%------------------------------------------------------------------------------
% File     : cvc5---1.0.5
% Problem  : SWW246+1 : TPTP v8.1.2. Released v5.2.0.
% Transfm  : none
% Format   : tptp
% Command  : do_cvc5 %s %d

% Computer : n006.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Fri Sep  1 00:20:07 EDT 2023

% Result   : Theorem 1.61s 1.86s
% Output   : Proof 1.69s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.26  % Problem    : SWW246+1 : TPTP v8.1.2. Released v5.2.0.
% 0.12/0.27  % Command    : do_cvc5 %s %d
% 0.27/0.48  % Computer : n006.cluster.edu
% 0.27/0.48  % Model    : x86_64 x86_64
% 0.27/0.48  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.27/0.48  % Memory   : 8042.1875MB
% 0.27/0.48  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.27/0.48  % CPULimit   : 300
% 0.27/0.48  % WCLimit    : 300
% 0.27/0.48  % DateTime   : Sun Aug 27 21:32:52 EDT 2023
% 0.27/0.48  % CPUTime    : 
% 0.33/0.74  %----Proving TF0_NAR, FOF, or CNF
% 1.61/1.86  ------- convert to smt2 : /export/starexec/sandbox/tmp/tmp.bteDnuLdue/cvc5---1.0.5_10613.p...
% 1.61/1.86  ------- get file name : TPTP file name is SWW246+1
% 1.61/1.86  ------- cvc5-fof : /export/starexec/sandbox/solver/bin/cvc5---1.0.5_10613.smt2...
% 1.61/1.86  --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10...
% 1.61/1.86  % SZS status Theorem for SWW246+1
% 1.61/1.86  % SZS output start Proof for SWW246+1
% 1.61/1.86  (
% 1.61/1.86  (let ((_let_1 (tptp.class_Rings_Oidom tptp.t_a))) (let ((_let_2 (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Ocomm__semiring__0 T))))) (let ((_let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_5 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint _let_5))) (let ((_let_7 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_8 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint _let_7))) (let ((_let_9 (tptp.hAPP tptp.c_Nat_OSuc _let_7))) (let ((_let_10 (tptp.c_Polynomial_Opoly tptp.t_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly tptp.t_a))))) (let ((_let_11 (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant tptp.t_a tptp.t_a _let_10))) (let ((_let_12 (not _let_11))) (let ((_let_13 (=> _let_1 _let_12))) (let ((_let_14 (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant T_b T_c V_f_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_f_2 B_y))))))) (let ((_let_15 (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))) V_x) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (let ((_let_16 (tptp.hAPP _let_10 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_31))) (let ((_let_17 (tptp.hAPP _let_10 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_30))) (let ((_let_18 (= _let_17 _let_16))) (let ((_let_19 (tptp.c_Groups_Ozero__class_Ozero tptp.t_a))) (let ((_let_20 (= _let_19 (tptp.hAPP _let_10 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_29)))) (let ((_let_21 (= _let_19 _let_16))) (let ((_let_22 (= _let_19 _let_17))) (let ((_let_23 (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Opoly tptp.t_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly tptp.t_a))))) (= (tptp.hAPP _let_1 B_x) (tptp.hAPP _let_1 B_y)))))) (let ((_let_24 (not _let_18))) (let ((_let_25 (= _let_11 _let_23))) (let ((_let_26 (not _let_23))) (let ((_let_27 (_let_14))) (let ((_let_28 (ASSUME :args _let_27))) (let ((_let_29 (ASSUME :args (_let_1)))) (let ((_let_30 (not _let_1))) (let ((_let_31 (or))) (let ((_let_32 (_let_24))) (let ((_let_33 (_let_26))) (let ((_let_34 (tptp.class_Rings_Ocomm__semiring__0 tptp.t_a))) (let ((_let_35 (not _let_34))) (let ((_let_36 (or _let_35 _let_20))) (let ((_let_37 (forall ((V_x $$unsorted) (T_a $$unsorted)) (or (not (tptp.class_Rings_Ocomm__semiring__0 T_a)) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))) V_x) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (let ((_let_38 (EQ_RESOLVE (ASSUME :args (_let_15)) (MACRO_SR_EQ_INTRO :args (_let_15 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_39 (_let_37))) (let ((_let_40 ((tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))) V_x)))) (let ((_let_41 (or _let_30 _let_34))) (let ((_let_42 (forall ((T $$unsorted)) (or (not (tptp.class_Rings_Oidom T)) (tptp.class_Rings_Ocomm__semiring__0 T))))) (let ((_let_43 (EQ_RESOLVE (ASSUME :args (_let_2)) (MACRO_SR_EQ_INTRO :args (_let_2 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_44 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_41)) :args ((or _let_30 _let_34 (not _let_41)))) _let_29 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_43 :args (tptp.t_a QUANTIFIERS_INST_E_MATCHING_SIMPLE ((not (= (tptp.class_Rings_Oidom T) false))))) :args (_let_42))) _let_43 :args (_let_41 false _let_42)) :args (_let_34 false _let_1 false _let_41)))) (let ((_let_45 (or _let_35 _let_21))) (let ((_let_46 (or _let_35 _let_22))) (let ((_let_47 (ASSUME :args (_let_20)))) (let ((_let_48 (SYMM (SYMM _let_47)))) (let ((_let_49 (ASSUME :args (_let_21)))) (let ((_let_50 (ASSUME :args (_let_22)))) (let ((_let_51 (ASSUME :args _let_32))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (NOT_AND (MACRO_SR_PRED_TRANSFORM (SCOPE (AND_INTRO _let_47 _let_50 _let_49 _let_51) :args (_let_24 _let_20 _let_21 _let_22)) (SCOPE (MACRO_SR_PRED_ELIM (TRANS (SYMM (FALSE_INTRO _let_51)) (TRUE_INTRO (TRANS (SYMM _let_50) _let_48 (SYMM (TRANS (SYMM _let_49) _let_48)))))) :args (_let_20 _let_22 _let_21 _let_24)) :args ((not (and _let_24 _let_20 _let_21 _let_22)) SB_LITERAL))) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_24) _let_18))) (REFL :args ((not _let_20))) (REFL :args ((not _let_21))) (REFL :args ((not _let_22))) :args _let_31)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_46)) :args ((or _let_35 _let_22 (not _let_46)))) _let_44 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_38 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_30 tptp.t_a QUANTIFIERS_INST_E_MATCHING _let_40)) :args _let_39))) _let_38 :args (_let_46 false _let_37)) :args (_let_22 false _let_34 false _let_46)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_45)) :args ((or _let_35 _let_21 (not _let_45)))) _let_44 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_38 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_31 tptp.t_a QUANTIFIERS_INST_E_MATCHING _let_40)) :args _let_39))) _let_38 :args (_let_45 false _let_37)) :args (_let_21 false _let_34 false _let_45)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_36)) :args ((or _let_35 _let_20 (not _let_36)))) _let_44 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_38 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_29 tptp.t_a QUANTIFIERS_INST_E_MATCHING _let_40)) :args _let_39))) _let_38 :args (_let_36 false _let_37)) :args (_let_20 false _let_34 false _let_36)) (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (IMPLIES_ELIM (SCOPE (SKOLEMIZE (ASSUME :args _let_33)) :args _let_33)) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_26) _let_23))) (REFL :args _let_32) :args _let_31)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_EQUIV_POS2 :args (_let_25)) :args ((or _let_11 _let_26 (not _let_25)))) (MACRO_RESOLUTION_TRUST (REORDERING (IMPLIES_ELIM (ASSUME :args (_let_13))) :args ((or _let_12 _let_30))) _let_29 :args (_let_12 false _let_1)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_28 :args (_let_10 tptp.t_a tptp.t_a QUANTIFIERS_INST_E_MATCHING_SIMPLE ((tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant T_b T_c V_f_2)))) :args _let_27)) _let_28 :args (_let_25 false _let_14)) :args (_let_26 true _let_11 false _let_25)) :args (_let_24 true _let_23)) :args (false false _let_22 false _let_21 false _let_20 true _let_18)) :args ((forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))) (= V_f_2 V_g_2))) (=> _let_1 (not (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Oconstant tptp.t_a tptp.t_a (tptp.c_Polynomial_Opoly tptp.t_a tptp.v_p)))) (forall ((V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (= (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Opsize T_b V_pa_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= V_pa_2 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)))))) (forall ((V_x $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) V_p)) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_x)))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) V_n))) (let ((_let_3 (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (and (tptp.class_Power_Opower T_a) (tptp.class_Rings_Osemiring__0 T_a)) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (not _let_3) (= _let_2 _let_1)))))))) _let_15 (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oone__class_Oone _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Ozero__class_Ozero _let_1)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Groups_Ozero T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) _let_1) _let_1)))) (forall ((V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)))) (=> (tptp.class_Groups_Ozero T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_b) V_a_2) V_pa_2) _let_1) (and (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b)) (= V_pa_2 _let_1)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_p)) (tptp.hAPP _let_1 V_q)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_p) V_q)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Power_Opower T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_n_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (and (tptp.class_Power_Opower T_b) (tptp.class_Rings_Omult__zero T_b) (tptp.class_Rings_Ono__zero__divisors T_b) (tptp.class_Rings_Ozero__neq__one T_b)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_a_2) V_n_2) _let_1) (and (= V_a_2 _let_1) (not (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_2 V_m) V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_1)) V_m)))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_p)) V_q) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_p) V_q))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) V_n)))))) (forall ((V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus _let_1) V_p) (tptp.c_Groups_Ozero__class_Ozero _let_1)) V_p)))) (forall ((V_q $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus _let_1) (tptp.c_Groups_Ozero__class_Ozero _let_1)) V_q) V_q)))) (forall ((V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) V_p) _let_2) _let_2))))) (forall ((V_q $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) _let_2) V_q) _let_2))))) _let_14 (forall ((V_ry $$unsorted) (V_rx $$unsorted) (V_ly $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_ly)) (tptp.hAPP (tptp.hAPP _let_1 V_rx) V_ry)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_rx)) (tptp.hAPP (tptp.hAPP _let_1 V_ly) V_ry))))))) (forall ((V_ry $$unsorted) (V_rx $$unsorted) (V_ly $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_lx) V_ly)))) (let ((_let_3 (tptp.hAPP _let_1 V_rx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_ry)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_ry)))))))) (forall ((V_ry $$unsorted) (V_rx $$unsorted) (V_ly $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_rx) V_ry))) (let ((_let_3 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_ly)) _let_2) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_ly) _let_2)))))))) (forall ((V_rx $$unsorted) (V_ly $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_ly)) V_rx) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_rx)) V_ly)))))) (forall ((V_rx $$unsorted) (V_ly $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_ly)) V_rx) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_ly) V_rx))))))) (forall ((V_ry $$unsorted) (V_rx $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_rx) V_ry)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_rx)) V_ry)))))) (forall ((V_ry $$unsorted) (V_rx $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (let ((_let_3 (tptp.hAPP _let_1 V_rx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_ry)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_ry)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_c)) V_b)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_c)) V_d)))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (let ((_let_3 (tptp.hAPP _let_1 V_c))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_d)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_d)))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a))))) (forall ((V_q_2 $$unsorted) (V_b_2 $$unsorted) (V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_OpCons T_b))) (=> (tptp.class_Groups_Ozero T_b) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_pa_2) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_q_2)) (and (= V_a_2 V_b_2) (= V_pa_2 V_q_2)))))) (forall ((V_q_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Int_Oring__char__0 T_b) (tptp.class_Rings_Oidom T_b)) (= (= (tptp.c_Polynomial_Opoly T_b V_pa_2) (tptp.c_Polynomial_Opoly T_b V_q_2)) (= V_pa_2 V_q_2)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_1) V_a) _let_1)))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_b) (= (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_b_2) V_a_2)) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a) V_a))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (let ((_let_3 (tptp.hAPP _let_1 V_w_2))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_b) (= (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_y_2)) (tptp.hAPP _let_2 V_z_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_z_2)) (tptp.hAPP _let_2 V_y_2))) (or (= V_w_2 V_x_2) (= V_y_2 V_z_2))))))))) (forall ((V_b $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_m)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_m)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_m)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_b_2))) (let ((_let_3 (tptp.hAPP _let_1 V_a_2))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_b) (= (and (not (= V_a_2 V_b_2)) (not (= V_c_2 V_d_2))) (not (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) (tptp.hAPP _let_2 V_d_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_d_2)) (tptp.hAPP _let_2 V_c_2))))))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a) V_a))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_a) (=> (not (= V_a _let_1)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n) _let_1)))))) (forall ((V_q $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_q) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_q)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_q))))))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_n) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a) (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) V_n) _let_1)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_a))) (forall ((V_q $$unsorted) (V_b $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_p)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_q)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_p) V_q))))))) (forall ((V_x $$unsorted) (V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_x)) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_q) V_x))))) (forall ((V_x $$unsorted) (V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_x)) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_q) V_x))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.c_Groups_Oone__class_Oone (tptp.tc_Polynomial_Opoly T_a))) V_x) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_x $$unsorted) (V_n $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower (tptp.tc_Polynomial_Opoly T_a)) V_p) V_n)) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_x)) V_n)))) (forall ((V_pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)))) (=> (and (tptp.class_Int_Oring__char__0 T_b) (tptp.class_Rings_Oidom T_b)) (= (= (tptp.c_Polynomial_Opoly T_b V_pa_2) (tptp.c_Polynomial_Opoly T_b _let_1)) (= V_pa_2 _let_1))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (V_r $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_r))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (=> (not (= V_r (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (and (= V_a V_b) (not (= V_c V_d))) (not (= (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) (tptp.hAPP _let_1 V_d)))))))))) (forall ((V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_m)) V_m) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) (tptp.c_Groups_Oone__class_Oone T_a))) V_m)))))) (forall ((V_a $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) (tptp.hAPP (tptp.hAPP _let_2 V_a) V_m)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) (tptp.c_Groups_Oone__class_Oone T_a))) V_m)))))) _let_13 (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_x_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_y_2)) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1))))))) (forall ((V_a_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Oidom T_b) (= (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_b V_pa_2) V_a_2) (tptp.c_Groups_Ozero__class_Ozero T_b)) (or (= V_pa_2 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b))) (not (= (tptp.c_Polynomial_Oorder T_b V_a_2 V_pa_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Power_Opower T_b) (= (tptp.c_Power_Opower__class_Opower T_b) (tptp.c_Power_Opower_Opower T_b (tptp.c_Groups_Oone__class_Oone T_b) (tptp.c_Groups_Otimes__class_Otimes T_b))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (= V_m (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (or (= V_n (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (= V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n) V_m) (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_1) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n) V_n)) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a) V_a))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m)))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_m))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_n)) V_k) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes _let_1))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus _let_1))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_p) V_q)) V_r) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_p) V_r)) (tptp.hAPP (tptp.hAPP _let_2 V_q) V_r)))))))) (forall ((V_a_2 $$unsorted) (V_times_2 $$unsorted) (V_one_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower_Opower T_b V_one_2 V_times_2) V_a_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_one_2)) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Ozero T_b) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Groups_Oab__semigroup__mult T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Groups_Oab__semigroup__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_b) (= (= (tptp.hAPP _let_1 V_b_2) (tptp.hAPP _let_1 V_c_2)) (= V_b_2 V_c_2))))) (forall ((V_c_2 $$unsorted) (V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_b) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2) (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_a_2)) (= V_b_2 V_c_2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_c)) (= V_b V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a))) (=> (tptp.class_Groups_Ocancel__ab__semigroup__add T_a) (=> (= (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_c)) (= V_b V_c))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a)) (= V_b V_c))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Groups_Oone T_b) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) _let_1) _let_1))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_1) (or (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (or (= V_m_2 V_n_2) (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_k_2)) (or (= V_m_2 V_n_2) (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z))))))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_m))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_n)) V_k) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2)))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_k_2)) (= V_m_2 V_n_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) V_k) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n) V_n)) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2)) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_1) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a) V_a))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2)) (= V_a_2 _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a) V_a))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.c_Polynomial_Opcompose T_a (tptp.hAPP _let_2 V_p) V_q) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus _let_1) (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) V_q) (tptp.c_Polynomial_Opcompose T_a V_p V_q)))))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_u $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_i) V_u)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_j) V_u)) V_k)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_j)) V_u)) V_k))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (or (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= V_m_2 V_n_2))))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_b_H)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_b_H))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_xa $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_xa))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_b $$unsorted) (V_a_H $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_a_H)) V_b) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b))))))) (forall ((V_ya $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_ya) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_ya)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_ya))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_e $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Osemiring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_e)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_e)) V_c)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_e)) V_c)))))) (forall ((V_x $$unsorted) (V_p $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__ring__1 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes (tptp.tc_Polynomial_Opoly T_a)) (tptp.c_Polynomial_Omonom T_a (tptp.c_Groups_Oone__class_Oone T_a) V_n)) V_p)) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) V_n)) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_x))))) (forall ((V_b_2 $$unsorted) (V_n_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (= (tptp.c_Polynomial_Omonom T_b V_a_2 V_n_2) (tptp.c_Polynomial_Omonom T_b V_b_2 V_n_2)) (= V_a_2 V_b_2)))) (forall ((V_n_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (= (tptp.c_Polynomial_Omonom T_b V_a_2 V_n_2) (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b))) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (= (tptp.c_Polynomial_Omonom T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_n) (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))))) (forall ((V_q $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.c_Polynomial_Opcompose T_a _let_1 V_q) _let_1)))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) (tptp.c_Polynomial_Omonom T_a V_a V_n)) (tptp.c_Polynomial_Omonom T_a V_b V_n)) (tptp.c_Polynomial_Omonom T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b) V_n)))) (forall ((V_x $$unsorted) (V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.c_Polynomial_Opcompose T_a V_p V_q)) V_x) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_q) V_x))))) (forall ((V_x $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.c_Polynomial_Omonom T_a V_a V_n)) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) V_n))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (= (tptp.c_Polynomial_Omonom T_a V_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes (tptp.tc_Polynomial_Opoly T_a)) (tptp.c_Polynomial_Omonom T_a V_a V_m)) (tptp.c_Polynomial_Omonom T_a V_b V_n)) (tptp.c_Polynomial_Omonom T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ono__zero__divisors T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) _let_1) (or (= V_a _let_1) (= V_b _let_1)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ono__zero__divisors T_a) (=> (not (= V_a _let_1)) (=> (not (= V_b _let_1)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) _let_1))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Rings_Oring__no__zero__divisors T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_b_2) _let_1) (or (= V_a_2 _let_1) (= V_b_2 _let_1)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Omult__zero T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1) _let_1)))) (forall ((V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_1) V_y) _let_1)))) (forall ((V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_1) V_b) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Omult__zero T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_1) V_a) _let_1)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Ozero__neq__one T_a) (not (= (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Ozero__neq__one T_a) (not (= (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_q_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_b))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (let ((_let_3 (tptp.tc_fun _let_1 _let_1))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_b) (= (tptp.c_Polynomial_Opcompose T_b V_pa_2 V_q_2) (tptp.c_Polynomial_Opoly__rec _let_1 T_b _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_1 _let_3) T_b) (tptp.c_COMBK _let_3 _let_1)) (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_3 _let_3) T_b) (tptp.c_COMBB _let_1 _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_Groups_Oplus__class_Oplus _let_1)) (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1 (tptp.c_Polynomial_OpCons T_b)) _let_2)))) (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) V_q_2))) V_pa_2))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ozero T_c) (=> (= (tptp.hAPP (tptp.hAPP (tptp.hAPP V_f_2 (tptp.c_Groups_Ozero__class_Ozero T_c)) (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_c))) V_z_2) V_z_2) (= (tptp.c_Polynomial_Opoly__rec T_b T_c V_z_2 V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_c) V_a_2) V_pa_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_pa_2) (tptp.c_Polynomial_Opoly__rec T_b T_c V_z_2 V_f_2 V_pa_2)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_3 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_a) (tptp.hAPP (tptp.hAPP _let_3 _let_1) (tptp.c_Groups_Ozero__class_Ozero _let_2)))) V_n)) V_n) _let_1)))))) (forall ((V_pa_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (V_z_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ozero T_c) (= (tptp.c_Polynomial_Opoly__rec T_b T_c V_z_2 V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_c) V_a_2) V_pa_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_pa_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_If T_b) (tptp.hAPP (tptp.c_fequal V_pa_2) (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_c)))) V_z_2) (tptp.c_Polynomial_Opoly__rec T_b T_c V_z_2 V_f_2 V_pa_2)))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes _let_1))) (let ((_let_3 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_a) V_p)) V_q) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_q)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP _let_2 V_p) V_q))))))))) (forall ((V_q $$unsorted) (V_a $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) V_p))) (let ((_let_3 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_a) V_q)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_p)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP _let_2 V_q))))))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.c_Groups_Oone__class_Oone (tptp.tc_Polynomial_Opoly T_a))) V_n))) (let ((_let_2 (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)))))))) (forall ((V_h $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly T_a _let_1 V_h) _let_1)))) (forall ((V_q_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (= V_pa_2 V_q_2) (forall ((B_n $$unsorted)) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_b V_pa_2) B_n) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_b V_q_2) B_n)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (= (tptp.c_Polynomial_Ocoeff T_b V_x_2) (tptp.c_Polynomial_Ocoeff T_b V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_n $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_p)) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_p) V_n))))) (forall ((V_p $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Osmult T_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_p)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) V_p))))) (forall ((V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_p) V_p))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) _let_1) _let_1)))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_p) V_q)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_p)) (tptp.hAPP _let_1 V_q))))))) (forall ((V_q $$unsorted) (V_a $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes (tptp.tc_Polynomial_Opoly T_a)) V_p))) (let ((_let_2 (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_q)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_q))))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes (tptp.tc_Polynomial_Opoly T_a)))) (let ((_let_2 (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_p)) V_q) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_p) V_q))))))) (forall ((V_h $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly T_a V_p V_h))) (let ((_let_2 (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly T_a (tptp.hAPP _let_2 V_p) V_h) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_h) _let_1)) (tptp.hAPP _let_2 _let_1))))))) (forall ((V_h_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_b))) (let ((_let_2 (tptp.tc_fun _let_1 _let_1))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_b) (= (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly T_b V_pa_2 V_h_2) (tptp.c_Polynomial_Opoly__rec _let_1 T_b (tptp.c_Groups_Ozero__class_Ozero _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_1 _let_2) T_b) (tptp.c_COMBK _let_2 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) (tptp.hAPP (tptp.c_COMBS _let_1 _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 _let_1) (tptp.c_Groups_Oplus__class_Oplus _let_1)) (tptp.hAPP (tptp.c_Polynomial_Osmult T_b) V_h_2)))) (tptp.c_Polynomial_OpCons T_b))) V_pa_2)))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))) V_n) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_p) (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))))) (forall ((V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)))) (=> (tptp.class_Rings_Oidom T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_b) V_a_2) V_pa_2) _let_1) (or (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b)) (= V_pa_2 _let_1)))))) (forall ((V_p $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_p)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) (tptp.hAPP _let_1 V_p))))))) (forall ((V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) V_p)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_a))) (forall ((V_x $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_p)) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_x))))) (forall ((V_a $$unsorted) (V_p $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_c) V_p) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) V_p)) (= V_p (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))))) (forall ((V_a $$unsorted) (V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.c_Polynomial_Omonom T_a V_a V_m)) V_n))) (let ((_let_2 (= V_m V_n))) (=> (tptp.class_Groups_Ozero T_a) (and (=> _let_2 (= _let_1 V_a)) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)))))))) (forall ((V_n $$unsorted) (V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_p) V_n)) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_q) V_n))))) (forall ((V_p $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Osmult T_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)) V_p) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_p)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_p)))))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) (tptp.c_Polynomial_Omonom T_a V_b V_n)) (tptp.c_Polynomial_Omonom T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) V_n)))) (forall ((V_h $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly T_a _let_1 V_h) _let_1)))) (forall ((V_h_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_b) (= (= (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly T_b V_pa_2 V_h_2) _let_1) (= V_pa_2 _let_1))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) V_z)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))) (forall ((V_a $$unsorted) (V_p $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_c) V_p)) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) V_p)) _let_2) (= V_p _let_2)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_y) V_z)))))) (forall ((V_q_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_b))) (let ((_let_2 (tptp.tc_fun _let_1 _let_1))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) V_pa_2) V_q_2) (tptp.c_Polynomial_Opoly__rec _let_1 T_b (tptp.c_Groups_Ozero__class_Ozero _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_1 _let_2) T_b) (tptp.c_COMBK _let_2 _let_1)) (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.c_COMBB _let_1 _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 T_b) (tptp.c_Groups_Oplus__class_Oplus _let_1)) (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1 (tptp.c_Polynomial_Osmult T_b)) V_q_2)))) (tptp.hAPP (tptp.c_Polynomial_OpCons T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)))) V_pa_2)))))) (forall ((V_x $$unsorted) (V_h $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly T_a V_p V_h)) V_x) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_h) V_x))))) (forall ((T_b $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_c)))) (=> (tptp.class_Groups_Ozero T_c) (=> (= (tptp.hAPP (tptp.hAPP (tptp.hAPP V_f_2 (tptp.c_Groups_Ozero__class_Ozero T_c)) _let_1) V_z_2) V_z_2) (= (tptp.c_Polynomial_Opoly__rec T_b T_c V_z_2 V_f_2 _let_1) V_z_2))))) (forall ((V_c $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Osynthetic__div T_a V_p V_c))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) V_p) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_c) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_c)) _let_1))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_c $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) V_p) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_c) V_q)) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_r) V_q)) (and (= V_r (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_c)) (= V_q (tptp.c_Polynomial_Osynthetic__div T_a V_p V_c)))))) (forall ((V_q_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_b)) V_pa_2) V_q_2) (tptp.c_Polynomial_OAbs__poly T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) tptp.tc_Nat_Onat) (tptp.c_Groups_Oplus__class_Oplus T_b)) (tptp.c_Polynomial_Ocoeff T_b V_pa_2))) (tptp.c_Polynomial_Ocoeff T_b V_q_2)))))) (forall ((V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_b) V_a_2) V_pa_2) (tptp.c_Polynomial_OAbs__poly T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2)) (tptp.c_Polynomial_Ocoeff T_b V_pa_2)))))) (forall ((V_c $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.c_Polynomial_Osynthetic__div T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_p) V_c) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_c)) (tptp.c_Polynomial_Osynthetic__div T_a V_p V_c)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Ozero__class_Ozero _let_1)))) V_n)) V_n))))) (forall ((V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (tptp.c_Polynomial_Ocoeff T_b (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_b) V_a_2) V_pa_2)) (tptp.c_Nat_Onat_Onat__case T_b V_a_2 (tptp.c_Polynomial_Ocoeff T_b V_pa_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_4 (= V_m _let_3))) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n)))))))))) (forall ((V_n $$unsorted) (V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_p) V_n)) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_q) V_n))))) (forall ((V_p $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Osmult T_a))) (=> (tptp.class_Rings_Ocomm__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b)) V_p) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_Polynomial_Opoly T_a)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_p)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_p)))))) (forall ((V_w $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z1) V_z2)) V_w) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z1) V_w)) (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_w)))))) (forall ((V_z3 $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_z1))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_z2)) V_z3) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_z3)))))) (forall ((V_z2 $$unsorted) (V_z1 $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z1) V_z2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z1)) (tptp.hAPP _let_1 V_z2)))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z)))) (forall ((V_q_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_Polynomial_Opoly T_b)) V_pa_2) V_q_2) (tptp.c_Polynomial_OAbs__poly T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) tptp.tc_Nat_Onat) (tptp.c_Groups_Ominus__class_Ominus T_b)) (tptp.c_Polynomial_Ocoeff T_b V_pa_2))) (tptp.c_Polynomial_Ocoeff T_b V_q_2)))))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_z) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z)) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z) V_z)) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (tptp.class_Groups_Oab__group__add T_b) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2) (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_d_2)) (= (= V_a_2 V_b_2) (= V_c_2 V_d_2)))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_i))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_j)) V_k) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_k)) V_j))))) (forall ((V_c $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.c_Polynomial_Odegree T_a (tptp.c_Polynomial_Osynthetic__div T_a V_p V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.c_Polynomial_Odegree T_a V_p)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_x $$unsorted) (V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__ring T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_x)) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_q) V_x))))) (forall ((V_q $$unsorted) (V_b $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_Polynomial_Opoly T_a)))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_p)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_q)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_p) V_q))))))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_Polynomial_Opoly T_a)) (tptp.c_Polynomial_Omonom T_a V_a V_n)) (tptp.c_Polynomial_Omonom T_a V_b V_n)) (tptp.c_Polynomial_Omonom T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b) V_n)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_b)) (= V_a_2 V_b_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_b) (= (= V_a_2 V_b_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_ya $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_ya) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_ya)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_ya))))))) (forall ((V_b $$unsorted) (V_a_H $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_a_H)) V_b) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_xa $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_xa))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y))))))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_b_H)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_b_H))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)) V_b) V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b)) V_b) V_a))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n) _let_1) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_n) V_m) _let_1) (= V_m V_n)))))) (forall ((V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_k)) (tptp.hAPP (tptp.hAPP _let_2 V_n) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) (tptp.hAPP _let_2 V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_i))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_j)) V_k) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j) V_k)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)) V_n) V_m)) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)) V_n) V_m)) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) V_k) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((V_c_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_b) (= (= (tptp.c_Polynomial_Osynthetic__div T_b V_pa_2 V_c_2) (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b))) (= (tptp.c_Polynomial_Odegree T_b V_pa_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_h $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.c_Polynomial_Odegree T_a (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ooffset__poly T_a V_p V_h)) (tptp.c_Polynomial_Odegree T_a V_p)))) (forall ((V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_b) V_a_2) V_pa_2) (tptp.c_Polynomial_OAbs__poly T_b (tptp.c_Nat_Onat_Onat__case T_b V_a_2 (tptp.c_Polynomial_Ocoeff T_b V_pa_2)))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (tptp.c_Polynomial_OAbs__poly T_b (tptp.c_Polynomial_Ocoeff T_b V_x_2)) V_x_2))) (forall ((T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (= (tptp.c_Polynomial_Odegree T_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Rings_Oring T_b) (= (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_a_2) V_b_2)) V_e_2)) V_c_2) V_d_2)))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Rings_Oring T_b) (= (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2)) (= V_c_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_b_2) V_a_2)) V_e_2)) V_d_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_b))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_4 (tptp.hAPP _let_3 V_a))) (let ((_let_5 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_a)))) (let ((_let_6 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 V_x) V_y)) (tptp.hAPP _let_4 V_b)) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 _let_2)) (tptp.hAPP _let_5 V_b))) (tptp.hAPP _let_4 _let_2))))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Polynomial_Odegree T_a (tptp.c_Polynomial_Omonom T_a V_a V_n)) V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Onat_Onat__case T_b V_f1_2 V_f2_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_f1_2)) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Polynomial_Odegree T_a (tptp.c_Groups_Oone__class_Oone (tptp.tc_Polynomial_Opoly T_a))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.c_Polynomial_Osynthetic__div T_a _let_1 V_c) _let_1)))) (forall ((V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_p)))) (let ((_let_2 (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Oidom T_a) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_2) (= _let_1 (tptp.c_Polynomial_Odegree T_a V_p)))))))) (forall ((V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (=> (not (= V_p (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (not (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_p) (tptp.c_Polynomial_Odegree T_a V_p)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_b V_pa_2) (tptp.c_Polynomial_Odegree T_b V_pa_2)) (tptp.c_Groups_Ozero__class_Ozero T_b)) (= V_pa_2 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (= (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (=> (tptp.class_Rings_Oidom T_a) (=> (not (= V_p _let_2)) (=> (not (= V_q _let_2)) (= (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) V_p) V_q)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Polynomial_Odegree T_a V_p)) (tptp.c_Polynomial_Odegree T_a V_q))))))))) (forall ((V_p $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_p))) (let ((_let_3 (tptp.hAPP _let_2 V_m))) (let ((_let_4 (= V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_4 (= _let_3 _let_1)) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_p) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) _let_1))))))))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Groups_Ozero T_b) (= (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)) (tptp.c_Polynomial_OAbs__poly T_b (tptp.hAPP (tptp.c_COMBK T_b tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero T_b)))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Odegree T_a V_q))) (let ((_let_2 (tptp.c_Polynomial_Odegree T_a V_p))) (=> (tptp.class_Rings_Ocomm__semiring__0 T_a) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) _let_2) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_p) _let_2)) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_q) _let_1))))))) (forall ((V_m_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b T_b))) (=> (tptp.class_Groups_Ozero T_b) (= (tptp.c_Polynomial_Omonom T_b V_a_2 V_m_2) (tptp.c_Polynomial_OAbs__poly T_b (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat T_b T_b (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_1) tptp.tc_Nat_Onat) (tptp.c_If T_b)) (tptp.c_fequal V_m_2))) V_a_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))))))) (forall ((V_m $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_m))) (let ((_let_2 (tptp.hAPP _let_1 V_n))) (let ((_let_3 (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.class_Power_Opower T_a) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_m) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_x) V_x)) _let_1) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_x) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_x))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_y)) (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_b))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_a)) V_b)))))))) (forall ((V_p $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (let ((_let_3 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Rings_Ocomm__ring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_c)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Oone__class_Oone T_a)) _let_2))) (tptp.c_Polynomial_Osynthetic__div T_a V_p V_c))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_c)) _let_2)) V_p)))))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_d)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Opoly__gcd T_a V_x V_y))) (let ((_let_2 (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a _let_1) (tptp.c_Polynomial_Odegree T_a _let_1)))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (let ((_let_4 (and (= V_x _let_3) (= V_y _let_3)))) (=> (tptp.class_Fields_Ofield T_a) (and (=> _let_4 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_4) (= _let_2 (tptp.c_Groups_Oone__class_Oone T_a)))))))))) (forall ((V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Opsize T_a V_p))) (let ((_let_2 (= V_p (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))))) (=> (tptp.class_Groups_Ozero T_a) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_2) (= _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Polynomial_Odegree T_a V_p))))))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ominus T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_c T_b)) V_A_2) V_B_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (tptp.hAPP V_A_2 V_x_2)) (tptp.hAPP V_B_2 V_x_2))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z) V_w))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z) V_w) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)))) (forall ((V_q $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.c_Groups_Ozero__class_Ozero _let_1)) V_q) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_q))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_b))) (=> (tptp.class_Rings_Oidom