -class_Groups_Olinordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ouminus__class_Ouminus(A,B)) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_le__minus__self__iff) # label(axiom). [clausify(491)]. 3.53/3.77 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1169,a,1160,a)]. 3.53/3.77 1170 -class_Groups_Olinordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ouminus__class_Ouminus(A,B)) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_le__minus__self__iff) # label(axiom). [clausify(491)]. 3.53/3.77 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1170,a,1160,a)]. 3.53/3.77 1171 -class_Groups_Olinordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,B)) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(636)]. 3.53/3.77 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A)). [resolve(1171,a,1160,a)]. 3.53/3.77 1172 -class_Groups_Olinordered__ab__group__add(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,B)) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(636)]. 3.53/3.77 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A)). [resolve(1172,a,1160,a)]. 3.53/3.77 1173 -class_Groups_Olinordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,B)) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(685)]. 3.53/3.77 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A)). [resolve(1173,a,1160,a)]. 3.53/3.77 1174 -class_Groups_Olinordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,B)) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(685)]. 3.56/3.80 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A)). [resolve(1174,a,1160,a)]. 3.56/3.80 1175 -class_Groups_Olinordered__ab__group__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,B),c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(751)]. 3.56/3.80 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1175,a,1160,a)]. 3.56/3.80 1176 -class_Groups_Olinordered__ab__group__add(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,B),c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(751)]. 3.56/3.80 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1176,a,1160,a)]. 3.56/3.80 1177 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ouminus__class_Ouminus(A,B) = B # label(fact_neg__equal__zero) # label(axiom). [clausify(782)]. 3.56/3.80 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) = A. [resolve(1177,a,1160,a)]. 3.56/3.80 1178 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ouminus__class_Ouminus(A,B) != B # label(fact_neg__equal__zero) # label(axiom). [clausify(782)]. 3.56/3.80 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) != A. [resolve(1178,a,1160,a)]. 3.56/3.80 1179 class_Groups_Olinordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 3.56/3.80 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1179,a,1161,a)]. 3.56/3.80 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A) != c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1179,a,1162,a)]. 3.56/3.80 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),A). [resolve(1179,a,1163,a)]. 3.56/3.80 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),A). [resolve(1179,a,1164,a)]. 3.56/3.80 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1179,a,1165,a)]. 3.56/3.80 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1179,a,1166,a)]. 3.56/3.80 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),A) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A). [resolve(1179,a,1167,a)]. 3.64/3.89 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A). [resolve(1179,a,1168,a)]. 3.64/3.89 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1179,a,1169,a)]. 3.64/3.89 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1179,a,1170,a)]. 3.64/3.89 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A)). [resolve(1179,a,1171,a)]. 3.64/3.89 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A)). [resolve(1179,a,1172,a)]. 3.64/3.89 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A)). [resolve(1179,a,1173,a)]. 3.64/3.89 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A)). [resolve(1179,a,1174,a)]. 3.64/3.89 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1179,a,1175,a)]. 3.64/3.89 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1179,a,1176,a)]. 3.64/3.89 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) = A. [resolve(1179,a,1177,a)]. 3.64/3.89 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) != A. [resolve(1179,a,1178,a)]. 3.64/3.89 1180 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ouminus__class_Ouminus(A,B) = B # label(fact_equal__neg__zero) # label(axiom). [clausify(915)]. 3.64/3.89 1181 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ouminus__class_Ouminus(A,B) != B # label(fact_equal__neg__zero) # label(axiom). [clausify(915)]. 3.64/3.89 1182 -class_Groups_Olinordered__ab__group__add(A) | 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(962)]. 3.64/3.89 1183 -class_Groups_Olinordered__ab__group__add(A) | 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(962)]. 3.64/3.89 1184 class_Groups_Omonoid__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Omonoid__mult) # label(axiom). [assumption]. 3.64/3.89 1185 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),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(21)]. 3.64/3.89 1186 -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(62)]. 3.71/3.93 1187 -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(65)]. 3.71/3.93 1188 -class_Groups_Omonoid__mult(A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | hAPP(hAPP(c_Power_Opower__class_Opower(A),C),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,B,c_Groups_Oone__class_Oone(tc_Nat_Onat)))),C) # label(fact_realpow__minus__mult) # label(axiom). [clausify(87)]. 3.71/3.93 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)))),B). [resolve(1188,a,1184,a)]. 3.71/3.93 1189 class_Groups_Omonoid__mult(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Omonoid__mult) # label(axiom). [assumption]. 3.71/3.93 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),B),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),B),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)))),B). [resolve(1189,a,1188,a)]. 3.71/3.93 1190 -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(116)]. 3.71/3.93 1191 class_Groups_Omonoid__mult(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Omonoid__mult) # label(axiom). [assumption]. 3.71/3.93 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),B),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)))),B). [resolve(1191,a,1188,a)]. 3.71/3.93 1192 -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(188)]. 3.71/3.93 1193 -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(330)]. 3.71/3.93 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(1193,a,1184,a)]. 3.71/3.93 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(1193,a,1189,a)]. 3.71/3.93 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(1193,a,1191,a)]. 3.71/3.93 1194 class_Groups_Omonoid__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__mult) # label(axiom). [assumption]. 3.71/3.93 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),c_Groups_Ominus__class_Ominus(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)))),B). [resolve(1194,a,1188,a)]. 3.71/3.93 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(1194,a,1193,a)]. 3.71/3.93 1195 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),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(625)]. 3.82/4.06 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(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(1195,a,1184,a)]. 3.82/4.06 1196 -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(794)]. 3.82/4.06 1197 class_Rings_Oordered__cancel__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 3.82/4.06 1198 -class_Rings_Oordered__cancel__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | 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(23)]. 3.82/4.06 1199 -class_Rings_Oordered__cancel__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | 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(288)]. 3.82/4.06 1200 -class_Rings_Oordered__cancel__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | 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(288)]. 3.82/4.06 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | 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)). [resolve(1197,a,1198,a)]. 3.82/4.06 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | 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(1197,a,1199,a)]. 3.82/4.06 1201 class_Rings_Oordered__cancel__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 3.82/4.06 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(1201,a,1198,a)]. 3.82/4.06 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | 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(1201,a,1199,a)]. 3.82/4.06 1202 -class_Rings_Oordered__cancel__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | 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(529)]. 3.94/4.20 1203 -class_Rings_Oordered__cancel__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | 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(600)]. 3.94/4.20 1204 -class_Rings_Oordered__cancel__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | 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(874)]. 3.94/4.20 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | 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(1204,a,1197,a)]. 3.94/4.20 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | 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(1204,a,1201,a)]. 3.94/4.20 1205 class_Rings_Oordered__cancel__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 3.94/4.20 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | 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)). [resolve(1205,a,1198,a)]. 3.94/4.20 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | 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(1205,a,1199,a)]. 3.94/4.20 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | 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(1205,a,1204,a)]. 3.94/4.20 1206 class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 3.94/4.20 1207 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,D,B),c_Groups_Oplus__class_Oplus(A,D,C)) # label(fact_add__strict__left__mono) # label(axiom). [clausify(26)]. 3.94/4.20 1208 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,E)) # label(fact_add__le__less__mono) # label(axiom). [clausify(176)]. 3.94/4.20 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,D)). [resolve(1208,a,1206,a)]. 3.94/4.20 1209 class_Groups_Oordered__cancel__ab__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 3.94/4.20 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,A),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,B)). [resolve(1209,a,1207,a)]. 4.04/4.33 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,D) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,D)). [resolve(1209,a,1208,a)]. 4.04/4.33 1210 class_Groups_Oordered__cancel__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 4.04/4.33 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,A),c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,B)). [resolve(1210,a,1207,a)]. 4.04/4.33 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,D) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,D)). [resolve(1210,a,1208,a)]. 4.04/4.33 1211 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,E) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,E)) # label(fact_add__less__le__mono) # label(axiom). [clausify(430)]. 4.04/4.33 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,D)). [resolve(1211,a,1206,a)]. 4.04/4.33 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,D) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,D)). [resolve(1211,a,1209,a)]. 4.04/4.33 1212 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_add__strict__right__mono) # label(axiom). [clausify(434)]. 4.04/4.33 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C)). [resolve(1212,a,1209,a)]. 4.04/4.33 1213 -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,E) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,E)) # label(fact_add__strict__mono) # label(axiom). [clausify(968)]. 4.04/4.33 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,D)). [resolve(1213,a,1206,a)]. 4.04/4.33 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,D) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,D)). [resolve(1213,a,1209,a)]. 4.04/4.33 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,D) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,D)). [resolve(1213,a,1210,a)]. 4.04/4.33 1214 class_Groups_Oordered__comm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 4.04/4.33 1215 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,C)) # label(fact_add__nonneg__pos) # label(axiom). [clausify(29)]. 4.11/4.34 1216 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_add__nonpos__neg) # label(axiom). [clausify(35)]. 4.11/4.34 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B)). [resolve(1214,a,1215,a)]. 4.11/4.34 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,B,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1214,a,1216,a)]. 4.11/4.34 1217 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,C)) # label(fact_add__pos__nonneg) # label(axiom). [clausify(214)]. 4.11/4.34 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B)). [resolve(1217,a,1214,a)]. 4.11/4.34 1218 class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 4.11/4.34 1219 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,D) | c_Orderings_Oord__class_Oless(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_add__strict__increasing) # label(axiom). [clausify(290)]. 4.11/4.34 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,C) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C)). [resolve(1219,a,1214,a)]. 4.11/4.34 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)). [resolve(1219,a,1218,a)]. 4.11/4.34 1220 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | 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(369)]. 4.11/4.34 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | 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(1220,a,1214,a)]. 4.11/4.34 1221 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | 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(369)]. 4.11/4.34 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | 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(1221,a,1214,a)]. 4.15/4.39 1222 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | 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(369)]. 4.15/4.39 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | 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(1222,a,1214,a)]. 4.15/4.39 1223 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_add__nonpos__nonpos) # label(axiom). [clausify(458)]. 4.15/4.39 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1223,a,1214,a)]. 4.15/4.39 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(1223,a,1218,a)]. 4.15/4.39 1224 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_add__neg__nonpos) # label(axiom). [clausify(657)]. 4.15/4.39 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1224,a,1214,a)]. 4.15/4.39 1225 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,D) | c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_add__increasing) # label(axiom). [clausify(705)]. 4.15/4.39 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,C) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C)). [resolve(1225,a,1214,a)]. 4.15/4.39 1226 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,C)) # label(fact_add__pos__pos) # label(axiom). [clausify(759)]. 4.15/4.39 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B)). [resolve(1226,a,1214,a)]. 4.15/4.41 1227 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,D) | c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Oplus__class_Oplus(A,D,B)) # label(fact_add__increasing2) # label(axiom). [clausify(832)]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,C) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,A)). [resolve(1227,a,1214,a)]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,A)). [resolve(1227,a,1218,a)]. 4.15/4.41 1228 class_Groups_Oordered__comm__monoid__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B)). [resolve(1228,a,1215,a)]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1228,a,1216,a)]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B)). [resolve(1228,a,1217,a)]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,C) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C)). [resolve(1228,a,1219,a)]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | 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(1228,a,1220,a)]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | 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(1228,a,1221,a)]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | 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(1228,a,1222,a)]. 4.15/4.41 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1228,a,1223,a)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1228,a,1224,a)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,C) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C)). [resolve(1228,a,1225,a)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B)). [resolve(1228,a,1226,a)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,C) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,A)). [resolve(1228,a,1227,a)]. 4.15/4.42 1229 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,C,D) | c_Orderings_Oord__class_Oless(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_add__strict__increasing2) # label(axiom). [clausify(886)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,B,C) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C)). [resolve(1229,a,1214,a)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)). [resolve(1229,a,1218,a)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,C) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C)). [resolve(1229,a,1228,a)]. 4.15/4.42 1230 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Ozero__class_Ozero(A)) # label(fact_add__neg__neg) # label(axiom). [clausify(894)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,B,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1230,a,1214,a)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(1230,a,1218,a)]. 4.15/4.42 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1230,a,1228,a)]. 4.15/4.42 1231 -class_Groups_Oordered__comm__monoid__add(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oplus__class_Oplus(A,B,C)) # label(fact_add__nonneg__nonneg) # label(axiom). [clausify(926)]. 4.54/4.80 1232 -class_RealVector_Odist__norm(A) | c_RealVector_Odist__class_Odist(A,B,C) = c_RealVector_Onorm__class_Onorm(A,c_Groups_Ominus__class_Ominus(A,B,C)) # label(fact_dist__norm) # label(axiom). [clausify(542)]. 4.54/4.80 1233 class_RealVector_Odist__norm(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Odist__norm) # label(axiom). [assumption]. 4.54/4.80 1234 class_RealVector_Odist__norm(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Odist__norm) # label(axiom). [assumption]. 4.54/4.80 Derived: c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B) = c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B)). [resolve(1232,a,1234,a)]. 4.54/4.80 1235 class_Groups_Ocomm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 4.54/4.80 1236 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ozero__class_Ozero(A)) = B # label(fact_add_Ocomm__neutral) # label(axiom). [clausify(34)]. 4.54/4.80 1237 class_Groups_Ocomm__monoid__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 4.54/4.80 1238 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ozero__class_Ozero(A),B) = B # label(fact_add__0) # label(axiom). [clausify(873)]. 4.54/4.80 1239 class_Groups_Ocomm__monoid__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 4.54/4.80 1240 class_Groups_Ocomm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 4.54/4.80 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(1240,a,1236,a)]. 4.54/4.80 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) = A. [resolve(1240,a,1238,a)]. 4.54/4.80 1241 class_Groups_Ogroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ogroup__add) # label(axiom). [assumption]. 4.54/4.80 1242 -class_Groups_Ogroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oplus__class_Oplus(A,c_Groups_Ouminus__class_Ouminus(A,B),C)) = C # label(fact_add__minus__cancel) # label(axiom). [clausify(37)]. 4.54/4.80 1243 -class_Groups_Ogroup__add(A) | c_Groups_Ominus__class_Ominus(A,c_Groups_Oplus__class_Oplus(A,B,C),C) = B # label(fact_add__diff__cancel) # label(axiom). [clausify(38)]. 4.54/4.80 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),B)) = B. [resolve(1241,a,1242,a)]. 4.54/4.80 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),B) = A. [resolve(1241,a,1243,a)]. 4.54/4.80 1244 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,c_Groups_Oplus__class_Oplus(A,B,C)) = c_Groups_Oplus__class_Oplus(A,c_Groups_Ouminus__class_Ouminus(A,C),c_Groups_Ouminus__class_Ouminus(A,B)) # label(fact_minus__add) # label(axiom). [clausify(89)]. 4.54/4.80 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B)) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)). [resolve(1244,a,1241,a)]. 4.54/4.80 1245 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,c_Groups_Ouminus__class_Ouminus(A,B)) = B # label(fact_minus__minus) # label(axiom). [clausify(170)]. 4.54/4.80 1246 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,B) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_neg__0__equal__iff__equal) # label(axiom). [clausify(238)]. 4.54/4.80 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(1246,a,1241,a)]. 4.54/4.80 1247 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,B) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_neg__0__equal__iff__equal) # label(axiom). [clausify(238)]. 4.54/4.81 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(1247,a,1241,a)]. 4.54/4.81 1248 -class_Groups_Ogroup__add(A) | B != C | c_Groups_Ouminus__class_Ouminus(A,B) = c_Groups_Ouminus__class_Ouminus(A,C) # label(fact_neg__equal__iff__equal) # label(axiom). [clausify(271)]. 4.54/4.81 Derived: A != B | c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) = c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B). [resolve(1248,a,1241,a)]. 4.54/4.81 1249 -class_Groups_Ogroup__add(A) | B = C | c_Groups_Ouminus__class_Ouminus(A,B) != c_Groups_Ouminus__class_Ouminus(A,C) # label(fact_neg__equal__iff__equal) # label(axiom). [clausify(271)]. 4.54/4.81 Derived: A = B | c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) != c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B). [resolve(1249,a,1241,a)]. 4.54/4.81 1250 -class_Groups_Ogroup__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Oplus__class_Oplus(A,B,C)) = C # label(fact_minus__add__cancel) # label(axiom). [clausify(275)]. 4.54/4.81 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B)) = B. [resolve(1250,a,1241,a)]. 4.54/4.81 1251 class_Groups_Ogroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Ogroup__add) # label(axiom). [assumption]. 4.54/4.81 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),B)) = B. [resolve(1251,a,1242,a)]. 4.54/4.81 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),B) = A. [resolve(1251,a,1243,a)]. 4.54/4.81 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)). [resolve(1251,a,1244,a)]. 4.54/4.81 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)) = A. [resolve(1251,a,1245,a)]. 4.54/4.81 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1251,a,1246,a)]. 4.54/4.81 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1251,a,1247,a)]. 4.54/4.81 Derived: A != B | c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B). [resolve(1251,a,1248,a)]. 4.54/4.81 Derived: A = B | c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) != c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B). [resolve(1251,a,1249,a)]. 4.54/4.81 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B)) = B. [resolve(1251,a,1250,a)]. 4.54/4.81 1252 -class_Groups_Ogroup__add(A) | c_Groups_Ominus__class_Ominus(A,B,c_Groups_Ouminus__class_Ouminus(A,C)) = c_Groups_Oplus__class_Oplus(A,B,C) # label(fact_diff__minus__eq__add) # label(axiom). [clausify(415)]. 4.54/4.81 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B). [resolve(1252,a,1241,a)]. 4.54/4.81 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B). [resolve(1252,a,1251,a)]. 4.54/4.81 1253 -class_Groups_Ogroup__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ominus__class_Ominus(A,B,C),C) = B # label(fact_diff__add__cancel) # label(axiom). [clausify(423)]. 4.54/4.81 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B),B) = A. [resolve(1253,a,1241,a)]. 4.54/4.81 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B),B) = A. [resolve(1253,a,1251,a)]. 4.60/4.83 1254 -class_Groups_Ogroup__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ouminus__class_Ouminus(A,B),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_left__minus) # label(axiom). [clausify(439)]. 4.60/4.83 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1254,a,1251,a)]. 4.60/4.83 1255 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,B) != C | c_Groups_Oplus__class_Oplus(A,C,B) = c_Groups_Ozero__class_Ozero(A) # label(fact_eq__neg__iff__add__eq__0) # label(axiom). [clausify(493)]. 4.60/4.83 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) != B | c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1255,a,1241,a)]. 4.60/4.83 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) != B | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1255,a,1251,a)]. 4.60/4.83 1256 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,B) = C | c_Groups_Oplus__class_Oplus(A,C,B) != c_Groups_Ozero__class_Ozero(A) # label(fact_eq__neg__iff__add__eq__0) # label(axiom). [clausify(493)]. 4.60/4.83 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) = B | c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,A) != c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1256,a,1241,a)]. 4.60/4.83 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) = B | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,A) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1256,a,1251,a)]. 4.60/4.83 1257 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,B) != C | c_Groups_Ouminus__class_Ouminus(A,C) = B # label(fact_minus__equation__iff) # label(axiom). [clausify(500)]. 4.60/4.83 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) != B | c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B) = A. [resolve(1257,a,1241,a)]. 4.60/4.83 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) != B | c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B) = A. [resolve(1257,a,1251,a)]. 4.60/4.83 1258 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,B) = C | c_Groups_Ouminus__class_Ouminus(A,C) != B # label(fact_minus__equation__iff) # label(axiom). [clausify(500)]. 4.60/4.83 1259 -class_Groups_Ogroup__add(A) | c_Groups_Ominus__class_Ominus(A,B,c_Groups_Ozero__class_Ozero(A)) = B # label(fact_diff__0__right) # label(axiom). [clausify(525)]. 4.60/4.83 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(1259,a,1241,a)]. 4.60/4.83 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = A. [resolve(1259,a,1251,a)]. 4.60/4.83 1260 -class_Groups_Ogroup__add(A) | c_Groups_Ominus__class_Ominus(A,B,B) = c_Groups_Ozero__class_Ozero(A) # label(fact_diff__self) # label(axiom). [clausify(531)]. 4.60/4.83 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1260,a,1241,a)]. 4.60/4.83 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1260,a,1251,a)]. 4.60/4.83 1261 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,c_Groups_Ozero__class_Ozero(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_minus__zero) # label(axiom). [clausify(610)]. 4.60/4.83 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1261,a,1241,a)]. 4.60/4.83 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1261,a,1251,a)]. 4.60/4.83 1262 -class_Groups_Ogroup__add(A) | c_Groups_Ominus__class_Ominus(A,c_Groups_Ozero__class_Ozero(A),B) = c_Groups_Ouminus__class_Ouminus(A,B) # label(fact_diff__0) # label(axiom). [clausify(633)]. 4.60/4.83 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) = c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A). [resolve(1262,a,1241,a)]. 4.60/4.83 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A). [resolve(1262,a,1251,a)]. 4.62/4.86 1263 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,B) != C | c_Groups_Ouminus__class_Ouminus(A,C) = B # label(fact_equation__minus__iff) # label(axiom). [clausify(668)]. 4.62/4.86 1264 -class_Groups_Ogroup__add(A) | c_Groups_Ouminus__class_Ouminus(A,B) = C | c_Groups_Ouminus__class_Ouminus(A,C) != B # label(fact_equation__minus__iff) # label(axiom). [clausify(668)]. 4.62/4.86 1265 -class_Groups_Ogroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ouminus__class_Ouminus(A,B) = C # label(fact_minus__unique) # label(axiom). [clausify(701)]. 4.62/4.86 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) = B. [resolve(1265,a,1241,a)]. 4.62/4.86 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) = B. [resolve(1265,a,1251,a)]. 4.62/4.86 1266 -class_Groups_Ogroup__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) = c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ouminus__class_Ouminus(A,C)) # label(fact_diff__def) # label(axiom). [clausify(728)]. 4.62/4.86 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)). [resolve(1266,a,1241,a)]. 4.62/4.86 1267 -class_Groups_Ogroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ouminus__class_Ouminus(A,B) = C # label(fact_add__eq__0__iff) # label(axiom). [clausify(768)]. 4.62/4.86 1268 -class_Groups_Ogroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ouminus__class_Ouminus(A,B) != C # label(fact_add__eq__0__iff) # label(axiom). [clausify(768)]. 4.62/4.86 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) != B. [resolve(1268,a,1241,a)]. 4.62/4.86 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) != B. [resolve(1268,a,1251,a)]. 4.62/4.86 1269 -class_Groups_Ogroup__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) != c_Groups_Ozero__class_Ozero(A) | B = C # label(fact_right__minus__eq) # label(axiom). [clausify(786)]. 4.62/4.86 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | A = B. [resolve(1269,a,1241,a)]. 4.62/4.86 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | A = B. [resolve(1269,a,1251,a)]. 4.62/4.86 1270 -class_Groups_Ogroup__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) = c_Groups_Ozero__class_Ozero(A) | B != C # label(fact_right__minus__eq) # label(axiom). [clausify(786)]. 4.62/4.86 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | A != B. [resolve(1270,a,1241,a)]. 4.62/4.86 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | A != B. [resolve(1270,a,1251,a)]. 4.62/4.86 1271 class_Groups_Ogroup__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Ogroup__add) # label(axiom). [assumption]. 4.62/4.86 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A),B)) = B. [resolve(1271,a,1242,a)]. 4.62/4.86 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B),B) = A. [resolve(1271,a,1243,a)]. 4.62/4.86 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B)) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A)). [resolve(1271,a,1244,a)]. 4.62/4.86 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A)) = A. [resolve(1271,a,1245,a)]. 4.62/4.87 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A. [resolve(1271,a,1246,a)]. 4.62/4.87 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A. [resolve(1271,a,1247,a)]. 4.62/4.87 Derived: A != B | c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) = c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B). [resolve(1271,a,1248,a)]. 4.62/4.87 Derived: A = B | c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) != c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B). [resolve(1271,a,1249,a)]. 4.62/4.87 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B)) = B. [resolve(1271,a,1250,a)]. 4.62/4.87 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B)) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B). [resolve(1271,a,1252,a)]. 4.62/4.87 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,B),B) = A. [resolve(1271,a,1253,a)]. 4.62/4.87 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A),A) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1271,a,1254,a)]. 4.62/4.87 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) != B | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,A) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1271,a,1255,a)]. 4.62/4.87 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) = B | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,A) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1271,a,1256,a)]. 4.62/4.87 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) != B | c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B) = A. [resolve(1271,a,1257,a)]. 4.62/4.87 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)) = A. [resolve(1271,a,1259,a)]. 4.62/4.87 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,A) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1271,a,1260,a)]. 4.62/4.87 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1271,a,1261,a)]. 4.62/4.87 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex),A) = c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A). [resolve(1271,a,1262,a)]. 4.62/4.87 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) = B. [resolve(1271,a,1265,a)]. 4.62/4.87 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,B) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B)). [resolve(1271,a,1266,a)]. 4.62/4.87 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) != B. [resolve(1271,a,1268,a)]. 4.62/4.87 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,B) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | A = B. [resolve(1271,a,1269,a)]. 4.62/4.87 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,B) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | A != B. [resolve(1271,a,1270,a)]. 4.62/4.87 1272 -class_Groups_Ogroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ouminus__class_Ouminus(A,B)) = c_Groups_Ozero__class_Ozero(A) # label(fact_right__minus) # label(axiom). [clausify(870)]. 4.62/4.87 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1272,a,1241,a)]. 4.62/4.87 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1272,a,1251,a)]. 4.88/5.13 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A)) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1272,a,1271,a)]. 4.88/5.13 1273 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ouminus__class_Ouminus(A,B) = c_Groups_Ozero__class_Ozero(A) # label(fact_neg__equal__0__iff__equal) # label(axiom). [clausify(890)]. 4.88/5.13 1274 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ouminus__class_Ouminus(A,B) != c_Groups_Ozero__class_Ozero(A) # label(fact_neg__equal__0__iff__equal) # label(axiom). [clausify(890)]. 4.88/5.13 1275 class_Rings_Olinordered__semidom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semidom) # label(axiom). [assumption]. 4.88/5.13 1276 -class_Rings_Olinordered__semidom(A) | 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(43)]. 4.88/5.13 1277 -class_Rings_Olinordered__semidom(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),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(163)]. 4.88/5.13 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),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(1277,a,1275,a)]. 4.88/5.13 1278 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | 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(186)]. 4.88/5.13 1279 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),D) | 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(252)]. 4.88/5.13 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(1279,a,1275,a)]. 4.88/5.13 1280 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A),D) | 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(273)]. 4.88/5.13 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),C) | 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(1280,a,1275,a)]. 4.88/5.13 1281 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | -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)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) # label(fact_power__le__imp__le__exp) # label(axiom). [clausify(274)]. 4.88/5.13 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | -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)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C). [resolve(1281,a,1275,a)]. 4.91/5.14 1282 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | 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)) # label(fact_power__increasing__iff) # label(axiom). [clausify(284)]. 4.91/5.14 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | 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)). [resolve(1282,a,1275,a)]. 4.91/5.14 1283 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | -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)) # label(fact_power__increasing__iff) # label(axiom). [clausify(284)]. 4.91/5.14 1284 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | 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(309)]. 4.91/5.14 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | 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(1284,a,1275,a)]. 4.91/5.14 1285 -class_Rings_Olinordered__semidom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Nat_OSuc(C)) != hAPP(hAPP(c_Power_Opower__class_Opower(A),D),c_Nat_OSuc(C)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | B = D # label(fact_power__inject__base) # label(axiom). [clausify(348)]. 4.91/5.14 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Nat_OSuc(B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),c_Nat_OSuc(B)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | A = C. [resolve(1285,a,1275,a)]. 4.91/5.14 1286 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | -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)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) # label(fact_power__less__imp__less__exp) # label(axiom). [clausify(355)]. 4.91/5.14 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | -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)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C). [resolve(1286,a,1275,a)]. 4.91/5.14 1287 -class_Rings_Olinordered__semidom(A) | 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(382)]. 4.91/5.14 1288 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(437)]. 4.91/5.14 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | 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(1288,a,1275,a)]. 4.91/5.14 1289 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,C,D) | c_Orderings_Oord__class_Oless(A,C,c_Groups_Oplus__class_Oplus(A,B,D)) # label(fact_pos__add__strict) # label(axiom). [clausify(452)]. 4.91/5.15 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,B,C) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C)). [resolve(1289,a,1275,a)]. 4.91/5.15 1290 class_Rings_Olinordered__semidom(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semidom) # label(axiom). [assumption]. 4.91/5.15 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),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(1290,a,1277,a)]. 4.91/5.15 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(1290,a,1279,a)]. 4.91/5.15 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),C) | 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(1290,a,1280,a)]. 4.91/5.15 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | -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)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C). [resolve(1290,a,1281,a)]. 4.91/5.15 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | 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)). [resolve(1290,a,1282,a)]. 4.91/5.15 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Nat_OSuc(B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),c_Nat_OSuc(B)) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | A = C. [resolve(1290,a,1285,a)]. 4.91/5.15 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | -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)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C). [resolve(1290,a,1286,a)]. 4.91/5.15 Derived: 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(1290,a,1287,a)]. 4.91/5.15 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | 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(1290,a,1288,a)]. 4.91/5.15 1291 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | 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(488)]. 4.91/5.15 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | 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(1291,a,1275,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | 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(1291,a,1290,a)]. 4.91/5.16 1292 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),D) | -c_Orderings_Oord__class_Oless(A,D,c_Groups_Oone__class_Oone(A)) | 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(569)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,c_Groups_Oone__class_Oone(tc_Int_Oint)) | 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(1292,a,1275,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,c_Groups_Oone__class_Oone(tc_Nat_Onat)) | 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(1292,a,1290,a)]. 4.91/5.16 1293 -class_Rings_Olinordered__semidom(A) | -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)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | c_Orderings_Oord__class_Oless(A,B,D) # label(fact_power__less__imp__less__base) # label(axiom). [clausify(575)]. 4.91/5.16 Derived: -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)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | c_Orderings_Oord__class_Oless(tc_Int_Oint,A,C). [resolve(1293,a,1275,a)]. 4.91/5.16 Derived: -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)) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,C). [resolve(1293,a,1290,a)]. 4.91/5.16 1294 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Nat_OSuc(C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),c_Nat_OSuc(C))) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | c_Orderings_Oord__class_Oless__eq(A,B,D) # label(fact_power__le__imp__le__base) # label(axiom). [clausify(592)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Nat_OSuc(B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),c_Nat_OSuc(B))) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,C). [resolve(1294,a,1275,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Nat_OSuc(B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),c_Nat_OSuc(B))) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,C). [resolve(1294,a,1290,a)]. 4.91/5.16 1295 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | C != D | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) # label(fact_power__inject__exp) # label(axiom). [clausify(642)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | B != C | 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(1295,a,1275,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | B != C | 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(1295,a,1290,a)]. 4.91/5.16 1296 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | C = D | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) # label(fact_power__inject__exp) # label(axiom). [clausify(642)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | B = C | 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(1296,a,1275,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | B = C | 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(1296,a,1290,a)]. 4.91/5.16 1297 class_Rings_Olinordered__semidom(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semidom) # label(axiom). [assumption]. 4.91/5.16 Derived: 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(1297,a,1276,a)]. 4.91/5.16 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),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(1297,a,1277,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | 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(1297,a,1278,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(1297,a,1279,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),C) | 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(1297,a,1280,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | -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)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C). [resolve(1297,a,1281,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | 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)). [resolve(1297,a,1282,a)]. 4.91/5.16 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | 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(1297,a,1284,a)]. 4.91/5.17 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),c_Nat_OSuc(B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),c_Nat_OSuc(B)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | A = C. [resolve(1297,a,1285,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | -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)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C). [resolve(1297,a,1286,a)]. 4.91/5.17 Derived: 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(1297,a,1287,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | 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(1297,a,1288,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,C) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C)). [resolve(1297,a,1289,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | 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(1297,a,1291,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) | 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(1297,a,1292,a)]. 4.91/5.17 Derived: -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)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,C). [resolve(1297,a,1293,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),c_Nat_OSuc(B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),C),c_Nat_OSuc(B))) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,C). [resolve(1297,a,1294,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | B != C | 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(1297,a,1295,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | B = C | 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(1297,a,1296,a)]. 