T_b) (= (= (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) _let_1)) (or (= V_x_2 V_y_2) (= V_x_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_y_2)))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_a)) (tptp.c_Polynomial_Omonom T_a V_a V_n)) (tptp.c_Polynomial_Omonom T_a (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a) V_n)))) (forall ((V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_a)))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_p)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.hAPP _let_1 V_p))))))) (forall ((V_p $$unsorted) (V_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_b)))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_b))) (=> (tptp.class_Groups_Oab__group__add T_b) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_p)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a)) (tptp.hAPP _let_1 V_p))))))) (forall ((V_x $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__ring T_a) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_a (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_a)) V_p)) V_x) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP (tptp.c_Polynomial_Opoly T_a V_p) V_x))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Polynomial_Opoly__gcd T_a V_a V_b) (tptp.c_Polynomial_Opoly__gcd T_a V_b V_a)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Polynomial_Opoly__gcd T_a V_x (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_a)) V_y)) (tptp.c_Polynomial_Opoly__gcd T_a V_x V_y)))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Polynomial_Opoly__gcd T_a V_b (tptp.c_Polynomial_Opoly__gcd T_a V_a V_c)) (tptp.c_Polynomial_Opoly__gcd T_a V_a (tptp.c_Polynomial_Opoly__gcd T_a V_b V_c))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Polynomial_Opoly__gcd T_a (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_a)) V_x) V_y) (tptp.c_Polynomial_Opoly__gcd T_a V_x V_y)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Polynomial_Opoly__gcd T_a (tptp.c_Polynomial_Opoly__gcd T_a V_a V_b) V_c) (tptp.c_Polynomial_Opoly__gcd T_a V_a (tptp.c_Polynomial_Opoly__gcd T_a V_b V_c))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Lattices_Oboolean__algebra T_b) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ouminus T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_c T_b)) V_A_2) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) (tptp.hAPP V_A_2 V_x_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_x)) V_x)))) (forall ((V_n $$unsorted)) (not (= V_n (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_n) V_n))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ouminus T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_c T_b)) V_A_2) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) (tptp.hAPP V_A_2 V_x_2))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_a)) V_a)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= V_a_2 (tptp.hAPP _let_1 V_b_2)) (= V_b_2 (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= (tptp.hAPP _let_1 V_a_2) V_b_2) (= (tptp.hAPP _let_1 V_b_2) V_a_2))))) (forall ((V_nat_H_2 $$unsorted) (V_nat_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Nat_OSuc V_nat_2) (tptp.hAPP tptp.c_Nat_OSuc V_nat_H_2)) (= V_nat_2 V_nat_H_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_b_2)) (= V_a_2 V_b_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (= (tptp.hAPP tptp.c_Nat_OSuc V_x) (tptp.hAPP tptp.c_Nat_OSuc V_y)) (= V_x V_y))) (forall ((V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (not (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z)) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z) V_w)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z)) (tptp.hAPP _let_1 V_w)))))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z) V_z)) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z))))))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_z)) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (forall ((V_z3 $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_z1))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_z2)) V_z3) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_z3)))))) (forall ((V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Osmult T_a))) (=> (tptp.class_Rings_Ocomm__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_p) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_a)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_p)))))) (forall ((V_n $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_a)) V_p)) V_n) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_p) V_n))))) (not (= _let_4 _let_3)) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) _let_1) _let_1)))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) (= _let_1 V_a_2))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (= V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2) V_a_2) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b))))))) (forall ((V_x $$unsorted) (V_xa $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_xa))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_x)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_x))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) V_b) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_x)) V_y) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_b) (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Oidom T_b) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_a_2) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_b_2)) (or (= V_a_2 V_b_2) (= V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_b_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_a))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_b)) V_b)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_b)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_m $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)))) (forall ((V_nat_H $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_nat_H)))) (forall ((V_m $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_nat_H_1 $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_nat_H_1) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)))) (forall ((V_m $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_x_2) V_m_2) _let_2) (or (= V_m_2 _let_1) (= V_x_2 _let_2)))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_z2 $$unsorted) (V_z1 $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z1) V_z2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z1)) (tptp.hAPP _let_1 V_z2)))))) (forall ((V_w $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z1) V_z2)) V_w) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z1) V_w)) (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_w)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_m) (tptp.hAPP tptp.c_Nat_OSuc V_n))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_z)) V_w) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w)))))) (forall ((V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_p) (tptp.c_Groups_Ozero__class_Ozero _let_1)) V_p)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) _let_2) _let_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n)) (tptp.hAPP tptp.c_Nat_OSuc V_k)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) V_k)))) (forall ((V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_a)) V_p)) (tptp.c_Polynomial_Odegree T_a V_p)))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Rings_Ocomm__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_p) V_q)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_p)) (tptp.hAPP _let_1 V_q))))))) (forall ((V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Rings_Ocomm__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_p)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_p))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_x_2) V_y_2) (tptp.c_Groups_Ozero__class_Ozero T_b)) (= V_y_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_x_2))))) (forall ((V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_b) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_Polynomial_Opoly T_b)) V_pa_2) (tptp.c_Polynomial_OAbs__poly T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b tptp.tc_Nat_Onat) (tptp.c_Groups_Ouminus__class_Ouminus T_b)) (tptp.c_Polynomial_Ocoeff T_b V_pa_2)))))) (forall ((V_nat_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Onat_Onat__case T_b V_f1_2 V_f2_2) (tptp.hAPP tptp.c_Nat_OSuc V_nat_2)) (tptp.hAPP V_f2_2 V_nat_2))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Polynomial_Opoly__gcd T_a _let_1 _let_1) _let_1)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)))) (=> (tptp.class_Fields_Ofield T_b) (= (= (tptp.c_Polynomial_Opoly__gcd T_b V_x_2 V_y_2) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))) (forall ((V_n_2 $$unsorted) (V_a_2 $$unsorted) (V_times_2 $$unsorted) (V_one_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower_Opower T_b V_one_2 V_times_2) V_a_2))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hAPP (tptp.hAPP V_times_2 V_a_2) (tptp.hAPP _let_1 V_n_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Polynomial_Opoly__gcd T_a V_x _let_1) _let_1)))) (forall ((V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Polynomial_Opoly__gcd T_a _let_1 V_y) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_b_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b) (tptp.c_Groups_Ozero__class_Ozero T_a)) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a) V_b)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Ocomm__ring__1 T_a) (= (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone T_a))) V_x))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_x_2) V_x_2) _let_1) (or (= V_x_2 _let_1) (= V_x_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) _let_1))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__ring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_x) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_x) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_y))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_b))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (and (tptp.class_Power_Opower T_a) (tptp.class_Rings_Osemiring__0 T_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.hAPP tptp.c_Nat_OSuc V_n)) _let_1)))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Power_Opower T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP _let_1 V_n)))))) (forall ((V_q $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_q)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) (tptp.hAPP _let_1 V_q)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_n)) V_a))))) (forall ((V_q $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) (tptp.hAPP _let_1 V_q)) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_q)))))) (forall ((V_q $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_q)) V_x) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_q)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2)) (or (and (= V_m_2 _let_2) (= V_n_2 _let_1)) (and (= V_m_2 _let_1) (= V_n_2 _let_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_2) (or (and (= V_m_2 _let_2) (= V_n_2 _let_1)) (and (= V_m_2 _let_1) (= V_n_2 _let_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_1) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))) (= _let_5 _let_9) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) (tptp.hAPP _let_1 V_n))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Nat_OSuc V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n))) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Nat_OSuc V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_m))) (= (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n))))) (forall ((V_n $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) V_p)) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_p) V_n)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m_2))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hAPP (tptp.c_Nat_Onat_Onat__case tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_COMBI tptp.tc_Nat_Onat)) (tptp.hAPP _let_1 V_n_2))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a))) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a) V_a))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) V_x) V_x))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ominus T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_c T_b)) V_A_2) V_B_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (tptp.hAPP V_A_2 V_x_2)) (tptp.hAPP V_B_2 V_x_2))))) (forall ((V_a $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (=> (not (= V_p (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (= (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) V_p)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Polynomial_Odegree T_a V_p)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (= (tptp.c_Polynomial_Omonom T_a V_a (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Polynomial_Omonom T_a V_a V_n))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Ocomm__ring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 V_x) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_x) V_y)))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))) (let ((_let_3 (= V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_3 (= _let_2 V_n)) (=> (not _let_3) (= _let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n))))))))) (forall ((V_a $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a) V_p)))) (let ((_let_2 (= V_p (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))))) (=> (tptp.class_Groups_Ozero T_a) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_2) (= _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Polynomial_Odegree T_a V_p))))))))) (forall ((V_a $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (let ((_let_3 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Rings_Oidom T_a) (=> (not (= V_p _let_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower _let_1) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Oone__class_Oone T_a)) _let_2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Polynomial_Oorder T_a V_a V_p)))) V_p))))))))) (forall ((V_a $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Oorder T_a V_a V_p))) (let ((_let_2 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero _let_2))) (let ((_let_4 (tptp.c_Polynomial_OpCons T_a))) (let ((_let_5 (tptp.hAPP (tptp.c_Power_Opower__class_Opower _let_2) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_Groups_Oone__class_Oone T_a)) _let_3))))) (let ((_let_6 (tptp.c_Rings_Odvd__class_Odvd _let_2))) (=> (tptp.class_Rings_Oidom T_a) (=> (not (= V_p _let_3)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 _let_1)) V_p)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) V_p))))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_a))) (=> (tptp.class_Rings_Oidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower _let_1) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Ozero__class_Ozero _let_1)))) (tptp.c_Polynomial_Oorder T_a V_a V_p))) V_p)))))) (forall ((V_c_2 $$unsorted) (V_pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_b))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_b))) (=> (tptp.class_Rings_Oidom T_b) (= (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_b V_pa_2) V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd _let_1) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.c_Groups_Ozero__class_Ozero _let_1)))) V_pa_2))))))) (forall ((V_pa_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_b))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_b))) (=> (tptp.class_Rings_Oidom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_c_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.c_Groups_Ozero__class_Ozero _let_1)))) V_pa_2)) (= (tptp.hAPP (tptp.c_Polynomial_Opoly T_b V_pa_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_c_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Polynomial_Opoly__gcd T_a V_x (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a))) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_x) (tptp.c_Polynomial_Odegree T_a V_x)))) V_x)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_a)) (tptp.c_Polynomial_Opoly__gcd T_a V_x V_y)) V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_a)) (tptp.c_Polynomial_Opoly__gcd T_a V_x V_y)) V_x)))) (forall ((V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_z)) V_z))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) _let_4) _let_4) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (=> (= (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (= V_a V_b))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.hBOOL (tptp.hAPP _let_2 V_c)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_b))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_b) (= (= (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_b_2)) (= V_a_2 V_b_2))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_a)) V_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) V_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) _let_1) V_a)) (= V_a _let_1))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)))))) (forall ((V_k $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odvd T_a) (=> (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_b) V_a))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_c)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_c))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_c))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_c)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_b) V_c)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (= (tptp.hAPP _let_2 V_a) (tptp.hAPP _let_2 V_b)) (=> (not (= V_a _let_1)) (=> (not (= V_b _let_1)) (= V_a V_b)))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a) _let_1) (= V_a _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_a)) V_a))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a _let_1)) (not (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a) _let_1)))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_b) (= (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_b) V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) _let_1) _let_1)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_x))) (=> (tptp.class_Rings_Ocomm__ring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_y) V_z)))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_b))) (=> (tptp.class_Rings_Ocomm__ring__1 T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_x_2)) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_x_2))) (=> (tptp.class_Rings_Ocomm__ring__1 T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_y_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)))))) (forall ((V_n $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_n)))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_n)))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_p)) V_q)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_p) V_q)))))) (forall ((V_a $$unsorted) (V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_a)) V_p))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_q)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_q))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_k $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_a)) V_k))) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Polynomial_Opoly__gcd T_a V_x V_y)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_b)) V_k_2))) (=> (tptp.class_Fields_Ofield T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Polynomial_Opoly__gcd T_b V_x_2 V_y_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_y_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Oidom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (or (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Oidom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2))) (or (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a _let_3)) (=> (not (= V_b _let_3)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_a)))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) (tptp.c_Groups_Oone__class_Oone T_a)) (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a) V_b)))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_n))))))) (forall ((V_q $$unsorted) (V_a $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_a)) V_p))) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_q))) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP _let_1 V_q))))))) (forall ((V_a $$unsorted) (V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_p) V_q)) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_p)) V_q))))))) (forall ((V_q_2 $$unsorted) (V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_b)) V_pa_2))) (=> (tptp.class_Fields_Ofield T_b) (=> (not (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_b) V_a_2) V_q_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_q_2))))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Rings_Oidom T_a) (=> (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_p) (tptp.c_Polynomial_Odegree T_a V_p)) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_q) (tptp.c_Polynomial_Odegree T_a V_q))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_p) V_q)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_q) V_p)) (= V_p V_q))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_4 (tptp.hAPP _let_1 V_a))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_6 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a _let_6)) (=> (not (= V_b _let_6)) (= (tptp.hAPP (tptp.hAPP _let_3 _let_4) _let_2) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_5 _let_4) (tptp.hAPP (tptp.hAPP _let_3 V_a) V_b))) _let_2)))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_4 (tptp.hAPP _let_1 V_a))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_6 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a _let_6)) (=> (not (= V_b _let_6)) (= (tptp.hAPP (tptp.hAPP _let_3 _let_4) _let_2) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_5 _let_4) (tptp.hAPP (tptp.hAPP _let_3 V_b) V_a))) _let_2)))))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a)) V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a)) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_q_2 $$unsorted) (V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_b))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd _let_1))) (let ((_let_3 (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b)))) (=> (tptp.class_Fields_Ofield T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_b) V_a_2) V_pa_2)) V_q_2)) (and (=> _let_3 (= V_q_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (=> (not _let_3) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_pa_2) V_q_2)))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_4 (tptp.hAPP _let_1 V_a))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_6 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a _let_6)) (=> (not (= V_b _let_6)) (= (tptp.hAPP (tptp.hAPP _let_3 _let_4) _let_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_5 _let_4) (tptp.hAPP (tptp.hAPP _let_3 V_a) V_b))) _let_2))))))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a)) V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_6 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_a _let_6)) (=> (not (= V_b _let_6)) (= (tptp.hAPP (tptp.hAPP _let_4 _let_3) _let_2) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_a) V_b)) _let_3)) _let_2)))))))))))) (forall ((V_t_2 $$unsorted) (V_D_2 $$unsorted) (V_d_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Rings_Ocomm__ring T_b) (tptp.class_Rings_Odvd T_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_d_2) V_D_2)) (forall ((B_x $$unsorted) (B_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_d_2))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) V_t_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) B_k) V_D_2))) V_t_2)))))))))) (forall ((V_t_2 $$unsorted) (V_D_2 $$unsorted) (V_d_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Rings_Ocomm__ring