4.91/5.17 1298 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oone__class_Oone(A),B) | 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(649)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | 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(1298,a,1275,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | 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(1298,a,1290,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | 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(1298,a,1297,a)]. 4.91/5.17 1299 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),D) | 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(655)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),C) | c_Orderings_Oord__class_Oless(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(1299,a,1275,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),C) | c_Orderings_Oord__class_Oless(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(1299,a,1290,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),C) | c_Orderings_Oord__class_Oless(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(1299,a,1297,a)]. 4.91/5.17 1300 -class_Rings_Olinordered__semidom(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oone__class_Oone(A))) # label(fact_less__add__one) # label(axiom). [clausify(659)]. 4.91/5.17 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(1300,a,1275,a)]. 4.91/5.17 Derived: c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat))). [resolve(1300,a,1290,a)]. 4.91/5.17 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))). [resolve(1300,a,1297,a)]. 4.91/5.17 1301 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | 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(712)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | 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(1301,a,1275,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | 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(1301,a,1290,a)]. 4.91/5.17 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | 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(1301,a,1297,a)]. 4.91/5.19 1302 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Oone__class_Oone(A)) | 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(776)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Oone__class_Oone(tc_Int_Oint)) | 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(1302,a,1275,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)) | 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(1302,a,1290,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) | 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(1302,a,1297,a)]. 4.91/5.19 1303 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Nat_OSuc(C))) # label(fact_power__gt1) # label(axiom). [clausify(790)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | 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),c_Nat_OSuc(B))). [resolve(1303,a,1275,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | 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),c_Nat_OSuc(B))). [resolve(1303,a,1290,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | 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),c_Nat_OSuc(B))). [resolve(1303,a,1297,a)]. 4.91/5.19 1304 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),C) | 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(850)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),B) | 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(1304,a,1275,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),B) | 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(1304,a,1290,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),B) | 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(1304,a,1297,a)]. 4.91/5.19 1305 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | -c_Orderings_Oord__class_Oless__eq(A,D,c_Groups_Oone__class_Oone(A)) | 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(868)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,c_Groups_Oone__class_Oone(tc_Int_Oint)) | 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(1305,a,1275,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,c_Groups_Oone__class_Oone(tc_Nat_Onat)) | 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(1305,a,1290,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) | 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(1305,a,1297,a)]. 4.91/5.19 1306 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | 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(896)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),A) | 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(1306,a,1275,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A) | 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(1306,a,1290,a)]. 4.91/5.19 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) | 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(1306,a,1297,a)]. 4.91/5.19 1307 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | -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)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) # label(fact_power__strict__increasing__iff) # label(axiom). [clausify(904)]. 4.91/5.19 1308 -class_Rings_Olinordered__semidom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oone__class_Oone(A),B) | 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)) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,C,D) # label(fact_power__strict__increasing__iff) # label(axiom). [clausify(904)]. 5.05/5.31 1309 -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) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | -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(921)]. 5.05/5.31 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) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | A = C. [resolve(1309,a,1275,a)]. 5.05/5.31 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) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | A = C. [resolve(1309,a,1290,a)]. 5.05/5.31 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) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | A = C. [resolve(1309,a,1297,a)]. 5.05/5.31 1310 -class_Rings_Olinordered__semidom(A) | -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(960)]. 5.05/5.31 Derived: -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(1310,a,1275,a)]. 5.05/5.31 Derived: -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(1310,a,1290,a)]. 5.05/5.31 Derived: -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(1310,a,1297,a)]. 5.05/5.31 1311 -class_Rings_Olinordered__semidom(A) | -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(965)]. 5.05/5.31 Derived: -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(1311,a,1275,a)]. 5.05/5.31 Derived: -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(1311,a,1290,a)]. 5.05/5.31 Derived: -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(1311,a,1297,a)]. 5.05/5.31 1312 class_Rings_Olinordered__ring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 5.05/5.31 1313 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,C,c_Groups_Ozero__class_Ozero(A)) | 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(44)]. 5.05/5.31 1314 -class_Rings_Olinordered__ring__strict(A) | 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) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(55)]. 5.05/5.31 1315 -class_Rings_Olinordered__ring__strict(A) | 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) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(55)]. 5.05/5.31 1316 -class_Rings_Olinordered__ring__strict(A) | 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) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(55)]. 5.05/5.31 1317 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -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)) | c_Orderings_Oord__class_Oless(A,D,C) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(206)]. 5.05/5.31 1318 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | 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)) | -c_Orderings_Oord__class_Oless(A,D,C) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(206)]. 5.05/5.31 1319 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -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)) | c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_mult__le__cancel__left__pos) # label(axiom). [clausify(209)]. 5.05/5.31 1320 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | 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)) | -c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_mult__le__cancel__left__pos) # label(axiom). [clausify(209)]. 5.05/5.31 1321 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) # label(fact_mult__le__0__iff) # label(axiom). [clausify(219)]. 5.05/5.31 1322 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_mult__le__0__iff) # label(axiom). [clausify(219)]. 5.05/5.31 1323 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) # label(fact_mult__le__0__iff) # label(axiom). [clausify(219)]. 5.05/5.31 1324 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_mult__le__0__iff) # label(axiom). [clausify(219)]. 5.05/5.31 1325 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) # label(fact_mult__le__0__iff) # label(axiom). [clausify(219)]. 5.05/5.31 1326 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_mult__le__0__iff) # label(axiom). [clausify(219)]. 5.05/5.31 1327 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C | c_Orderings_Oord__class_Oless__eq(A,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(253)]. 5.05/5.31 1328 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | -c_Orderings_Oord__class_Oless__eq(A,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(253)]. 5.05/5.31 1329 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | -c_Orderings_Oord__class_Oless__eq(A,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(253)]. 5.05/5.31 1330 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),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(363)]. 5.05/5.31 1331 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),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(363)]. 5.05/5.31 1332 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),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(363)]. 5.05/5.31 1333 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,D) | 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(391)]. 5.05/5.31 1334 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless__eq(A,C,D) | -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(391)]. 5.05/5.31 1335 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),D) | 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(402)]. 5.05/5.31 1336 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,c_Groups_Ozero__class_Ozero(A)) | 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(402)]. 5.05/5.31 1337 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) | -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(402)]. 5.05/5.31 1338 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,D,c_Groups_Ozero__class_Ozero(A)) | -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(402)]. 5.05/5.31 1339 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless(A,C,D) | -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(402)]. 5.05/5.31 1340 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -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(402)]. 5.05/5.31 1341 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -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)) | c_Orderings_Oord__class_Oless(A,C,D) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(519)]. 5.05/5.31 1342 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | 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)) | -c_Orderings_Oord__class_Oless(A,C,D) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(519)]. 5.05/5.31 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,B,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | 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(1312,a,1313,a)]. 5.05/5.31 Derived: 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) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(1312,a,1314,a)]. 5.05/5.31 Derived: 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) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(1312,a,1315,a)]. 5.05/5.31 Derived: 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) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(1312,a,1316,a)]. 5.05/5.31 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -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)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,C,B). [resolve(1312,a,1317,a)]. 5.05/5.31 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | 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)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,B). [resolve(1312,a,1318,a)]. 5.05/5.32 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | 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)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,C). [resolve(1312,a,1320,a)]. 5.05/5.32 Derived: -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)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1312,a,1322,a)]. 5.05/5.32 Derived: -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)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B). [resolve(1312,a,1323,a)]. 5.05/5.32 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,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(1312,a,1327,a)]. 5.05/5.32 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,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(1312,a,1328,a)]. 5.05/5.32 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,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(1312,a,1329,a)]. 5.05/5.32 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),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(1312,a,1330,a)]. 5.05/5.32 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),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(1312,a,1331,a)]. 5.05/5.32 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),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(1312,a,1332,a)]. 5.05/5.32 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,C) | 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(1312,a,1333,a)]. 5.05/5.32 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,C) | -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(1312,a,1334,a)]. 5.05/5.32 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,A) | -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(1312,a,1337,a)]. 5.05/5.34 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,C,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -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(1312,a,1338,a)]. 5.05/5.34 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,C) | -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(1312,a,1339,a)]. 5.05/5.34 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -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(1312,a,1340,a)]. 5.05/5.34 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -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)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,C). [resolve(1312,a,1341,a)]. 5.05/5.34 1343 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,c_Groups_Ozero__class_Ozero(A)) | 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(606)]. 5.05/5.34 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | 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(1343,a,1312,a)]. 5.05/5.34 1344 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(708)]. 5.05/5.34 1345 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(708)]. 5.05/5.34 Derived: -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)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1345,a,1312,a)]. 5.05/5.34 1346 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(708)]. 5.05/5.34 Derived: -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)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1346,a,1312,a)]. 5.05/5.34 1347 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(708)]. 5.05/5.34 1348 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(708)]. 5.05/5.34 1349 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(708)]. 5.05/5.34 Derived: 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)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1349,a,1312,a)]. 5.05/5.34 1350 class_Rings_Olinordered__ring__strict(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 5.05/5.34 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | 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(1350,a,1313,a)]. 5.05/5.34 Derived: 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) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1350,a,1314,a)]. 5.05/5.34 Derived: 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) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B. [resolve(1350,a,1315,a)]. 5.05/5.34 Derived: 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) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B. [resolve(1350,a,1316,a)]. 5.05/5.34 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -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)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,B). [resolve(1350,a,1317,a)]. 5.05/5.34 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | 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)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,B). [resolve(1350,a,1318,a)]. 5.05/5.34 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | 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)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,C). [resolve(1350,a,1320,a)]. 5.05/5.34 Derived: -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)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1350,a,1322,a)]. 5.05/5.34 Derived: -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)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B). [resolve(1350,a,1323,a)]. 5.05/5.34 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,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(1350,a,1327,a)]. 5.05/5.34 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,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(1350,a,1328,a)]. 5.05/5.34 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,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(1350,a,1329,a)]. 5.05/5.34 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),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(1350,a,1330,a)]. 5.05/5.34 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),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(1350,a,1331,a)]. 5.05/5.34 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),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(1350,a,1332,a)]. 5.05/5.34 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,C) | 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(1350,a,1333,a)]. 5.05/5.34 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,C) | -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(1350,a,1334,a)]. 5.05/5.34 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,A) | -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(1350,a,1337,a)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -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(1350,a,1338,a)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,C) | -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(1350,a,1339,a)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -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(1350,a,1340,a)]. 5.05/5.35 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -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)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,C). [resolve(1350,a,1341,a)]. 5.05/5.35 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | 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(1350,a,1343,a)]. 5.05/5.35 Derived: -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)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1350,a,1345,a)]. 5.05/5.35 Derived: -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)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1350,a,1346,a)]. 5.05/5.35 Derived: 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)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1350,a,1349,a)]. 5.05/5.35 1351 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,C,D) | c_Orderings_Oord__class_Oless(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(734)]. 5.05/5.35 1352 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless(A,C,D) | 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(734)]. 5.05/5.35 1353 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | -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(734)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -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(1353,a,1312,a)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -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(1353,a,1350,a)]. 5.05/5.35 1354 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,C,D) | -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(734)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,C) | -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(1354,a,1312,a)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,C) | -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(1354,a,1350,a)]. 5.05/5.35 1355 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),D) | -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(734)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | -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(1355,a,1312,a)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -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(1355,a,1350,a)]. 5.05/5.35 1356 -class_Rings_Olinordered__ring__strict(A) | c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,C,B) | -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(734)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,B,A) | -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(1356,a,1312,a)]. 5.05/5.35 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,B,A) | -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(1356,a,1350,a)]. 5.42/5.67 1357 -class_Rings_Olinordered__ring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,c_Groups_Ozero__class_Ozero(A)) | 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(844)]. 5.42/5.67 1358 -class_Groups_Oordered__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,D,B),c_Groups_Oplus__class_Oplus(A,D,C)) # label(fact_add__left__mono) # label(axiom). [clausify(113)]. 5.42/5.67 1359 class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 5.42/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B)). [resolve(1358,a,1359,a)]. 5.42/5.67 1360 class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 5.42/5.67 1361 -class_Groups_Oordered__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,E) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,E)) # label(fact_add__mono) # label(axiom). [clausify(538)]. 5.42/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,D) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,D)). [resolve(1361,a,1360,a)]. 5.42/5.67 1362 class_Groups_Oordered__ab__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 5.42/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,A),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,B)). [resolve(1362,a,1358,a)]. 5.42/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,D) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,D)). [resolve(1362,a,1361,a)]. 5.42/5.67 1363 -class_Groups_Oordered__ab__semigroup__add(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_add__right__mono) # label(axiom). [clausify(910)]. 5.42/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B,C)). [resolve(1363,a,1359,a)]. 5.42/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,C)). [resolve(1363,a,1360,a)]. 5.42/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C)). [resolve(1363,a,1362,a)]. 5.42/5.67 1364 class_Groups_Oordered__ab__group__add__abs(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 5.42/5.67 1365 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,B),C) # label(fact_abs__leI) # label(axiom). [clausify(51)]. 5.42/5.67 1366 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Oabs__class_Oabs(A,C))),c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,B,C))) # label(fact_abs__triangle__ineq3) # label(axiom). [clausify(70)]. 5.42/5.67 1367 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Oabs__class_Oabs(A,B)) # label(fact_abs__ge__minus__self) # label(axiom). [clausify(104)]. 5.42/5.67 1368 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Oabs__class_Oabs(A,c_Groups_Ouminus__class_Ouminus(A,B)) = c_Groups_Oabs__class_Oabs(A,B) # label(fact_abs__minus__cancel) # label(axiom). [clausify(114)]. 5.42/5.67 1369 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),B) | c_Groups_Oabs__class_Oabs(A,B) = B # label(fact_abs__of__pos) # label(axiom). [clausify(122)]. 5.42/5.67 1370 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,E))),c_Groups_Oplus__class_Oplus(A,c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,B,D)),c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,C,E)))) # label(fact_abs__diff__triangle__ineq) # label(axiom). [clausify(127)]. 5.42/5.67 1371 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Oabs__class_Oabs(A,c_Groups_Ozero__class_Ozero(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_abs__zero) # label(axiom). [clausify(196)]. 5.42/5.67 1372 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,B),C) | c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_abs__le__D1) # label(axiom). [clausify(212)]. 5.42/5.67 1373 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,c_Groups_Oabs__class_Oabs(A,B)),c_Groups_Ozero__class_Ozero(A)) # label(fact_abs__minus__le__zero) # label(axiom). [clausify(251)]. 5.42/5.67 1374 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oabs__class_Oabs(A,B)) # label(fact_abs__ge__zero) # label(axiom). [clausify(255)]. 5.42/5.67 1375 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ominus__class_Ominus(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Oabs__class_Oabs(A,C)),c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,C,B))) # label(fact_abs__triangle__ineq2__sym) # label(axiom). [clausify(283)]. 5.42/5.67 1376 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Ozero__class_Ozero(A)) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_abs__le__zero__iff) # label(axiom). [clausify(292)]. 5.42/5.67 1377 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Ozero__class_Ozero(A)) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_abs__le__zero__iff) # label(axiom). [clausify(292)]. 5.42/5.67 1378 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Oabs__class_Oabs(A,B) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_abs__eq__0) # label(axiom). [clausify(347)]. 5.42/5.67 1379 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Oabs__class_Oabs(A,B) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_abs__eq__0) # label(axiom). [clausify(347)]. 5.42/5.67 1380 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Oabs__class_Oabs(A,c_Groups_Oabs__class_Oabs(A,B)) = c_Groups_Oabs__class_Oabs(A,B) # label(fact_abs__idempotent) # label(axiom). [clausify(368)]. 5.42/5.67 1381 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Groups_Oabs__class_Oabs(A,B) = B # label(fact_abs__of__nonneg) # label(axiom). [clausify(503)]. 5.42/5.67 1382 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oabs__class_Oabs(A,B)) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_zero__less__abs__iff) # label(axiom). [clausify(512)]. 5.42/5.67 1383 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),c_Groups_Oabs__class_Oabs(A,B)) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_zero__less__abs__iff) # label(axiom). [clausify(512)]. 5.42/5.67 1384 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Ozero__class_Ozero(A)) # label(fact_abs__not__less__zero) # label(axiom). [clausify(591)]. 5.42/5.67 1385 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Oabs__class_Oabs(A,c_Groups_Oplus__class_Oplus(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Oabs__class_Oabs(A,C))) = c_Groups_Oplus__class_Oplus(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Oabs__class_Oabs(A,C)) # label(fact_abs__add__abs) # label(axiom). [clausify(594)]. 5.42/5.67 1386 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,B),C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),C) # label(fact_abs__le__iff) # label(axiom). [clausify(602)]. 5.42/5.67 1387 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,B),C) | c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_abs__le__iff) # label(axiom). [clausify(602)]. 5.42/5.67 1388 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,B),C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),C) | -c_Orderings_Oord__class_Oless__eq(A,B,C) # label(fact_abs__le__iff) # label(axiom). [clausify(602)]. 5.42/5.67 1389 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,B),C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ouminus__class_Ouminus(A,B),C) # label(fact_abs__le__D2) # label(axiom). [clausify(613)]. 5.42/5.67 1390 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,B,C)) = c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,C,B)) # label(fact_abs__minus__commute) # label(axiom). [clausify(650)]. 5.42/5.67 1391 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Groups_Oabs__class_Oabs(A,B) = c_Groups_Ouminus__class_Ouminus(A,B) # label(fact_abs__of__neg) # label(axiom). [clausify(710)]. 5.42/5.67 1392 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,c_Groups_Oplus__class_Oplus(A,B,C)),c_Groups_Oplus__class_Oplus(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Oabs__class_Oabs(A,C))) # label(fact_abs__triangle__ineq) # label(axiom). [clausify(726)]. 5.42/5.67 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),B). [resolve(1364,a,1365,a)]. 5.42/5.67 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint,B))),c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B))). [resolve(1364,a,1366,a)]. 5.42/5.67 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)). [resolve(1364,a,1367,a)]. 5.42/5.67 Derived: c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)) = c_Groups_Oabs__class_Oabs(tc_Int_Oint,A). [resolve(1364,a,1368,a)]. 5.42/5.67 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Groups_Oabs__class_Oabs(tc_Int_Oint,A) = A. [resolve(1364,a,1369,a)]. 5.42/5.67 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,D))),c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,C)),c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,B,D)))). [resolve(1364,a,1370,a)]. 5.42/5.68 Derived: c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1364,a,1371,a)]. 5.42/5.68 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B). [resolve(1364,a,1372,a)]. 5.42/5.68 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1364,a,1373,a)]. 5.42/5.68 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)). [resolve(1364,a,1374,a)]. 5.42/5.68 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint,B)),c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,B,A))). [resolve(1364,a,1375,a)]. 5.42/5.68 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(1364,a,1376,a)]. 5.42/5.68 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(1364,a,1377,a)]. 5.42/5.68 Derived: c_Groups_Oabs__class_Oabs(tc_Int_Oint,A) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(1364,a,1378,a)]. 5.42/5.68 Derived: c_Groups_Oabs__class_Oabs(tc_Int_Oint,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(1364,a,1379,a)]. 5.42/5.68 Derived: c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)) = c_Groups_Oabs__class_Oabs(tc_Int_Oint,A). [resolve(1364,a,1380,a)]. 5.42/5.68 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Groups_Oabs__class_Oabs(tc_Int_Oint,A) = A. [resolve(1364,a,1381,a)]. 5.42/5.68 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(1364,a,1382,a)]. 5.42/5.68 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(1364,a,1383,a)]. 5.42/5.68 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1364,a,1384,a)]. 5.42/5.68 Derived: c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint,B))) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint,B)). [resolve(1364,a,1385,a)]. 5.42/5.68 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),B) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),B). [resolve(1364,a,1386,a)]. 5.42/5.68 Derived: c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B)) = c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,B,A)). [resolve(1364,a,1390,a)]. 5.42/5.68 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Groups_Oabs__class_Oabs(tc_Int_Oint,A) = c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A). [resolve(1364,a,1391,a)]. 5.42/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B)),c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint,B))). [resolve(1364,a,1392,a)]. 5.42/5.69 1393 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ominus__class_Ominus(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Oabs__class_Oabs(A,C)),c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,B,C))) # label(fact_abs__triangle__ineq2) # label(axiom). [clausify(808)]. 5.42/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint,B)),c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B))). [resolve(1393,a,1364,a)]. 5.42/5.69 1394 class_Groups_Oordered__ab__group__add__abs(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 5.42/5.69 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),B). [resolve(1394,a,1365,a)]. 5.42/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,B))),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B))). [resolve(1394,a,1366,a)]. 5.42/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)). [resolve(1394,a,1367,a)]. 5.42/5.69 Derived: c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A). [resolve(1394,a,1368,a)]. 5.42/5.69 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A) = A. [resolve(1394,a,1369,a)]. 5.42/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,D))),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,C)),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,D)))). [resolve(1394,a,1370,a)]. 5.42/5.69 Derived: c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1394,a,1371,a)]. 5.42/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1394,a,1373,a)]. 5.42/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)). [resolve(1394,a,1374,a)]. 5.42/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,B)),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,A))). [resolve(1394,a,1375,a)]. 5.42/5.69 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1394,a,1376,a)]. 5.42/5.69 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1394,a,1377,a)]. 5.42/5.71 Derived: c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1394,a,1378,a)]. 5.42/5.71 Derived: c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1394,a,1379,a)]. 5.42/5.71 Derived: c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A). [resolve(1394,a,1380,a)]. 5.42/5.71 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A) = A. [resolve(1394,a,1381,a)]. 5.42/5.71 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1394,a,1382,a)]. 5.42/5.71 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1394,a,1383,a)]. 5.42/5.71 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1394,a,1384,a)]. 5.42/5.71 Derived: c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,B))) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,B)). [resolve(1394,a,1385,a)]. 5.42/5.71 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),B). [resolve(1394,a,1386,a)]. 5.42/5.71 Derived: c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B)) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,A)). [resolve(1394,a,1390,a)]. 5.42/5.71 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A). [resolve(1394,a,1391,a)]. 5.42/5.71 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,B))). [resolve(1394,a,1392,a)]. 5.42/5.71 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,B)),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B))). [resolve(1394,a,1393,a)]. 5.42/5.71 1395 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Oabs__class_Oabs(A,B)) # label(fact_abs__ge__self) # label(axiom). [clausify(935)]. 5.42/5.71 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)). [resolve(1395,a,1364,a)]. 5.42/5.71 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)). [resolve(1395,a,1394,a)]. 5.42/5.71 1396 -class_Groups_Oordered__ab__group__add__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Groups_Oabs__class_Oabs(A,B) = c_Groups_Ouminus__class_Ouminus(A,B) # label(fact_abs__of__nonpos) # label(axiom). [clausify(938)]. 5.42/5.71 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Groups_Oabs__class_Oabs(tc_Int_Oint,A) = c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A). [resolve(1396,a,1364,a)]. 5.61/5.84 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A). [resolve(1396,a,1394,a)]. 5.61/5.84 1397 -class_Groups_Oordered__ab__group__add__abs(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,B,C)),c_Groups_Oplus__class_Oplus(A,c_Groups_Oabs__class_Oabs(A,B),c_Groups_Oabs__class_Oabs(A,C))) # label(fact_abs__triangle__ineq4) # label(axiom). [clausify(975)]. 5.61/5.84 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B)),c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint,B))). [resolve(1397,a,1364,a)]. 5.61/5.84 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B)),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,B))). [resolve(1397,a,1394,a)]. 5.61/5.84 1398 class_Rings_Oordered__ring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oordered__ring) # label(axiom). [assumption]. 5.61/5.84 1399 -class_Rings_Oordered__ring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,c_Groups_Ozero__class_Ozero(A)) | 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(52)]. 5.61/5.84 1400 -class_Rings_Oordered__ring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,c_Groups_Ozero__class_Ozero(A)) | 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(301)]. 5.61/5.84 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | 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(1398,a,1399,a)]. 5.61/5.84 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | 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(1398,a,1400,a)]. 5.61/5.84 1401 -class_Rings_Oordered__ring(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D),c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F)) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,B,E)),C),D),F) # label(fact_less__add__iff1) # label(axiom). [clausify(632)]. 5.61/5.84 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,D)),B),C),E). [resolve(1401,a,1398,a)]. 5.61/5.84 1402 -class_Rings_Oordered__ring(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D),c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,B,E)),C),D),F) # label(fact_less__add__iff1) # label(axiom). [clausify(632)]. 5.61/5.85 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,D)),B),C),E). [resolve(1402,a,1398,a)]. 5.61/5.85 1403 -class_Rings_Oordered__ring(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D),c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F)) | c_Orderings_Oord__class_Oless__eq(A,D,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,E,B)),C),F)) # label(fact_le__add__iff2) # label(axiom). [clausify(671)]. 5.61/5.85 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,D,A)),B),E)). [resolve(1403,a,1398,a)]. 5.61/5.85 1404 -class_Rings_Oordered__ring(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D),c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F)) | -c_Orderings_Oord__class_Oless__eq(A,D,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,E,B)),C),F)) # label(fact_le__add__iff2) # label(axiom). [clausify(671)]. 5.61/5.85 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E)) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,D,A)),B),E)). [resolve(1404,a,1398,a)]. 5.61/5.85 1405 -class_Rings_Oordered__ring(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,B,C)),D),E),F) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D),E),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(711)]. 5.61/5.85 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B)),C),D),E) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C),D),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C),E)). [resolve(1405,a,1398,a)]. 5.61/5.85 1406 -class_Rings_Oordered__ring(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,B,C)),D),E),F) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D),E),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(711)]. 5.61/5.86 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B)),C),D),E) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C),D),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C),E)). [resolve(1406,a,1398,a)]. 5.61/5.86 1407 class_Rings_Oordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__ring) # label(axiom). [assumption]. 5.61/5.86 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | 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(1407,a,1399,a)]. 5.61/5.86 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | 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(1407,a,1400,a)]. 5.61/5.86 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,D)),B),C),E). [resolve(1407,a,1401,a)]. 5.61/5.86 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,D)),B),C),E). [resolve(1407,a,1402,a)]. 5.61/5.86 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,D,A)),B),E)). [resolve(1407,a,1403,a)]. 5.61/5.86 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,D,A)),B),E)). [resolve(1407,a,1404,a)]. 5.61/5.86 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B)),C),D),E) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C),D),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C),E)). [resolve(1407,a,1405,a)]. 5.61/5.86 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B)),C),D),E) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C),D),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C),E)). [resolve(1407,a,1406,a)]. 5.61/5.87 1408 -class_Rings_Oordered__ring(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D),c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F)) | c_Orderings_Oord__class_Oless(A,D,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,E,B)),C),F)) # label(fact_less__add__iff2) # label(axiom). [clausify(738)]. 5.61/5.87 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,D,A)),B),E)). [resolve(1408,a,1398,a)]. 5.61/5.87 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,C,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,D,A)),B),E)). [resolve(1408,a,1407,a)]. 5.61/5.87 1409 -class_Rings_Oordered__ring(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D),c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F)) | -c_Orderings_Oord__class_Oless(A,D,c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,E,B)),C),F)) # label(fact_less__add__iff2) # label(axiom). [clausify(738)]. 5.61/5.87 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,D,A)),B),E)). [resolve(1409,a,1398,a)]. 5.61/5.87 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,D,A)),B),E)). [resolve(1409,a,1407,a)]. 5.61/5.87 1410 -class_Rings_Oordered__ring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | 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(833)]. 5.61/5.87 1411 -class_Rings_Oordered__ring(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | 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(872)]. 5.61/5.87 1412 -class_Rings_Oordered__ring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | 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(872)]. 5.76/6.02 1413 class_Groups_Oab__group__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Oab__group__add) # label(axiom). [assumption]. 5.76/6.02 1414 -class_Groups_Oab__group__add(A) | c_Groups_Ouminus__class_Ouminus(A,c_Groups_Oplus__class_Oplus(A,B,C)) = c_Groups_Oplus__class_Oplus(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Ouminus__class_Ouminus(A,C)) # label(fact_minus__add__distrib) # label(axiom). [clausify(53)]. 5.76/6.02 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B)) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B)). [resolve(1413,a,1414,a)]. 5.76/6.02 1415 -class_Groups_Oab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) = c_Groups_Ouminus__class_Ouminus(A,c_Groups_Ominus__class_Ominus(A,C,B)) # label(fact_minus__diff__eq) # label(axiom). [clausify(378)]. 5.76/6.02 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,B) = c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,B,A)). [resolve(1415,a,1413,a)]. 5.76/6.02 1416 -class_Groups_Oab__group__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Ouminus__class_Ouminus(A,B),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_ab__left__minus) # label(axiom). [clausify(424)]. 5.76/6.02 1417 -class_Groups_Oab__group__add(A) | c_Groups_Ominus__class_Ominus(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,E)) = c_Groups_Oplus__class_Oplus(A,c_Groups_Ominus__class_Ominus(A,B,D),c_Groups_Ominus__class_Ominus(A,C,E)) # label(fact_Deriv_Oadd__diff__add) # label(axiom). [clausify(537)]. 5.76/6.02 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,C,D)) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,C),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,B,D)). [resolve(1417,a,1413,a)]. 5.76/6.02 1418 -class_Groups_Oab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) != c_Groups_Ozero__class_Ozero(A) | B = C # label(fact_eq__iff__diff__eq__0) # label(axiom). [clausify(541)]. 5.76/6.02 1419 -class_Groups_Oab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) = c_Groups_Ozero__class_Ozero(A) | B != C # label(fact_eq__iff__diff__eq__0) # label(axiom). [clausify(541)]. 5.76/6.02 1420 -class_Groups_Oab__group__add(A) | c_Groups_Ominus__class_Ominus(A,c_Groups_Ouminus__class_Ouminus(A,B),c_Groups_Ouminus__class_Ouminus(A,C)) = c_Groups_Ouminus__class_Ouminus(A,c_Groups_Ominus__class_Ominus(A,B,C)) # label(fact_Limits_Ominus__diff__minus) # label(axiom). [clausify(697)]. 5.76/6.02 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B)) = c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,B)). [resolve(1420,a,1413,a)]. 5.76/6.02 1421 class_Groups_Oab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__group__add) # label(axiom). [assumption]. 5.76/6.02 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) = c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,B,A)). [resolve(1421,a,1415,a)]. 5.76/6.02 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(1421,a,1416,a)]. 5.76/6.02 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,D)) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,C),c_Groups_Ominus__class_Ominus(tc_Int_Oint,B,D)). [resolve(1421,a,1417,a)]. 5.76/6.02 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)) = c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B)). [resolve(1421,a,1420,a)]. 5.82/6.15 1422 class_Groups_Oab__group__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oab__group__add) # label(axiom). [assumption]. 5.82/6.15 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)). [resolve(1422,a,1414,a)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,A)). [resolve(1422,a,1415,a)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,D)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,C),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,D)). [resolve(1422,a,1417,a)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B)). [resolve(1422,a,1420,a)]. 5.82/6.15 1423 -class_Groups_Oab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) != c_Groups_Ominus__class_Ominus(A,D,E) | B != C | D = E # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(906)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,B) != c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,C,D) | A != B | C = D. [resolve(1423,a,1413,a)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) != c_Groups_Ominus__class_Ominus(tc_Int_Oint,C,D) | A != B | C = D. [resolve(1423,a,1421,a)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) != c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,C,D) | A != B | C = D. [resolve(1423,a,1422,a)]. 5.82/6.15 1424 -class_Groups_Oab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) != c_Groups_Ominus__class_Ominus(A,D,E) | B = C | D != E # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(906)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,B) != c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,C,D) | A = B | C != D. [resolve(1424,a,1413,a)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B) != c_Groups_Ominus__class_Ominus(tc_Int_Oint,C,D) | A = B | C != D. [resolve(1424,a,1421,a)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) != c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,C,D) | A = B | C != D. [resolve(1424,a,1422,a)]. 5.82/6.15 1425 -class_Groups_Oab__group__add(A) | c_Groups_Ominus__class_Ominus(A,B,C) = c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ouminus__class_Ouminus(A,C)) # label(fact_ab__diff__minus) # label(axiom). [clausify(945)]. 5.82/6.15 Derived: c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)). [resolve(1425,a,1422,a)]. 5.82/6.15 1426 class_Groups_Ocancel__semigroup__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 5.82/6.15 1427 -class_Groups_Ocancel__semigroup__add(A) | B != C | c_Groups_Oplus__class_Oplus(A,D,B) = c_Groups_Oplus__class_Oplus(A,D,C) # label(fact_add__left__cancel) # label(axiom). [clausify(54)]. 5.82/6.15 1428 -class_Groups_Ocancel__semigroup__add(A) | B = C | c_Groups_Oplus__class_Oplus(A,D,B) != c_Groups_Oplus__class_Oplus(A,D,C) # label(fact_add__left__cancel) # label(axiom). [clausify(54)]. 5.82/6.15 1429 -class_Groups_Ocancel__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Oplus__class_Oplus(A,D,C) | B = D # label(fact_add__right__cancel) # label(axiom). [clausify(60)]. 5.82/6.15 1430 -class_Groups_Ocancel__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) = c_Groups_Oplus__class_Oplus(A,D,C) | B != D # label(fact_add__right__cancel) # label(axiom). [clausify(60)]. 5.82/6.15 Derived: A != B | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,C,A) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,C,B). [resolve(1426,a,1427,a)]. 6.02/6.33 Derived: A = B | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,C,A) != c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,C,B). [resolve(1426,a,1428,a)]. 6.02/6.33 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B) != c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,C,B) | A = C. [resolve(1426,a,1429,a)]. 6.02/6.33 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,C,B) | A != C. [resolve(1426,a,1430,a)]. 6.02/6.33 1431 -class_Groups_Ocancel__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Oplus__class_Oplus(A,D,C) | B = D # label(fact_add__right__imp__eq) # label(axiom). [clausify(217)]. 6.02/6.33 1432 class_Groups_Ocancel__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 6.02/6.33 Derived: A != B | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,A) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,B). [resolve(1432,a,1427,a)]. 6.02/6.33 Derived: A = B | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,A) != c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,B). [resolve(1432,a,1428,a)]. 6.02/6.33 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B) != c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,B) | A = C. [resolve(1432,a,1429,a)]. 6.02/6.33 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,B) | A != C. [resolve(1432,a,1430,a)]. 6.02/6.33 1433 class_Groups_Ocancel__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 6.02/6.33 Derived: A != B | c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,A) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,B). [resolve(1433,a,1427,a)]. 6.02/6.33 Derived: A = B | c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,A) != c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,B). [resolve(1433,a,1428,a)]. 6.02/6.33 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B) != c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,B) | A = C. [resolve(1433,a,1429,a)]. 6.02/6.33 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,B) | A != C. [resolve(1433,a,1430,a)]. 6.02/6.33 1434 class_Groups_Ocancel__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 6.02/6.33 Derived: A != B | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,A) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B). [resolve(1434,a,1427,a)]. 6.02/6.33 Derived: A = B | c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,A) != c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B). [resolve(1434,a,1428,a)]. 6.02/6.33 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) != c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B) | A = C. [resolve(1434,a,1429,a)]. 6.02/6.33 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B) | A != C. [resolve(1434,a,1430,a)]. 6.02/6.33 1435 -class_Groups_Ocancel__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Oplus__class_Oplus(A,B,D) | C = D # label(fact_add__left__imp__eq) # label(axiom). [clausify(964)]. 6.02/6.33 1436 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) # label(fact_field__power__not__zero) # label(axiom). [clausify(520)]. 6.02/6.33 1437 class_Rings_Oring__1__no__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 6.02/6.33 1438 class_Rings_Oring__1__no__zero__divisors(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 6.02/6.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B). [resolve(1436,a,1437,a)]. 6.02/6.33 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B). [resolve(1436,a,1438,a)]. 6.02/6.33 1439 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Oone__class_Oone(A) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) | c_Groups_Ouminus__class_Ouminus(A,c_Groups_Oone__class_Oone(A)) = B | c_Groups_Oone__class_Oone(A) = B # label(fact_square__eq__1__iff) # label(axiom). [clausify(730)]. 6.21/6.48 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) | c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)) = A | c_Groups_Oone__class_Oone(tc_Int_Oint) = A. [resolve(1439,a,1437,a)]. 6.21/6.48 Derived: c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),A) | c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)) = A | c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = A. [resolve(1439,a,1438,a)]. 6.21/6.48 1440 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) | c_Groups_Ouminus__class_Ouminus(A,c_Groups_Oone__class_Oone(A)) != B # label(fact_square__eq__1__iff) # label(axiom). [clausify(730)]. 6.21/6.48 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) | c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)) != A. [resolve(1440,a,1437,a)]. 6.21/6.48 Derived: c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),A) | c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)) != A. [resolve(1440,a,1438,a)]. 6.21/6.48 1441 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) | c_Groups_Oone__class_Oone(A) != B # label(fact_square__eq__1__iff) # label(axiom). [clausify(730)]. 6.21/6.48 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) | c_Groups_Oone__class_Oone(tc_Int_Oint) != A. [resolve(1441,a,1437,a)]. 6.21/6.48 Derived: c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),A) | c_Groups_Oone__class_Oone(tc_Complex_Ocomplex) != A. [resolve(1441,a,1438,a)]. 6.21/6.48 1442 class_Rings_Oring__1__no__zero__divisors(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 6.21/6.48 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B). [resolve(1442,a,1436,a)]. 6.21/6.48 Derived: c_Groups_Oone__class_Oone(tc_RealDef_Oreal) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A) | c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) = A | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = A. [resolve(1442,a,1439,a)]. 6.21/6.48 Derived: c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A) | c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) != A. [resolve(1442,a,1440,a)]. 6.21/6.48 Derived: c_Groups_Oone__class_Oone(tc_RealDef_Oreal) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A) | c_Groups_Oone__class_Oone(tc_RealDef_Oreal) != A. [resolve(1442,a,1441,a)]. 6.21/6.48 1443 class_Rings_Oring__1(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Oring__1) # label(axiom). [assumption]. 6.21/6.48 1444 -class_Rings_Oring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ouminus__class_Ouminus(A,B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ouminus__class_Ouminus(A,c_Groups_Oone__class_Oone(A))),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_power__minus) # label(axiom). [clausify(64)]. 6.21/6.48 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),A),B)). [resolve(1443,a,1444,a)]. 6.36/6.63 1445 class_Rings_Oring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1) # label(axiom). [assumption]. 6.36/6.63 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint))),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)). [resolve(1445,a,1444,a)]. 6.36/6.63 1446 class_Rings_Oring__1(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oring__1) # label(axiom). [assumption]. 6.36/6.63 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)). [resolve(1446,a,1444,a)]. 6.36/6.63 1447 -class_Rings_Oring__1(A) | 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),c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oone__class_Oone(A))),c_Groups_Ominus__class_Ominus(A,B,c_Groups_Oone__class_Oone(A))) # label(fact_real__squared__diff__one__factored) # label(axiom). [clausify(820)]. 6.36/6.63 Derived: 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),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))). [resolve(1447,a,1443,a)]. 6.36/6.63 Derived: 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),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,c_Groups_Oone__class_Oone(tc_Int_Oint))),c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(1447,a,1445,a)]. 6.36/6.63 Derived: 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),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))). [resolve(1447,a,1446,a)]. 6.36/6.63 1448 -class_Rings_Olinordered__idom(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ominus__class_Ominus(A,C,D),B) | c_Orderings_Oord__class_Oless(A,c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,B,C)),D) # label(fact_abs__diff__less__iff) # label(axiom). [clausify(77)]. 6.36/6.63 1449 class_Rings_Olinordered__idom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__idom) # label(axiom). [assumption]. 6.36/6.63 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,C)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,B,C),A) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B)),C). [resolve(1448,a,1449,a)]. 6.36/6.63 1450 -class_Rings_Olinordered__idom(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,B,C)),D) # label(fact_abs__diff__less__iff) # label(axiom). [clausify(77)]. 6.36/6.63 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Oplus__class_Oplus(tc_Int_Oint,B,C)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B)),C). [resolve(1450,a,1449,a)]. 6.36/6.65 1451 -class_Rings_Olinordered__idom(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Ominus__class_Ominus(A,B,C),D) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oabs__class_Oabs(A,c_Groups_Ominus__class_Ominus(A,D,B)),C) # label(fact_abs__diff__less__iff) # label(axiom). [clausify(77)]. 6.36/6.65 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,B),C) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Ominus__class_Ominus(tc_Int_Oint,C,A)),B). [resolve(1451,a,1449,a)]. 6.36/6.65 1452 -class_Rings_Olinordered__idom(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Oone__class_Oone(A)) | c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),B) # label(fact_mult__right__le__one__le) # label(axiom). [clausify(224)]. 6.36/6.65 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Oone__class_Oone(tc_Int_Oint)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),A). [resolve(1452,a,1449,a)]. 6.36/6.65 1453 -class_Rings_Olinordered__idom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oabs__class_Oabs(A,B),C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oabs__class_Oabs(A,D),E) | c_Orderings_Oord__class_Oless(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oabs__class_Oabs(A,B)),c_Groups_Oabs__class_Oabs(A,D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)) # label(fact_abs__mult__less) # label(axiom). [clausify(296)]. 6.36/6.65 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,C),D) | c_Orderings_Oord__class_Oless(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)),c_Groups_Oabs__class_Oabs(tc_Int_Oint,C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)). [resolve(1453,a,1449,a)]. 6.36/6.65 1454 -class_Rings_Olinordered__idom(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Oabs__class_Oabs(A,B)),C)) # label(fact_zero__le__power__abs) # label(axiom). [clausify(329)]. 6.36/6.65 1455 -class_Rings_Olinordered__idom(A) | c_Groups_Oabs__class_Oabs(A,c_Groups_Oone__class_Oone(A)) = c_Groups_Oone__class_Oone(A) # label(fact_abs__one) # label(axiom). [clausify(341)]. 6.36/6.65 Derived: c_Groups_Oabs__class_Oabs(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Groups_Oone__class_Oone(tc_Int_Oint). [resolve(1455,a,1449,a)]. 6.36/6.65 1456 -class_Rings_Olinordered__idom(A) | c_Groups_Oabs__class_Oabs(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Oabs__class_Oabs(A,B)),C) # label(fact_power__abs) # label(axiom). [clausify(579)]. 6.36/6.65 Derived: c_Groups_Oabs__class_Oabs(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_Groups_Oabs__class_Oabs(tc_Int_Oint,A)),B). [resolve(1456,a,1449,a)]. 6.36/6.65 1457 class_Rings_Olinordered__idom(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__idom) # label(axiom). [assumption]. 6.36/6.65 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,C),A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B)),C). [resolve(1457,a,1448,a)]. 6.42/6.66 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B)),C). [resolve(1457,a,1450,a)]. 6.42/6.66 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,C,A)),B). [resolve(1457,a,1451,a)]. 6.42/6.66 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),A). [resolve(1457,a,1452,a)]. 6.42/6.66 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,C),D) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),D)). [resolve(1457,a,1453,a)]. 6.42/6.66 Derived: 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),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)),B)). [resolve(1457,a,1454,a)]. 6.42/6.66 Derived: c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal). [resolve(1457,a,1455,a)]. 6.42/6.66 Derived: c_Groups_Oabs__class_Oabs(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_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)),B). [resolve(1457,a,1456,a)]. 6.42/6.66 1458 -class_Rings_Olinordered__idom(A) | c_Groups_Oabs__class_Oabs(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oabs__class_Oabs(A,B)),c_Groups_Oabs__class_Oabs(A,C)) # label(fact_abs__mult) # label(axiom). [clausify(639)]. 6.42/6.66 Derived: c_Groups_Oabs__class_Oabs(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_Groups_Oabs__class_Oabs(tc_Int_Oint,A)),c_Groups_Oabs__class_Oabs(tc_Int_Oint,B)). [resolve(1458,a,1449,a)]. 6.42/6.66 Derived: c_Groups_Oabs__class_Oabs(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_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,B)). [resolve(1458,a,1457,a)]. 6.42/6.66 1459 -class_Rings_Olinordered__idom(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,B),c_Groups_Ozero__class_Ozero(A)) # label(fact_even__less__0__iff) # label(axiom). [clausify(687)]. 6.42/6.66 1460 -class_Rings_Olinordered__idom(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,B),c_Groups_Ozero__class_Ozero(A)) # label(fact_even__less__0__iff) # label(axiom). [clausify(687)]. 6.42/6.66 1461 -class_Rings_Olinordered__idom(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Groups_Oabs__class_Oabs(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oabs__class_Oabs(A,C)),B) # label(fact_abs__mult__pos) # label(axiom). [clausify(709)]. 6.42/6.66 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | c_Groups_Oabs__class_Oabs(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_Groups_Oabs__class_Oabs(tc_Int_Oint,B)),A). [resolve(1461,a,1449,a)]. 6.42/6.68 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | c_Groups_Oabs__class_Oabs(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_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,B)),A). [resolve(1461,a,1457,a)]. 6.42/6.68 1462 -class_Rings_Olinordered__idom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oabs__class_Oabs(A,B),C) | c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,B),C) # label(fact_abs__less__iff) # label(axiom). [clausify(770)]. 6.42/6.68 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),B). [resolve(1462,a,1449,a)]. 6.42/6.68 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),B). [resolve(1462,a,1457,a)]. 6.42/6.68 1463 -class_Rings_Olinordered__idom(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oabs__class_Oabs(A,B),C) | c_Orderings_Oord__class_Oless(A,B,C) # label(fact_abs__less__iff) # label(axiom). [clausify(770)]. 6.42/6.68 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),B) | c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B). [resolve(1463,a,1449,a)]. 6.42/6.68 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B). [resolve(1463,a,1457,a)]. 6.42/6.68 1464 -class_Rings_Olinordered__idom(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Oabs__class_Oabs(A,B),C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ouminus__class_Ouminus(A,B),C) | -c_Orderings_Oord__class_Oless(A,B,C) # label(fact_abs__less__iff) # label(axiom). [clausify(770)]. 6.42/6.68 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oabs__class_Oabs(tc_Int_Oint,A),B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A),B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B). [resolve(1464,a,1449,a)]. 6.42/6.68 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A),B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B). [resolve(1464,a,1457,a)]. 6.42/6.68 1465 -class_Rings_Olinordered__idom(A) | c_Groups_Oabs__class_Oabs(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ouminus__class_Ouminus(A,B)),C)) = c_Groups_Oabs__class_Oabs(A,hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_abs__power__minus) # label(axiom). [clausify(791)]. 6.42/6.68 Derived: c_Groups_Oabs__class_Oabs(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)),B)) = c_Groups_Oabs__class_Oabs(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)). [resolve(1465,a,1449,a)]. 6.42/6.68 Derived: c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)),B)) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),A),B)). [resolve(1465,a,1457,a)]. 6.42/6.68 1466 -class_Rings_Olinordered__idom(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oabs__class_Oabs(A,B)),c_Groups_Oabs__class_Oabs(A,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) # label(fact_abs__mult__self) # label(axiom). [clausify(857)]. 6.42/6.68 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)),c_Groups_Oabs__class_Oabs(tc_Int_Oint,A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A). [resolve(1466,a,1449,a)]. 6.42/6.68 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)),c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),A). [resolve(1466,a,1457,a)]. 6.73/6.97 1467 -class_Rings_Olinordered__idom(A) | B = C | c_Orderings_Oord__class_Oless(A,C,B) | c_Orderings_Oord__class_Oless(A,B,C) # label(fact_linorder__neqE__linordered__idom) # label(axiom). [clausify(954)]. 6.73/6.97 1468 -class_Rings_Olinordered__idom(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Oone__class_Oone(A)) | c_Orderings_Oord__class_Oless__eq(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B),B) # label(fact_mult__left__le__one__le) # label(axiom). [clausify(958)]. 6.73/6.97 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,c_Groups_Oone__class_Oone(tc_Int_Oint)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A),A). [resolve(1468,a,1449,a)]. 6.73/6.97 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A),A). [resolve(1468,a,1457,a)]. 6.73/6.97 1469 -class_Rings_Olinordered__idom(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ouminus__class_Ouminus(A,B)) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_less__minus__self__iff) # label(axiom). [clausify(970)]. 6.73/6.97 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1469,a,1449,a)]. 6.73/6.97 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1469,a,1457,a)]. 6.73/6.97 1470 -class_Rings_Olinordered__idom(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ouminus__class_Ouminus(A,B)) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) # label(fact_less__minus__self__iff) # label(axiom). [clausify(970)]. 6.73/6.97 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(1470,a,1449,a)]. 6.73/6.97 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1470,a,1457,a)]. 6.73/6.97 1471 class_Rings_Olinordered__comm__semiring__strict(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 6.73/6.97 1472 -class_Rings_Olinordered__comm__semiring__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Ozero__class_Ozero(A),D) | 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(73)]. 6.73/6.97 1473 class_Rings_Olinordered__comm__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 6.73/6.97 1474 class_Rings_Olinordered__comm__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 6.73/6.97 1475 class_RealVector_Ometric__space(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Ometric__space) # label(axiom). [assumption]. 6.73/6.97 1476 -class_RealVector_Ometric__space(A) | c_RealVector_Odist__class_Odist(A,B,C) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | C = B # label(fact_dist__eq__0__iff) # label(axiom). [clausify(80)]. 6.73/6.97 1477 -class_RealVector_Ometric__space(A) | c_RealVector_Odist__class_Odist(A,B,C) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | C != B # label(fact_dist__eq__0__iff) # label(axiom). [clausify(80)]. 6.73/6.97 1478 -class_RealVector_Ometric__space(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(A,B,C)) # label(fact_zero__le__dist) # label(axiom). [clausify(164)]. 6.73/6.97 1479 -class_RealVector_Ometric__space(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(A,B,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(A,B,D),c_RealVector_Odist__class_Odist(A,C,D))) # label(fact_dist__triangle2) # label(axiom). [clausify(167)]. 6.73/6.97 1480 -class_RealVector_Ometric__space(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(A,B,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(A,D,B),c_RealVector_Odist__class_Odist(A,D,C))) # label(fact_dist__triangle3) # label(axiom). [clausify(168)]. 6.73/6.97 1481 -class_RealVector_Ometric__space(A) | c_RealVector_Odist__class_Odist(A,B,B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) # label(fact_dist__self) # label(axiom). [clausify(225)]. 6.73/6.97 1482 -class_RealVector_Ometric__space(A) | c_RealVector_Odist__class_Odist(A,B,C) = c_RealVector_Odist__class_Odist(A,C,B) # label(fact_dist__commute) # label(axiom). [clausify(234)]. 6.73/6.97 Derived: c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | B = A. [resolve(1475,a,1476,a)]. 6.73/6.97 Derived: c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | B != A. [resolve(1475,a,1477,a)]. 6.73/6.97 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B)). [resolve(1475,a,1478,a)]. 6.73/6.97 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,C),c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,B,C))). [resolve(1475,a,1479,a)]. 6.73/6.97 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,C,A),c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,C,B))). [resolve(1475,a,1480,a)]. 6.73/6.97 Derived: c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1475,a,1481,a)]. 6.73/6.97 Derived: c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B) = c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,B,A). [resolve(1475,a,1482,a)]. 6.73/6.97 1483 class_RealVector_Ometric__space(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Ometric__space) # label(axiom). [assumption]. 6.73/6.97 Derived: c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B) != c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | B = A. [resolve(1483,a,1476,a)]. 6.73/6.97 Derived: c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | B != A. [resolve(1483,a,1477,a)]. 6.73/6.97 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B)). [resolve(1483,a,1478,a)]. 6.73/6.97 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,C),c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,B,C))). [resolve(1483,a,1479,a)]. 6.73/6.97 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,C,A),c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,C,B))). [resolve(1483,a,1480,a)]. 6.73/7.01 Derived: c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1483,a,1481,a)]. 6.73/7.01 Derived: c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B) = c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,B,A). [resolve(1483,a,1482,a)]. 6.73/7.01 1484 -class_RealVector_Ometric__space(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(A,B,C),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) # label(fact_dist__not__less__zero) # label(axiom). [clausify(612)]. 6.73/7.01 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1484,a,1475,a)]. 6.73/7.01 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1484,a,1483,a)]. 6.73/7.01 1485 -class_RealVector_Ometric__space(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(A,B,C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(A,B,D),c_RealVector_Odist__class_Odist(A,D,C))) # label(fact_dist__triangle) # label(axiom). [clausify(756)]. 6.73/7.01 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,C),c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,C,B))). [resolve(1485,a,1475,a)]. 6.73/7.01 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,C),c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,C,B))). [resolve(1485,a,1483,a)]. 6.73/7.01 1486 -class_RealVector_Ometric__space(A) | B != C | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(A,C,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) # label(fact_dist__le__zero__iff) # label(axiom). [clausify(911)]. 6.73/7.01 Derived: A != B | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,B,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1486,a,1475,a)]. 6.73/7.01 Derived: A != B | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,B,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1486,a,1483,a)]. 6.73/7.01 1487 -class_RealVector_Ometric__space(A) | B = C | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(A,C,B),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) # label(fact_dist__le__zero__iff) # label(axiom). [clausify(911)]. 6.73/7.01 Derived: A = B | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,B,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1487,a,1475,a)]. 6.73/7.01 Derived: A = B | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,B,A),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)). [resolve(1487,a,1483,a)]. 6.73/7.01 1488 -class_RealVector_Ometric__space(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(A,B,C)) | C != B # label(fact_zero__less__dist__iff) # label(axiom). [clausify(932)]. 6.73/7.01 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B)) | B != A. [resolve(1488,a,1475,a)]. 6.73/7.01 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B)) | B != A. [resolve(1488,a,1483,a)]. 6.73/7.01 1489 -class_RealVector_Ometric__space(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(A,B,C)) | C = B # label(fact_zero__less__dist__iff) # label(axiom). [clausify(932)]. 7.04/7.28 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(tc_RealDef_Oreal,A,B)) | B = A. [resolve(1489,a,1475,a)]. 7.04/7.28 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),c_RealVector_Odist__class_Odist(tc_Complex_Ocomplex,A,B)) | B = A. [resolve(1489,a,1483,a)]. 7.04/7.28 1490 class_Groups_Ocomm__monoid__mult(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 7.04/7.28 1491 -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(81)]. 7.04/7.28 1492 -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(222)]. 7.04/7.28 1493 class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 7.04/7.28 1494 -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(361)]. 7.04/7.28 1495 class_Groups_Ocomm__monoid__mult(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 7.04/7.28 1496 class_Groups_Ocomm__monoid__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 7.04/7.28 1497 -class_Rings_Osemiring(A) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oplus__class_Oplus(A,B,C)),D),E) = c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D),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(110)]. 7.04/7.28 1498 class_Rings_Osemiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Osemiring) # label(axiom). [assumption]. 7.04/7.28 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B)),C),D) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),C),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C),D)). [resolve(1497,a,1498,a)]. 7.04/7.28 1499 class_Rings_Osemiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring) # label(axiom). [assumption]. 7.04/7.28 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B)),C),D) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C),c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C),D)). [resolve(1499,a,1497,a)]. 7.04/7.28 1500 class_Rings_Osemiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring) # label(axiom). [assumption]. 7.04/7.28 Derived: c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B)),C),D) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C),D)). [resolve(1500,a,1497,a)]. 7.04/7.28 1501 class_Rings_Osemiring(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Osemiring) # label(axiom). [assumption]. 7.04/7.28 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,A,B)),C),D) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),C),c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),C),D)). [resolve(1501,a,1497,a)]. 7.22/7.45 1502 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]. 7.22/7.45 1503 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_add__le__imp__le__left) # label(axiom). [clausify(84)]. 7.22/7.45 1504 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_add__le__cancel__left) # label(axiom). [clausify(337)]. 7.22/7.45 1505 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | -c_Orderings_Oord__class_Oless__eq(A,C,D) # label(fact_add__le__cancel__left) # label(axiom). [clausify(337)]. 7.22/7.45 1506 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,C)) | c_Orderings_Oord__class_Oless(A,B,D) # label(fact_add__less__cancel__right) # label(axiom). [clausify(384)]. 7.22/7.45 1507 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,C)) | -c_Orderings_Oord__class_Oless(A,B,D) # label(fact_add__less__cancel__right) # label(axiom). [clausify(384)]. 7.22/7.45 1508 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,C)) | c_Orderings_Oord__class_Oless__eq(A,B,D) # label(fact_add__le__imp__le__right) # label(axiom). [clausify(396)]. 7.22/7.45 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,C)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,B,C). [resolve(1502,a,1503,a)]. 7.22/7.45 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,B)) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,C). [resolve(1502,a,1506,a)]. 7.22/7.45 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,B)) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,C). [resolve(1502,a,1508,a)]. 7.22/7.45 1509 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless(A,B,C) | c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,D,B),c_Groups_Oplus__class_Oplus(A,D,C)) # label(fact_add__less__cancel__left) # label(axiom). [clausify(404)]. 7.22/7.45 1510 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,D,B),c_Groups_Oplus__class_Oplus(A,D,C)) # label(fact_add__less__cancel__left) # label(axiom). [clausify(404)]. 7.22/7.45 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,A),c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,C,B)). [resolve(1510,a,1502,a)]. 7.22/7.45 1511 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]. 7.22/7.45 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,C). [resolve(1511,a,1503,a)]. 7.34/7.62 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,C)) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,B,C). [resolve(1511,a,1505,a)]. 7.34/7.62 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,B)) | c_Orderings_Oord__class_Oless(tc_Int_Oint,A,C). [resolve(1511,a,1506,a)]. 7.34/7.62 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,A,B),c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,B)) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,C). [resolve(1511,a,1508,a)]. 7.34/7.62 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,A),c_Groups_Oplus__class_Oplus(tc_Int_Oint,C,B)). [resolve(1511,a,1510,a)]. 7.34/7.62 1512 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,D,C)) | c_Orderings_Oord__class_Oless(A,B,D) # label(fact_add__less__imp__less__right) # label(axiom). [clausify(656)]. 7.34/7.62 1513 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless(A,c_Groups_Oplus__class_Oplus(A,B,C),c_Groups_Oplus__class_Oplus(A,B,D)) | c_Orderings_Oord__class_Oless(A,C,D) # label(fact_add__less__imp__less__left) # label(axiom). [clausify(723)]. 7.34/7.62 1514 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]. 7.34/7.62 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C). [resolve(1514,a,1503,a)]. 7.34/7.62 Derived: -c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B)) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,C). [resolve(1514,a,1506,a)]. 7.34/7.62 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,B),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,C,B)) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,C). [resolve(1514,a,1508,a)]. 7.34/7.62 1515 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_add__le__cancel__right) # label(axiom). [clausify(875)]. 7.34/7.62 1516 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Oplus__class_Oplus(A,B,D),c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_add__le__cancel__right) # label(axiom). [clausify(875)]. 7.34/7.62 1517 -class_Rings_Olinordered__semiring__1(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),E) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),F) | c_Groups_Oplus__class_Oplus(A,E,F) != c_Groups_Oone__class_Oone(A) | c_Orderings_Oord__class_Oless__eq(A,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(571)]. 7.34/7.62 1518 class_Rings_Olinordered__semiring__1(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 7.34/7.62 1519 class_Rings_Olinordered__semiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 7.34/7.62 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),D) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),E) | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,D,E) != c_Groups_Oone__class_Oone(tc_RealDef_Oreal) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,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(1517,a,1518,a)]. 7.64/7.92 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),D) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),E) | c_Groups_Oplus__class_Oplus(tc_Int_Oint,D,E) != c_Groups_Oone__class_Oone(tc_Int_Oint) | c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,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(1517,a,1519,a)]. 7.64/7.92 1520 -class_Rings_Olinordered__semiring(A) | -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)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | c_Orderings_Oord__class_Oless(A,C,D) # label(fact_mult__left__less__imp__less) # label(axiom). [clausify(572)]. 7.64/7.92 1521 class_Rings_Olinordered__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring) # label(axiom). [assumption]. 7.64/7.92 1522 class_Rings_Olinordered__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring) # label(axiom). [assumption]. 7.64/7.92 1523 class_Rings_Olinordered__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semiring) # label(axiom). [assumption]. 7.64/7.92 1524 -class_Rings_Olinordered__semiring(A) | -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)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Orderings_Oord__class_Oless(A,B,D) # label(fact_mult__right__less__imp__less) # label(axiom). [clausify(877)]. 7.64/7.92 Derived: -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)) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | c_Orderings_Oord__class_Oless(tc_Nat_Onat,A,C). [resolve(1524,a,1522,a)]. 7.64/7.92 1525 -class_RealVector_Oreal__normed__vector(A) | c_SEQ_OBseq(A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f14(B,A,C))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) # label(fact_Bseq__def) # label(axiom). [clausify(277)]. 7.64/7.92 1526 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f8(B,A)) # label(fact_BseqD) # label(axiom). [clausify(97)]. 7.64/7.92 1527 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,C)),f8(B,A)) # label(fact_BseqD) # label(axiom). [clausify(97)]. 7.64/7.92 1528 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,C)),f13(B,A)) # label(fact_Bseq__def) # label(axiom). [clausify(277)]. 7.64/7.92 1529 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f13(B,A)) # label(fact_Bseq__def) # label(axiom). [clausify(277)]. 7.64/7.92 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f14(B,A,C))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f8(B,A)). [resolve(1525,b,1526,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f14(B,A,C))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,D)),f8(B,A)). [resolve(1525,b,1527,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f14(B,A,C))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,D)),f13(B,A)). [resolve(1525,b,1528,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f14(B,A,C))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f13(B,A)). [resolve(1525,b,1529,b)]. 7.64/7.93 1530 -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,f27(C,B,A))),B) | c_SEQ_OBseq(A,C) # label(fact_BseqI) # label(axiom). [clausify(420)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,f27(C,B,A))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f8(C,A)). [resolve(1530,d,1526,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,f27(C,B,A))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,D)),f8(C,A)). [resolve(1530,d,1527,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,f27(C,B,A))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,D)),f13(C,A)). [resolve(1530,d,1528,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,f27(C,B,A))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f13(C,A)). [resolve(1530,d,1529,b)]. 7.64/7.93 1531 -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f32(B,A,C,D)),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,D)))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | c_SEQ_OBseq(A,B) # label(fact_Bseq__iff3) # label(axiom). [clausify(470)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f32(B,A,C,D)),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,D)))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f8(B,A)). [resolve(1531,d,1526,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f32(B,A,C,D)),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,D)))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,E)),f8(B,A)). [resolve(1531,d,1527,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f32(B,A,C,D)),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,D)))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,E)),f13(B,A)). [resolve(1531,d,1528,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f32(B,A,C,D)),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,D)))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f13(B,A)). [resolve(1531,d,1529,b)]. 7.64/7.93 1532 -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,C),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,f34(B,A))))),f33(B,A)) | -c_SEQ_OBseq(A,B) # label(fact_Bseq__iff3) # label(axiom). [clausify(470)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,C),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,f34(B,A))))),f33(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f14(B,A,D))),D) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),D). [resolve(1532,c,1525,b)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,C),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,f34(B,A))))),f33(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),D) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f27(B,D,A))),D). [resolve(1532,c,1530,d)]. 7.64/7.93 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,C),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,f34(B,A))))),f33(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f32(B,A,D,E)),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,E)))),D) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),D). [resolve(1532,c,1531,d)]. 7.64/7.93 1533 -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f33(B,A)) | -c_SEQ_OBseq(A,B) # label(fact_Bseq__iff3) # label(axiom). [clausify(470)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f33(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f14(B,A,C))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C). [resolve(1533,c,1525,b)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f33(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f27(B,C,A))),C). [resolve(1533,c,1530,d)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f33(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f32(B,A,C,D)),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,D)))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C). [resolve(1533,c,1531,d)]. 7.72/7.95 1534 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f51(B,A)) # label(fact_Bseq__iff2) # label(axiom). [clausify(722)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f51(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f14(B,A,C))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C). [resolve(1534,b,1525,b)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f51(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f27(B,C,A))),C). [resolve(1534,b,1530,d)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f51(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f32(B,A,C,D)),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,D)))),C) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C). [resolve(1534,b,1531,d)]. 7.72/7.95 1535 -class_RealVector_Oreal__normed__vector(A) | -c_SEQ_OBseq(A,B) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,C),c_Groups_Ouminus__class_Ouminus(A,f52(B,A)))),f51(B,A)) # label(fact_Bseq__iff2) # label(axiom). [clausify(722)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,C),c_Groups_Ouminus__class_Ouminus(A,f52(B,A)))),f51(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f14(B,A,D))),D) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),D). [resolve(1535,b,1525,b)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,C),c_Groups_Ouminus__class_Ouminus(A,f52(B,A)))),f51(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),D) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f27(B,D,A))),D). [resolve(1535,b,1530,d)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,C),c_Groups_Ouminus__class_Ouminus(A,f52(B,A)))),f51(B,A)) | -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f32(B,A,D,E)),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,E)))),D) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),D). [resolve(1535,b,1531,d)]. 7.72/7.95 1536 -class_RealVector_Oreal__normed__vector(A) | c_SEQ_OBseq(A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,f53(B,A,C,D)),c_Groups_Ouminus__class_Ouminus(A,D))),C) # label(fact_Bseq__iff2) # label(axiom). [clausify(722)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,f53(C,A,B,D)),c_Groups_Ouminus__class_Ouminus(A,D))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f8(C,A)). [resolve(1536,b,1526,b)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,f53(C,A,B,D)),c_Groups_Ouminus__class_Ouminus(A,D))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,E)),f8(C,A)). [resolve(1536,b,1527,b)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,f53(C,A,B,D)),c_Groups_Ouminus__class_Ouminus(A,D))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(C,E)),f13(C,A)). [resolve(1536,b,1528,b)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,f53(C,A,B,D)),c_Groups_Ouminus__class_Ouminus(A,D))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f13(C,A)). [resolve(1536,b,1529,b)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,f53(C,A,B,D)),c_Groups_Ouminus__class_Ouminus(A,D))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,E),c_Groups_Ouminus__class_Ouminus(A,hAPP(C,f34(C,A))))),f33(C,A)). [resolve(1536,b,1532,c)]. 7.72/7.95 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,f53(C,A,B,D)),c_Groups_Ouminus__class_Ouminus(A,D))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f33(C,A)). [resolve(1536,b,1533,c)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,f53(C,A,B,D)),c_Groups_Ouminus__class_Ouminus(A,D))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f51(C,A)). [resolve(1536,b,1534,b)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,f53(C,A,B,D)),c_Groups_Ouminus__class_Ouminus(A,D))),B) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(C,E),c_Groups_Ouminus__class_Ouminus(A,f52(C,A)))),f51(C,A)). [resolve(1536,b,1535,b)]. 7.72/7.96 1537 -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,f60(C,D,B,A)) | c_SEQ_OBseq(A,D) # label(fact_BseqI2_H) # label(axiom). [clausify(929)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,f60(C,D,B,A)) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f8(D,A)). [resolve(1537,c,1526,b)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,f60(C,D,B,A)) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(D,E)),f8(D,A)). [resolve(1537,c,1527,b)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,f60(C,D,B,A)) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(D,E)),f13(D,A)). [resolve(1537,c,1528,b)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,f60(C,D,B,A)) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f13(D,A)). [resolve(1537,c,1529,b)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,f60(C,D,B,A)) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(D,E),c_Groups_Ouminus__class_Ouminus(A,hAPP(D,f34(D,A))))),f33(D,A)). [resolve(1537,c,1532,c)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,f60(C,D,B,A)) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f33(D,A)). [resolve(1537,c,1533,c)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,f60(C,D,B,A)) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f51(D,A)). [resolve(1537,c,1534,b)]. 7.72/7.96 Derived: -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,f60(C,D,B,A)) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(D,E),c_Groups_Ouminus__class_Ouminus(A,f52(D,A)))),f51(D,A)). [resolve(1537,c,1535,b)]. 7.90/8.13 1538 -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f60(C,B,D,A))),C) | c_SEQ_OBseq(A,B) # label(fact_BseqI2_H) # label(axiom). [clausify(929)]. 7.90/8.13 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f60(C,B,D,A))),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f8(B,A)). [resolve(1538,c,1526,b)]. 7.90/8.13 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f60(C,B,D,A))),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,E)),f8(B,A)). [resolve(1538,c,1527,b)]. 7.90/8.13 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f60(C,B,D,A))),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,E)),f13(B,A)). [resolve(1538,c,1528,b)]. 7.90/8.13 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f60(C,B,D,A))),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f13(B,A)). [resolve(1538,c,1529,b)]. 7.90/8.13 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f60(C,B,D,A))),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,E),c_Groups_Ouminus__class_Ouminus(A,hAPP(B,f34(B,A))))),f33(B,A)). [resolve(1538,c,1532,c)]. 7.90/8.13 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f60(C,B,D,A))),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f33(B,A)). [resolve(1538,c,1533,c)]. 7.90/8.13 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f60(C,B,D,A))),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),f51(B,A)). [resolve(1538,c,1534,b)]. 7.90/8.13 Derived: -class_RealVector_Oreal__normed__vector(A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,hAPP(B,f60(C,B,D,A))),C) | -class_RealVector_Oreal__normed__vector(A) | c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_RealVector_Onorm__class_Onorm(A,c_Groups_Oplus__class_Oplus(A,hAPP(B,E),c_Groups_Ouminus__class_Ouminus(A,f52(B,A)))),f51(B,A)). [resolve(1538,c,1535,b)]. 7.90/8.13 1539 class_Rings_Ocomm__ring__1(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Ocomm__ring__1) # label(axiom). [assumption]. 7.90/8.13 1540 -class_Rings_Ocomm__ring__1(A) | c_Groups_Ominus__class_Ominus(A,B,C) = c_Groups_Oplus__class_Oplus(A,B,c_Groups_Ouminus__class_Ouminus(A,C)) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J) # label(axiom). [clausify(135)]. 7.90/8.13 1541 class_Rings_Ocomm__ring__1(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Ocomm__ring__1) # label(axiom). [assumption]. 7.90/8.13 1542 class_Rings_Ocomm__ring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__ring__1) # label(axiom). [assumption]. 7.90/8.13 1543 -class_Rings_Ocomm__ring__1(A) | c_Groups_Ouminus__class_Ouminus(A,B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ouminus__class_Ouminus(A,c_Groups_Oone__class_Oone(A))),B) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) # label(axiom). [clausify(950)]. 8.15/8.42 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))),A). [resolve(1543,a,1539,a)]. 8.15/8.42 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal))),A). [resolve(1543,a,1541,a)]. 8.15/8.42 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint))),A). [resolve(1543,a,1542,a)]. 8.15/8.42 1544 -class_Rings_Olinordered__semiring__1__strict(A) | -c_Orderings_Oord__class_Oless(A,B,C) | -c_Orderings_Oord__class_Oless(A,D,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),E) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),F) | c_Groups_Oplus__class_Oplus(A,E,F) != c_Groups_Oone__class_Oone(A) | c_Orderings_Oord__class_Oless(A,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(193)]. 8.15/8.42 1545 class_Rings_Olinordered__semiring__1__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 8.15/8.42 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless(tc_Int_Oint,C,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),D) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),E) | c_Groups_Oplus__class_Oplus(tc_Int_Oint,D,E) != c_Groups_Oone__class_Oone(tc_Int_Oint) | c_Orderings_Oord__class_Oless(tc_Int_Oint,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(1544,a,1545,a)]. 8.15/8.42 1546 class_Rings_Olinordered__semiring__1__strict(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 8.15/8.42 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,C,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),D) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),E) | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,D,E) != c_Groups_Oone__class_Oone(tc_RealDef_Oreal) | c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,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(1546,a,1544,a)]. 8.15/8.42 1547 -class_Rings_Oring__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B) # label(fact_mult__eq__0__iff) # label(axiom). [clausify(451)]. 8.15/8.42 1548 class_Rings_Oring__no__zero__divisors(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 8.15/8.42 1549 class_Rings_Oring__no__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 8.15/8.42 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),A). [resolve(1547,a,1548,a)]. 8.15/8.42 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A). [resolve(1547,a,1549,a)]. 8.15/8.42 1550 -class_Rings_Oring__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) # label(fact_mult__eq__0__iff) # label(axiom). [clausify(451)]. 8.51/8.75 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B). [resolve(1550,a,1548,a)]. 8.51/8.75 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B). [resolve(1550,a,1549,a)]. 8.51/8.75 1551 -class_Rings_Oring__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B) # label(fact_mult__eq__0__iff) # label(axiom). [clausify(451)]. 8.51/8.75 1552 class_Rings_Oring__no__zero__divisors(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 8.51/8.75 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),A). [resolve(1552,a,1547,a)]. 8.51/8.75 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B). [resolve(1552,a,1550,a)]. 8.51/8.75 1553 -class_Groups_Ocancel__ab__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,B,C) != c_Groups_Oplus__class_Oplus(A,B,D) | C = D # label(fact_add__imp__eq) # label(axiom). [clausify(540)]. 8.51/8.75 1554 class_Groups_Ocancel__ab__semigroup__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 8.51/8.75 1555 class_Groups_Ocancel__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 8.51/8.75 1556 class_Groups_Ocancel__ab__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 8.51/8.75 1557 class_Groups_Ocancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 8.51/8.75 1558 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B),C) = c_Groups_Ouminus__class_Ouminus(A,c_RealVector_OscaleR__class_OscaleR(A,B,C)) # label(fact_scaleR__minus__left) # label(axiom). [clausify(169)]. 8.51/8.75 1559 class_RealVector_Oreal__vector(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Oreal__vector) # label(axiom). [assumption]. 8.51/8.75 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),B) = c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B)). [resolve(1558,a,1559,a)]. 8.51/8.75 1560 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) = c_Groups_Oplus__class_Oplus(A,c_RealVector_OscaleR__class_OscaleR(A,B,C),c_RealVector_OscaleR__class_OscaleR(A,B,D)) # label(fact_scaleR__right_Oadd) # label(axiom). [clausify(211)]. 8.51/8.75 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,B,C)) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B),c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C)). [resolve(1560,a,1559,a)]. 8.51/8.75 1561 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C),D) = c_Groups_Oplus__class_Oplus(A,c_RealVector_OscaleR__class_OscaleR(A,B,D),c_RealVector_OscaleR__class_OscaleR(A,C,D)) # label(fact_scaleR__left_Oadd) # label(axiom). [clausify(216)]. 8.51/8.75 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),C) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C),c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,B,C)). [resolve(1561,a,1559,a)]. 8.51/8.75 1562 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,c_Groups_Ozero__class_Ozero(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_scaleR__right_Ozero) # label(axiom). [clausify(239)]. 8.51/8.76 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1562,a,1559,a)]. 8.51/8.76 1563 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),B) = B # label(fact_real__vector_Oscale__one) # label(axiom). [clausify(266)]. 8.51/8.76 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) = A. [resolve(1563,a,1559,a)]. 8.51/8.76 1564 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,c_Groups_Ominus__class_Ominus(A,C,D)) = c_Groups_Ominus__class_Ominus(A,c_RealVector_OscaleR__class_OscaleR(A,B,C),c_RealVector_OscaleR__class_OscaleR(A,B,D)) # label(fact_scaleR__right_Odiff) # label(axiom). [clausify(306)]. 8.51/8.76 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,B,C)) = c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B),c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C)). [resolve(1564,a,1559,a)]. 8.51/8.76 1565 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,C),D) = c_Groups_Ominus__class_Ominus(A,c_RealVector_OscaleR__class_OscaleR(A,B,D),c_RealVector_OscaleR__class_OscaleR(A,C,D)) # label(fact_scaleR__left_Odiff) # label(axiom). [clausify(316)]. 8.51/8.76 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B),C) = c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C),c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,B,C)). [resolve(1565,a,1559,a)]. 8.51/8.76 1566 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C),D) = c_Groups_Oplus__class_Oplus(A,c_RealVector_OscaleR__class_OscaleR(A,B,D),c_RealVector_OscaleR__class_OscaleR(A,C,D)) # label(fact_real__vector_Oscale__left__distrib) # label(axiom). [clausify(418)]. 8.51/8.76 1567 -class_RealVector_Oreal__vector(A) | B != C | c_RealVector_OscaleR__class_OscaleR(A,B,D) = c_RealVector_OscaleR__class_OscaleR(A,C,D) # label(fact_scaleR__cancel__right) # label(axiom). [clausify(472)]. 8.51/8.76 Derived: A != B | c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C) = c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,B,C). [resolve(1567,a,1559,a)]. 8.51/8.76 1568 -class_RealVector_Oreal__vector(A) | c_Groups_Ozero__class_Ozero(A) != B | c_RealVector_OscaleR__class_OscaleR(A,C,B) = c_RealVector_OscaleR__class_OscaleR(A,D,B) # label(fact_scaleR__cancel__right) # label(axiom). [clausify(472)]. 8.51/8.76 Derived: c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != A | c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,B,A) = c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,C,A). [resolve(1568,a,1559,a)]. 8.51/8.76 1569 -class_RealVector_Oreal__vector(A) | B = C | c_Groups_Ozero__class_Ozero(A) = D | c_RealVector_OscaleR__class_OscaleR(A,B,D) != c_RealVector_OscaleR__class_OscaleR(A,C,D) # label(fact_scaleR__cancel__right) # label(axiom). [clausify(472)]. 8.51/8.76 Derived: A = B | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = C | c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C) != c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,B,C). [resolve(1569,a,1559,a)]. 8.51/8.76 1570 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B),C) = c_Groups_Ouminus__class_Ouminus(A,c_RealVector_OscaleR__class_OscaleR(A,B,C)) # label(fact_scaleR__left_Ominus) # label(axiom). [clausify(477)]. 8.51/8.76 1571 -class_RealVector_Oreal__vector(A) | c_Groups_Ozero__class_Ozero(A) = B | c_RealVector_OscaleR__class_OscaleR(A,C,B) != c_RealVector_OscaleR__class_OscaleR(A,D,B) | C = D # label(fact_scaleR__right__imp__eq) # label(axiom). [clausify(517)]. 8.51/8.79 1572 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,c_RealVector_OscaleR__class_OscaleR(A,C,D)) = c_RealVector_OscaleR__class_OscaleR(A,C,c_RealVector_OscaleR__class_OscaleR(A,B,D)) # label(fact_scaleR__left__commute) # label(axiom). [clausify(532)]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,B,C)) = c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,B,c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C)). [resolve(1572,a,1559,a)]. 8.51/8.79 1573 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_scaleR__left_Ozero) # label(axiom). [clausify(731)]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex). [resolve(1573,a,1559,a)]. 8.51/8.79 1574 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,C),D) = c_Groups_Ominus__class_Ominus(A,c_RealVector_OscaleR__class_OscaleR(A,B,D),c_RealVector_OscaleR__class_OscaleR(A,C,D)) # label(fact_scaleR__left__diff__distrib) # label(axiom). [clausify(758)]. 8.51/8.79 1575 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,c_Groups_Ouminus__class_Ouminus(A,C)) = c_Groups_Ouminus__class_Ouminus(A,c_RealVector_OscaleR__class_OscaleR(A,B,C)) # label(fact_scaleR__minus__right) # label(axiom). [clausify(778)]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B)) = c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B)). [resolve(1575,a,1559,a)]. 8.51/8.79 1576 -class_RealVector_Oreal__vector(A) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B | c_RealVector_OscaleR__class_OscaleR(A,B,C) != c_RealVector_OscaleR__class_OscaleR(A,B,D) | C = D # label(fact_scaleR__left__imp__eq) # label(axiom). [clausify(824)]. 8.51/8.79 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B) != c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C) | B = C. [resolve(1576,a,1559,a)]. 8.51/8.79 1577 class_RealVector_Oreal__vector(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Oreal__vector) # label(axiom). [assumption]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A),B) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B)). [resolve(1577,a,1558,a)]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,B,C)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B),c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C)). [resolve(1577,a,1560,a)]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,A,B),C) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C),c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,B,C)). [resolve(1577,a,1561,a)]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1577,a,1562,a)]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal),A) = A. [resolve(1577,a,1563,a)]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,C)) = c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B),c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C)). [resolve(1577,a,1564,a)]. 8.51/8.79 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,B),C) = c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C),c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,B,C)). [resolve(1577,a,1565,a)]. 8.51/8.80 Derived: A != B | c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C) = c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,B,C). [resolve(1577,a,1567,a)]. 8.51/8.80 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A | c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,B,A) = c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,C,A). [resolve(1577,a,1568,a)]. 8.51/8.80 Derived: A = B | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = C | c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C) != c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,B,C). [resolve(1577,a,1569,a)]. 8.51/8.80 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,B,C)) = c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,B,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C)). [resolve(1577,a,1572,a)]. 8.51/8.80 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal). [resolve(1577,a,1573,a)]. 8.51/8.80 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B)). [resolve(1577,a,1575,a)]. 8.51/8.80 Derived: c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A | c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B) != c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C) | B = C. [resolve(1577,a,1576,a)]. 8.51/8.80 1578 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,C) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = C | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B # label(fact_scaleR__eq__0__iff) # label(axiom). [clausify(916)]. 8.51/8.80 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B) != c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = B | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = A. [resolve(1578,a,1559,a)]. 8.51/8.80 Derived: c_RealVector_OscaleR__class_OscaleR(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(1578,a,1577,a)]. 8.51/8.80 1579 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,C) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) != C # label(fact_scaleR__eq__0__iff) # label(axiom). [clausify(916)]. 8.51/8.80 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) != B. [resolve(1579,a,1559,a)]. 8.51/8.80 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B. [resolve(1579,a,1577,a)]. 8.51/8.80 1580 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,C) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B # label(fact_scaleR__eq__0__iff) # label(axiom). [clausify(916)]. 8.51/8.80 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B) = c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1580,a,1559,a)]. 8.51/8.80 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1580,a,1577,a)]. 8.51/8.80 1581 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),B),C),D) = c_RealVector_OscaleR__class_OscaleR(A,B,c_RealVector_OscaleR__class_OscaleR(A,C,D)) # label(fact_real__vector_Oscale__scale) # label(axiom). [clausify(953)]. 8.64/8.95 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C) = c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,B,C)). [resolve(1581,a,1559,a)]. 8.64/8.95 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C) = c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,B,C)). [resolve(1581,a,1577,a)]. 8.64/8.95 1582 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,C) != c_RealVector_OscaleR__class_OscaleR(A,B,D) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = B | C = D # label(fact_scaleR__cancel__left) # label(axiom). [clausify(955)]. 8.64/8.95 1583 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,C) = c_RealVector_OscaleR__class_OscaleR(A,B,D) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != B # label(fact_scaleR__cancel__left) # label(axiom). [clausify(955)]. 8.64/8.95 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B) = c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1583,a,1559,a)]. 8.64/8.95 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B) = c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) != A. [resolve(1583,a,1577,a)]. 8.64/8.95 1584 -class_RealVector_Oreal__vector(A) | c_RealVector_OscaleR__class_OscaleR(A,B,C) = c_RealVector_OscaleR__class_OscaleR(A,B,D) | C != D # label(fact_scaleR__cancel__left) # label(axiom). [clausify(955)]. 8.64/8.95 Derived: c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,B) = c_RealVector_OscaleR__class_OscaleR(tc_Complex_Ocomplex,A,C) | B != C. [resolve(1584,a,1559,a)]. 8.64/8.95 Derived: c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,B) = c_RealVector_OscaleR__class_OscaleR(tc_RealDef_Oreal,A,C) | B != C. [resolve(1584,a,1577,a)]. 8.64/8.95 1585 class_Rings_Osemiring__0(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Osemiring__0) # label(axiom). [assumption]. 8.64/8.95 1586 -class_Power_Opower(A) | -class_Rings_Osemiring__0(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),c_Nat_OSuc(B)) # label(fact_power__0__Suc) # label(axiom). [clausify(165)]. 8.64/8.95 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)),c_Nat_OSuc(A)). [resolve(1585,a,1586,b)]. 8.64/8.95 1587 -class_Power_Opower(A) | -class_Rings_Osemiring__0(A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_power__0__left) # label(axiom). [clausify(563)]. 8.64/8.95 Derived: -class_Power_Opower(tc_Complex_Ocomplex) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex),c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)),A). [resolve(1587,b,1585,a)]. 8.64/8.95 1588 -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(563)]. 8.64/8.95 1589 class_Rings_Osemiring__0(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring__0) # label(axiom). [assumption]. 8.64/8.95 Derived: -class_Power_Opower(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Nat_OSuc(A)). [resolve(1589,a,1586,b)]. 8.64/8.95 Derived: -class_Power_Opower(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(1589,a,1587,b)]. 9.19/9.45 1590 class_Rings_Osemiring__0(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring__0) # label(axiom). [assumption]. 9.19/9.45 Derived: -class_Power_Opower(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Nat_OSuc(A)). [resolve(1590,a,1586,b)]. 9.19/9.45 Derived: -class_Power_Opower(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A). [resolve(1590,a,1587,b)]. 9.19/9.45 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(1590,a,1588,b)]. 9.19/9.45 1591 class_Rings_Osemiring__0(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Osemiring__0) # label(axiom). [assumption]. 9.19/9.45 Derived: -class_Power_Opower(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),c_Nat_OSuc(A)). [resolve(1591,a,1586,b)]. 9.19/9.45 Derived: -class_Power_Opower(tc_RealDef_Oreal) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_RealDef_Oreal),c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)),A). [resolve(1591,a,1587,b)]. 9.19/9.45 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(1591,a,1588,b)]. 9.19/9.45 1592 class_Rings_Oring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring) # label(axiom). [assumption]. 9.19/9.45 1593 -class_Rings_Oring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ouminus__class_Ouminus(A,B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ouminus__class_Ouminus(A,C)) # label(fact_minus__mult__commute) # label(axiom). [clausify(178)]. 9.19/9.45 1594 -class_Rings_Oring(A) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D) != c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,B,E)),C),D) = F # label(fact_eq__add__iff1) # label(axiom). [clausify(182)]. 9.19/9.45 1595 -class_Rings_Oring(A) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D) = c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,B,E)),C),D) != F # label(fact_eq__add__iff1) # label(axiom). [clausify(182)]. 9.19/9.45 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)). [resolve(1592,a,1593,a)]. 9.19/9.45 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C) != c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E) | c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,D)),B),C) = E. [resolve(1592,a,1594,a)]. 9.19/9.45 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E) | c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,D)),B),C) != E. [resolve(1592,a,1595,a)]. 9.19/9.45 1596 -class_Rings_Oring(A) | c_Groups_Ouminus__class_Ouminus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ouminus__class_Ouminus(A,B)),C) # label(fact_minus__mult__left) # label(axiom). [clausify(560)]. 9.19/9.45 Derived: c_Groups_Ouminus__class_Ouminus(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_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)),B). [resolve(1596,a,1592,a)]. 9.19/9.45 1597 -class_Rings_Oring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ouminus__class_Ouminus(A,B)),c_Groups_Ouminus__class_Ouminus(A,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) # label(fact_minus__mult__minus) # label(axiom). [clausify(568)]. 9.19/9.45 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A)),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B). [resolve(1597,a,1592,a)]. 9.19/9.45 1598 -class_Rings_Oring(A) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D) != c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,E,B)),C),F) = D # label(fact_eq__add__iff2) # label(axiom). [clausify(611)]. 9.19/9.45 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C) != c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E) | c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,D,A)),B),E) = C. [resolve(1598,a,1592,a)]. 9.19/9.45 1599 -class_Rings_Oring(A) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D) = c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C),F) | c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,E,B)),C),F) != D # label(fact_eq__add__iff2) # label(axiom). [clausify(611)]. 9.19/9.45 Derived: c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B),E) | c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,D,A)),B),E) != C. [resolve(1599,a,1592,a)]. 9.19/9.45 1600 -class_Rings_Oring(A) | 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),E)) = c_Groups_Oplus__class_Oplus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ominus__class_Ominus(A,C,E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ominus__class_Ominus(A,B,D)),E)) # label(fact_mult__diff__mult) # label(axiom). [clausify(614)]. 9.19/9.45 Derived: c_Groups_Ominus__class_Ominus(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),D)) = c_Groups_Oplus__class_Oplus(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Groups_Ominus__class_Ominus(tc_Int_Oint,B,D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint,A,C)),D)). [resolve(1600,a,1592,a)]. 9.19/9.45 1601 class_Rings_Oring(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Oring) # label(axiom). [assumption]. 9.19/9.45 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B)). [resolve(1601,a,1593,a)]. 9.19/9.45 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B),C) != c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),D),B),E) | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,D)),B),C) = E. [resolve(1601,a,1594,a)]. 9.19/9.47 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B),C) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),D),B),E) | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,D)),B),C) != E. [resolve(1601,a,1595,a)]. 9.19/9.47 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A)),c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B). [resolve(1601,a,1597,a)]. 9.19/9.47 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B),C) != c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),D),B),E) | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,D,A)),B),E) = C. [resolve(1601,a,1598,a)]. 9.19/9.47 Derived: c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),B),C) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),D),B),E) | c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,D,A)),B),E) != C. [resolve(1601,a,1599,a)]. 9.19/9.47 Derived: 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),D)) = c_Groups_Oplus__class_Oplus(tc_Complex_Ocomplex,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,B,D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),c_Groups_Ominus__class_Ominus(tc_Complex_Ocomplex,A,C)),D)). [resolve(1601,a,1600,a)]. 9.19/9.48 1602 -class_Rings_Oring(A) | c_Groups_Ouminus__class_Ouminus(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ouminus__class_Ouminus(A,C)) # label(fact_minus__mult__right) # label(axiom). [clausify(928)]. 9.19/9.48 Derived: c_Groups_Ouminus__class_Ouminus(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_Groups_Ouminus__class_Ouminus(tc_Int_Oint,B)). [resolve(1602,a,1592,a)]. 9.19/9.48 1603 class_Rings_Oring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oring) # label(axiom). [assumption]. 9.19/9.48 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)). [resolve(1603,a,1593,a)]. 9.19/9.48 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C) != c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E) | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,D)),B),C) = E. [resolve(1603,a,1594,a)]. 9.19/9.48 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E) | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,D)),B),C) != E. [resolve(1603,a,1595,a)]. 9.19/9.48 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A)),c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B). [resolve(1603,a,1597,a)]. 9.35/9.63 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C) != c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E) | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,D,A)),B),E) = C. [resolve(1603,a,1598,a)]. 9.35/9.63 Derived: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),B),C) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),D),B),E) | c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,D,A)),B),E) != C. [resolve(1603,a,1599,a)]. 9.35/9.63 Derived: 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),D)) = c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),A),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,B,D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal),c_Groups_Ominus__class_Ominus(tc_RealDef_Oreal,A,C)),D)). [resolve(1603,a,1600,a)]. 9.35/9.63 1604 class_Rings_Olinordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring) # label(axiom). [assumption]. 9.35/9.63 1605 -class_Rings_Olinordered__ring(A) | 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(184)]. 9.35/9.63 1606 -class_Rings_Olinordered__ring(A) | c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),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(311)]. 9.35/9.63 1607 -class_Rings_Olinordered__ring(A) | -c_Orderings_Oord__class_Oless(A,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(593)]. 9.35/9.63 Derived: 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(1604,a,1605,a)]. 9.35/9.63 Derived: c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),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(1604,a,1606,a)]. 9.35/9.63 Derived: -c_Orderings_Oord__class_Oless(tc_Int_Oint,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(1604,a,1607,a)]. 9.35/9.63 1608 class_Rings_Olinordered__ring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Olinordered__ring) # label(axiom). [assumption]. 9.35/9.63 Derived: 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(1608,a,1605,a)]. 9.35/9.63 Derived: c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),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(1608,a,1606,a)]. 9.35/9.63 Derived: -c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,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(1608,a,1607,a)]. 9.64/9.92 1609 -class_Rings_Olinordered__ring(A) | -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(927)]. 9.64/9.92 Derived: -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(1609,a,1604,a)]. 9.64/9.92 Derived: -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(1609,a,1608,a)]. 9.64/9.92 1610 class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra) # label(axiom). [assumption]. 9.64/9.92 1611 -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(205)]. 9.64/9.92 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(1610,a,1611,a)]. 9.64/9.92 1612 -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(746)]. 9.64/9.92 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(1612,a,1610,a)]. 9.64/9.92 1613 class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra) # label(axiom). [assumption]. 9.64/9.92 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(1613,a,1611,a)]. 9.64/9.92 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(1613,a,1612,a)]. 9.64/9.92 1614 class_RealVector_Oreal__normed__algebra__1(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Oreal__normed__algebra__1) # label(axiom). [assumption]. 9.64/9.92 1615 -class_RealVector_Oreal__normed__algebra__1(A) | c_RealVector_Onorm__class_Onorm(A,c_Groups_Oone__class_Oone(A)) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal) # label(fact_norm__one) # label(axiom). [clausify(286)]. 9.64/9.92 Derived: c_RealVector_Onorm__class_Onorm(tc_RealDef_Oreal,c_Groups_Oone__class_Oone(tc_RealDef_Oreal)) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal). [resolve(1614,a,1615,a)]. 9.64/9.92 1616 -class_RealVector_Oreal__normed__algebra__1(A) | 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(453)]. 9.64/9.92 Derived: 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(1616,a,1614,a)]. 10.42/10.69 1617 class_RealVector_Oreal__normed__algebra__1(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Oreal__normed__algebra__1) # label(axiom). [assumption]. 10.42/10.69 Derived: c_RealVector_Onorm__class_Onorm(tc_Complex_Ocomplex,c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal). [resolve(1617,a,1615,a)]. 10.42/10.69 Derived: 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(1617,a,1616,a)]. 10.42/10.69 1618 class_Groups_Oab__semigroup__mult(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 10.42/10.69 1619 -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(300)]. 10.42/10.69 1620 class_Groups_Oab__semigroup__mult(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 10.42/10.69 1621 class_Groups_Oab__semigroup__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 10.42/10.69 1622 class_Groups_Oab__semigroup__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 10.42/10.69 1623 -class_Rings_Oordered__comm__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | 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(604)]. 10.42/10.69 1624 class_Rings_Oordered__comm__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 10.42/10.69 1625 class_Rings_Oordered__comm__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 10.42/10.69 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(1623,a,1624,a)]. 10.42/10.69 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(1623,a,1625,a)]. 10.42/10.69 1626 class_Rings_Oordered__comm__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 10.42/10.69 1627 class_RealVector_Oreal__algebra(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__RealVector_Oreal__algebra) # label(axiom). [assumption]. 10.42/10.69 1628 -class_RealVector_Oreal__algebra(A) | c_RealVector_OscaleR__class_OscaleR(A,B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),c_RealVector_OscaleR__class_OscaleR(A,B,D)) # label(fact_mult__scaleR__right) # label(axiom). [clausify(323)]. 10.42/10.69 1629 class_RealVector_Oreal__algebra(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__RealVector_Oreal__algebra) # label(axiom). [assumption]. 10.42/10.69 1630 -class_RealVector_Oreal__algebra(A) | c_RealVector_OscaleR__class_OscaleR(A,B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_RealVector_OscaleR__class_OscaleR(A,B,C)),D) # label(fact_mult__scaleR__left) # label(axiom). [clausify(785)]. 10.42/10.69 1631 class_Groups_Oab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 10.70/11.03 1632 -class_Groups_Oab__semigroup__add(A) | c_Groups_Oplus__class_Oplus(A,c_Groups_Oplus__class_Oplus(A,B,C),D) = c_Groups_Oplus__class_Oplus(A,B,c_Groups_Oplus__class_Oplus(A,C,D)) # label(fact_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(365)]. 10.70/11.03 1633 class_Groups_Oab__semigroup__add(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 10.70/11.03 1634 class_Groups_Oab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 10.70/11.03 1635 class_Groups_Oab__semigroup__add(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 10.70/11.03 1636 -class_Rings_Oidom(A) | c_Groups_Ouminus__class_Ouminus(A,B) != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) # label(fact_square__eq__iff) # label(axiom). [clausify(882)]. 10.70/11.03 1637 class_Rings_Oidom(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Oidom) # label(axiom). [assumption]. 10.70/11.03 1638 class_Rings_Oidom(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oidom) # label(axiom). [assumption]. 10.70/11.03 1639 class_Rings_Oidom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oidom) # label(axiom). [assumption]. 10.70/11.03 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) != B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),A). [resolve(1636,a,1637,a)]. 10.70/11.03 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) != B | 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(1636,a,1638,a)]. 10.70/11.03 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) != B | 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(1636,a,1639,a)]. 10.70/11.03 1640 -class_Rings_Oidom(A) | B != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C) # label(fact_square__eq__iff) # label(axiom). [clausify(882)]. 10.70/11.03 Derived: A != B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),B). [resolve(1640,a,1637,a)]. 10.70/11.03 Derived: A != B | 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(1640,a,1638,a)]. 10.70/11.03 Derived: A != B | 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(1640,a,1639,a)]. 10.70/11.03 1641 -class_Rings_Oidom(A) | c_Groups_Ouminus__class_Ouminus(A,B) = C | C = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) # label(fact_square__eq__iff) # label(axiom). [clausify(882)]. 10.70/11.03 Derived: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex,A) = B | B = A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),B),B) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex),A),A). [resolve(1641,a,1637,a)]. 10.70/11.03 Derived: c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal,A) = B | B = A | 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(1641,a,1638,a)]. 10.70/11.03 Derived: c_Groups_Ouminus__class_Ouminus(tc_Int_Oint,A) = B | B = A | 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(1641,a,1639,a)]. 10.70/11.03 1642 class_Rings_Oordered__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__semiring) # label(axiom). [assumption]. 10.70/11.03 1643 -class_Rings_Oordered__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,E) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | 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(448)]. 10.77/11.05 1644 -class_Rings_Oordered__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,D,E) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | 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(469)]. 10.77/11.05 1645 -class_Rings_Oordered__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | 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(527)]. 10.77/11.05 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(1642,a,1643,a)]. 10.77/11.05 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(1642,a,1644,a)]. 10.77/11.05 1646 class_Rings_Oordered__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oordered__semiring) # label(axiom). [assumption]. 10.77/11.05 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),A) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(1646,a,1643,a)]. 10.77/11.05 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(1646,a,1644,a)]. 10.77/11.05 1647 -class_Rings_Oordered__semiring(A) | -c_Orderings_Oord__class_Oless__eq(A,B,C) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),D) | 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(908)]. 10.77/11.05 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),C) | 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(1647,a,1642,a)]. 10.77/11.05 Derived: -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal),C) | 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(1647,a,1646,a)]. 11.08/11.36 1648 class_Rings_Oordered__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__semiring) # label(axiom). [assumption]. 11.08/11.36 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(1648,a,1643,a)]. 11.08/11.36 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(1648,a,1644,a)]. 11.08/11.36 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(1648,a,1645,a)]. 11.08/11.36 Derived: -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) | 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(1648,a,1647,a)]. 11.08/11.36 1649 -class_Rings_Ocomm__semiring(A) | 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),c_Groups_Oplus__class_Oplus(A,B,D)),C) # label(fact_comm__semiring__class_Odistrib) # label(axiom). [clausify(599)]. 11.08/11.36 1650 class_Rings_Ocomm__semiring(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Ocomm__semiring) # label(axiom). [assumption]. 11.08/11.36 1651 class_Rings_Ocomm__semiring(tc_Complex_Ocomplex) # label(arity_Complex__Ocomplex__Rings_Ocomm__semiring) # label(axiom). [assumption]. 11.08/11.36 1652 class_Rings_Ocomm__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__semiring) # label(axiom). [assumption]. 11.08/11.36 1653 class_Rings_Ocomm__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ocomm__semiring) # label(axiom). [assumption]. 11.08/11.36 Derived: c_Groups_Oplus__class_Oplus(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_Groups_Oplus__class_Oplus(tc_Nat_Onat,A,C)),B). [resolve(1653,a,1649,a)]. 11.08/11.36 1654 -class_Groups_Oabs__if(A) | c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Groups_Oabs__class_Oabs(A,B) = B # label(fact_abs__if) # label(axiom). [clausify(934)]. 11.08/11.36 1655 class_Groups_Oabs__if(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oabs__if) # label(axiom). [assumption]. 11.08/11.36 1656 class_Groups_Oabs__if(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Groups_Oabs__if) # label(axiom). [assumption]. 11.08/11.36 Derived: c_Orderings_Oord__class_Oless(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) | c_Groups_Oabs__class_Oabs(tc_Int_Oint,A) = A. [resolve(1654,a,1655,a)]. 11.08/11.36 Derived: c_Orderings_Oord__class_Oless(tc_RealDef_Oreal,A,c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) | c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal,A) = A. [resolve(1654,a,1656,a)]. 11.08/11.36 1657 -class_Groups_Oabs__if(A) | -c_Orderings_Oord__class_Oless(A,B,c_Groups_Ozero__class_Ozero(A)) | c_Groups_Oabs__class_Oabs(A,B) = c_Groups_Ouminus__class_Ouminus(A,B) # label(fact_abs__if) # label(axiom). [clausify(934)]. 11.47/11.80 1658 -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless__eq(A,hAPP(D,B),hAPP(D,C)) | -c_SEQ_Oincseq(A,D) # label(fact_incseq__def) # label(axiom). [clausify(583)]. 11.47/11.80 1659 -class_Orderings_Oorder(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f40(B,A),f41(B,A)) | c_SEQ_Oincseq(A,B) # label(fact_incseq__def) # label(axiom). [clausify(583)]. 11.47/11.80 1660 -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f40(B,A)),hAPP(B,f41(B,A))) | c_SEQ_Oincseq(A,B) # label(fact_incseq__def) # label(axiom). [clausify(583)]. 11.47/11.80 Derived: -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless__eq(A,hAPP(D,B),hAPP(D,C)) | -class_Orderings_Oorder(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f40(D,A),f41(D,A)). [resolve(1658,d,1659,c)]. 11.47/11.80 Derived: -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B,C) | c_Orderings_Oord__class_Oless__eq(A,hAPP(D,B),hAPP(D,C)) | -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless__eq(A,hAPP(D,f40(D,A)),hAPP(D,f41(D,A))). [resolve(1658,d,1660,c)]. 11.47/11.80 1661 -class_Rings_Oordered__ring__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Groups_Oabs__class_Oabs(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oabs__class_Oabs(A,B)),c_Groups_Oabs__class_Oabs(A,C)) # label(fact_abs__eq__mult) # label(axiom). [clausify(774)]. 11.47/11.80 1662 class_Rings_Oordered__ring__abs(tc_RealDef_Oreal) # label(arity_RealDef__Oreal__Rings_Oordered__ring__abs) # label(axiom). [assumption]. 11.47/11.80 1663 class_Rings_Oordered__ring__abs(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__ring__abs) # label(axiom). [assumption]. 11.47/11.80 1664 -class_Rings_Oordered__ring__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),B) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Groups_Oabs__class_Oabs(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oabs__class_Oabs(A,B)),c_Groups_Oabs__class_Oabs(A,C)) # label(fact_abs__eq__mult) # label(axiom). [clausify(774)]. 11.47/11.80 1665 -class_Rings_Oordered__ring__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,C,c_Groups_Ozero__class_Ozero(A)) | c_Groups_Oabs__class_Oabs(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oabs__class_Oabs(A,B)),c_Groups_Oabs__class_Oabs(A,C)) # label(fact_abs__eq__mult) # label(axiom). [clausify(774)]. 11.47/11.80 1666 -class_Rings_Oordered__ring__abs(A) | -c_Orderings_Oord__class_Oless__eq(A,B,c_Groups_Ozero__class_Ozero(A)) | -c_Orderings_Oord__class_Oless__eq(A,c_Groups_Ozero__class_Ozero(A),C) | c_Groups_Oabs__class_Oabs(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oabs__class_Oabs(A,B)),c_Groups_Oabs__class_Oabs(A,C)) # label(fact_abs__eq__mult) # label(axiom). [clausify(774)]. 11.47/11.80 1667 -class_Orderings_Oorder(A) | c_SEQ_Odecseq(A,B) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f54(B,A),f55(B,A)) # label(fact_decseq__def) # label(axiom). [clausify(745)]. 11.47/11.80 1668 -class_Orderings_Oorder(A) | -c_SEQ_Odecseq(A,B) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D),hAPP(B,C)) # label(fact_decseq__def) # label(axiom). [clausify(745)]. 11.47/11.80 Derived: -class_Orderings_Oorder(A) | c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,f54(B,A),f55(B,A)) | -class_Orderings_Oorder(A) | -c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,C,D) | c_Orderings_Oord__class_Oless__eq(A,hAPP(B,D),hAPP(B,C)). [resolve(1667,b,1668,b)]. 11.47/11.80 1669 -class_Orderings_Oorder(A) | c_SEQ_Odecseq(A,B) | -c_Orderings_Oord__class_Oless__eq(A,hAPP(B,f5Alarm clock 119.34/120.04 Prover9 interrupted 119.34/120.04 EOF