T_b) (tptp.class_Rings_Odvd T_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_d_2) V_D_2)) (forall ((B_x $$unsorted) (B_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_d_2))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) V_t_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) B_k) V_D_2))) V_t_2)))))))))) (forall ((V_l_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Rings_Odvd T_b) (tptp.class_Rings_Osemiring__0 T_b)) (= (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_l_2) B_x)))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_l_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) B_x) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x))))))) (forall ((V_k $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_k))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_m) V_n))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 V_m)))))) (forall ((V_t_2 $$unsorted) (V_d_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_d_2) V_t_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_d_2)) V_t_2))))) (forall ((V_t_2 $$unsorted) (V_d_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_d_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 V_t_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_t_2)))))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_k_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n_2) V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_n_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n))))))) (forall ((V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_m_2) _let_1)) (= V_m_2 _let_1)))) (forall ((V_t_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint))) (=> (not (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_t_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP _let_1 V_t_2)))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) (tptp.hAPP _let_2 V_n))) (=> (not (= V_k (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_k_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_n_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) V_m_2)))) (tptp.hBOOL (tptp.hAPP _let_1 V_n_2))))) (forall ((V_c_2 $$unsorted) (V_t_2 $$unsorted) (V_x_2 $$unsorted) (V_d_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_d_2)) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_t_2))) (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_c_2) V_d_2))) V_t_2))))))))) (forall ((V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_m_2) _let_1)) (= V_m_2 _let_1)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (or (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) _let_1) _let_1)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Fields_Ofield__inverse__zero T_b) (= (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_b) V_x_2) _let_1) (= V_x_2 _let_1))))) (forall ((V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_b))) (let ((_let_2 (tptp.c_Polynomial_OpCons T_b))) (=> (tptp.class_Rings_Oidom T_b) (= (tptp.c_Polynomial_Oorder T_b V_a_2 V_pa_2) (tptp.c_Orderings_Oord__class_OLeast tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_1 tptp.tc_HOL_Obool (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) (tptp.c_Rings_Odvd__class_Odvd _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_1 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Power_Opower__class_Opower _let_1) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.c_Groups_Ozero__class_Ozero _let_1))))) tptp.c_Nat_OSuc))) V_pa_2)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_d $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_d) (tptp.c_Polynomial_Odegree T_a V_d)))) (let ((_let_2 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero _let_2))) (let ((_let_4 (and (= V_x _let_3) (= V_y _let_3)))) (let ((_let_5 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd _let_2) V_d))) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.hBOOL (tptp.hAPP _let_5 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_5 V_y)) (=> (forall ((B_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd (tptp.tc_Polynomial_Opoly T_a)) B_k))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 V_d)))))) (=> (and (=> _let_4 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_4) (= _let_1 (tptp.c_Groups_Oone__class_Oone T_a)))) (= (tptp.c_Polynomial_Opoly__gcd T_a V_x V_y) V_d)))))))))))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_n))) (=> (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat))) (=> (not (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_n_2)) (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))))))) (forall ((V_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_x) V_x))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (=> (and _let_2 (not _let_3)) (not (and _let_3 (not _let_2)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_z))) (let ((_let_3 (tptp.hAPP _let_1 V_x))) (let ((_let_4 (tptp.hAPP _let_1 V_y))) (=> (and (tptp.hBOOL (tptp.hAPP _let_3 V_y)) (not (tptp.hBOOL (tptp.hAPP _let_4 V_x)))) (=> (and (tptp.hBOOL (tptp.hAPP _let_4 V_z)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_y)))) (and (tptp.hBOOL (tptp.hAPP _let_3 V_z)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_x))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)))) (=> (and _let_2 (not _let_3)) (not (and _let_3 (not _let_2)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (and (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (not (tptp.hBOOL (tptp.hAPP _let_3 V_x)))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_z)) (and (tptp.hBOOL (tptp.hAPP _let_2 V_z)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x)))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (and (tptp.hBOOL (tptp.hAPP _let_2 V_b)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)))) (=> (= V_b V_c) (and (tptp.hBOOL (tptp.hAPP _let_2 V_c)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (not (= V_y V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (not (= V_x V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (=> (and _let_2 (not _let_3)) (not (and _let_3 (not _let_2)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (=> (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) _let_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (=> (and _let_2 (not _let_3)) (not (and _let_3 (not _let_2)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (not (= V_x V_y))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_z))) (let ((_let_3 (tptp.hAPP _let_1 V_x))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_y)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_y)))) (and (tptp.hBOOL (tptp.hAPP _let_3 V_z)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_x)))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_c))) (=> (= V_a V_b) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_b)))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_a))))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)) (tptp.hBOOL (tptp.hAPP _let_2 V_z))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (= V_x V_y))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m)) (= V_m V_n))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP _let_1 V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (= V_a V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))) (=> _let_2 (=> (not (= V_a V_b)) (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (=> _let_2 (or (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (= V_x V_y)))))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (= V_x V_y) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_x) V_y)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))) (=> (not (= V_a V_b)) (=> _let_2 (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)))) (= (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)))) (and _let_2 (not (= V_x_2 V_y_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)))) (= _let_2 (or (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)))) (= V_x_2 V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (= (= V_x_2 V_y_2) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)))))) (forall ((V_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_x))) (forall ((V_n_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_n_2)) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.c_Orderings_Oord__class_OLeast tptp.tc_Nat_Onat V_P_2) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Orderings_Oord__class_OLeast tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_Nat_Onat) V_P_2) tptp.c_Nat_OSuc))))))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_n))) (=> (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint))) (=> (not (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_n_2)) (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))))))) (forall ((V_d_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Oinverse T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_d_2) (=> (not (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_b) _let_2) V_f_2) V_x_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_d_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) _let_1) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Polynomial_Opoly__gcd T_a V_x V_y))) (let ((_let_3 (= V_y (tptp.c_Groups_Ozero__class_Ozero _let_1)))) (=> (tptp.class_Fields_Ofield T_a) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_x) (tptp.c_Polynomial_Odegree T_a V_x)))) V_x))) (=> (not _let_3) (= _let_2 (tptp.c_Polynomial_Opoly__gcd T_a V_y (tptp.c_Divides_Odiv__class_Omod _let_1 V_x V_y)))))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (not (= V_x_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (tptp.c_Deriv_Oderiv T_b _let_1 V_x_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))) (forall ((V_k_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Owellorder T_b) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_k_2)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Orderings_Oord__class_OLeast T_b V_P_2)))))) (forall ((V_x_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.c_COMBK T_b T_b) V_k_2) V_x_2 (tptp.c_Groups_Ozero__class_Ozero T_b)))) (forall ((V_Db_2 $$unsorted) (V_Da_2 $$unsorted) (V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 (tptp.hAPP V_g_2 V_x_2) V_Da_2) (=> (tptp.c_Deriv_Oderiv T_b V_g_2 V_x_2 V_Db_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_b) V_f_2) V_g_2) V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_Da_2) V_Db_2)))))) (forall ((V_E_2 $$unsorted) (V_g_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (=> (tptp.c_Deriv_Oderiv T_b V_g_2 (tptp.hAPP V_f_2 V_x_2) V_E_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_b) V_g_2) V_f_2) V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_E_2) V_D_2)))))) (forall ((V_c_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_b) _let_1) V_f_2) V_x_2 (tptp.hAPP _let_1 V_D_2)))))) (forall ((V_x_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2) V_x_2 V_c_2))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a))) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Divides_Odiv__class_Omod _let_1 (tptp.hAPP _let_2 V_x) V_y) (tptp.hAPP _let_2 (tptp.c_Divides_Odiv__class_Omod _let_1 V_x V_y))))))) (forall ((V_E_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (=> (= V_D_2 V_E_2) (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_E_2))))) (forall ((V_Y_2 $$unsorted) (V_X_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_X_2) (=> (= V_X_2 V_Y_2) (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_Y_2))))) (forall ((V_E_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_E_2) (= V_D_2 V_E_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Divides_Odiv__class_Omod _let_1 V_x (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_y)) (tptp.c_Divides_Odiv__class_Omod _let_1 V_x V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.c_Divides_Odiv__class_Omod _let_1 (tptp.hAPP _let_2 V_x) V_y) (tptp.hAPP _let_2 (tptp.c_Divides_Odiv__class_Omod _let_1 V_x V_y))))))) (forall ((V_E_2 $$unsorted) (V_g_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (=> (tptp.c_Deriv_Oderiv T_b V_g_2 V_x_2 V_E_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_1) V_f_2)) V_g_2) V_x_2 (tptp.hAPP (tptp.hAPP _let_1 V_D_2) V_E_2))))))) (forall ((V_E_2 $$unsorted) (V_g_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (=> (tptp.c_Deriv_Oderiv T_b V_g_2 V_x_2 V_E_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_1) V_f_2)) V_g_2) V_x_2 (tptp.hAPP (tptp.hAPP _let_1 V_D_2) V_E_2))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (tptp.c_Deriv_Oderiv T_b (tptp.c_COMBI T_b) V_x_2 (tptp.c_Groups_Oone__class_Oone T_b)))) (forall ((V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_b) _let_1) V_f_2) V_x_2 (tptp.hAPP _let_1 V_D_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Omod _let_1 V_x (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a) V_y)) (tptp.c_Divides_Odiv__class_Omod _let_1 V_x V_y)))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_y (tptp.c_Groups_Ozero__class_Ozero _let_1))) (= (tptp.c_Polynomial_Opoly__gcd T_a V_x V_y) (tptp.c_Polynomial_Opoly__gcd T_a V_y (tptp.c_Divides_Odiv__class_Omod _let_1 V_x V_y))))))) (forall ((V_Db_2 $$unsorted) (V_g_2 $$unsorted) (V_Da_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_Da_2) (=> (tptp.c_Deriv_Oderiv T_b V_g_2 V_x_2 V_Db_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_1) V_f_2)) V_g_2) V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP (tptp.hAPP _let_1 V_Da_2) (tptp.hAPP V_g_2 V_x_2))) (tptp.hAPP (tptp.hAPP _let_1 V_Db_2) (tptp.hAPP V_f_2 V_x_2))))))))) (forall ((V_E_2 $$unsorted) (V_g_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (=> (tptp.c_Deriv_Oderiv T_b V_g_2 V_x_2 V_E_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_1) V_f_2)) V_g_2) V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_f_2 V_x_2)) V_E_2)) (tptp.hAPP (tptp.hAPP _let_1 V_D_2) (tptp.hAPP V_g_2 V_x_2))))))))) (forall ((V_E_2 $$unsorted) (V_g_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (=> (tptp.c_Deriv_Oderiv T_b V_g_2 V_x_2 V_E_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_b) _let_1) V_g_2)) V_x_2 (tptp.hAPP (tptp.hAPP _let_2 V_D_2) (tptp.hAPP _let_1 V_E_2))))))))) (forall ((V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Oinverse T_b))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (let ((_let_4 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (=> (not (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_b) _let_2) V_f_2) V_x_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_4 _let_3) V_D_2)) _let_3))))))))))) (forall ((V_P_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Owellorder T_b) (=> (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Orderings_Oord__class_OLeast T_b V_P_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_a_2) V_b_2)) (= (tptp.c_Divides_Odiv__class_Omod T_b V_b_2 V_a_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) V_b)) (= (tptp.c_Divides_Odiv__class_Omod T_a V_b V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c) V_b)) V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_c)) V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_f $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_f))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m V_n))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m V_n))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 V_m)))))) (forall ((V_m $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_x))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_y V_m)) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_y) V_m)))) (forall ((V_y $$unsorted) (V_m $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_m)) V_y) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) V_m)))) (forall ((V_m $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_m)) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_x) V_m)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) V_b))))) (forall ((V_a $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (forall ((V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_1 V_b) _let_1))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_b) V_c) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c)) V_c)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c)) V_c) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_b) V_c)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (= (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n))))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_k) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_n))) (forall ((V_y $$unsorted) (V_m $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_m)) V_y) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) V_m)))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_l)) _let_1)) (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l) _let_1))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_k) V_l) _let_1)) (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l) _let_1))))) (forall ((V_m $$unsorted) (V_n $$unsorted) (V_k $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k) V_n)) V_m) V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n))) (forall ((V_d_2 $$unsorted) (V_m_2 $$unsorted)) (= (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m_2 V_d_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (exists ((B_q $$unsorted)) (= V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_d_2) B_q))))) (forall ((V_d_2 $$unsorted) (V_m_2 $$unsorted)) (= (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m_2 V_d_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (exists ((B_q $$unsorted)) (= V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_d_2) B_q))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n))) (let ((_let_3 (= _let_1 V_n))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc _let_1)) _let_1))) (forall ((V_y_2 $$unsorted) (V_n_2 $$unsorted) (V_x_2 $$unsorted)) (= (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x_2 V_n_2) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_y_2 V_n_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_n_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_x_2) V_y_2))))) (forall ((V_m $$unsorted) (V_n $$unsorted) (V_k $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k) V_n)) V_m)) V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_a) V_b))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (= _let_1 _let_3))) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_b) _let_1))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_b)))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (= _let_1 _let_3))) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) _let_1) V_b))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a _let_1 V_b) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a V_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a _let_1 V_a) _let_1)))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a_H $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_a_H V_c)) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_b_H V_c)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b_H) V_c))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_c))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a_H $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_a_H V_c)) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_b_H V_c)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b_H) V_c))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_b) V_a) V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b) V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a_H $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_a_H V_c)) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_b_H V_c)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b_H) V_c))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a))) (=> (tptp.class_Divides_Oring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a) V_b) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_b))))) (forall ((V_a_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a_H V_b)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a) V_b) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a_H) V_b)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_k))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_m V_n))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 V_m))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_k))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_m V_n)))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_c) V_b)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_k_2))) (=> (tptp.class_Divides_Osemiring__div T_b) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n_2)) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_b V_m_2 V_n_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_m_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) V_b) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a) V_b) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a V_a (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_x $$unsorted) (V_a $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Odegree T_a V_y))) (let ((_let_2 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Polynomial_OpCons T_a) V_a))) (let ((_let_4 (tptp.hAPP _let_3 (tptp.c_Divides_Odiv__class_Omod _let_2 V_x V_y)))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_y (tptp.c_Groups_Ozero__class_Ozero _let_2))) (= (tptp.c_Divides_Odiv__class_Omod _let_2 (tptp.hAPP _let_3 V_x) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a _let_4) _let_1)) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_y) _let_1))) V_y)))))))))) (forall ((V_e_2 $$unsorted) (V_g_2 $$unsorted) (V_d_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_g_2 V_x_2))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_3 (tptp.c_Rings_Oinverse__class_Odivide T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_d_2) (=> (tptp.c_Deriv_Oderiv T_b V_g_2 V_x_2 V_e_2) (=> (not (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_3) V_f_2)) V_g_2) V_x_2 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (tptp.hAPP (tptp.hAPP _let_2 V_d_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_e_2) (tptp.hAPP V_f_2 V_x_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) _let_1) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))))))) (forall ((V_n_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Nat_Onat T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun tptp.tc_Nat_Onat T_b) T_b) _let_1) V_f_2)) V_n_2) V_x_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b V_n_2)) (tptp.hAPP (tptp.hAPP _let_2 V_D_2) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))))) (forall ((V_n_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Nat_Onat T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun tptp.tc_Nat_Onat T_b) T_b) _let_1) V_f_2)) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_x_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b V_n_2))) (tptp.hAPP (tptp.hAPP _let_2 V_D_2) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_f_2 V_x_2)) V_n_2))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_x) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_y)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_x V_y)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_a V_b)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_a) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_b)))) (forall ((V_c_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.c_COMBC T_b T_b T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_1) V_f_2)) V_c_2) V_x_2 (tptp.hAPP (tptp.hAPP _let_1 V_D_2) V_c_2)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) _let_1) _let_1)))) (forall ((V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_RealVector_Oreal__normed__field T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) _let_1) V_y) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) _let_1) V_a) _let_1)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) V_c)))))) (forall ((V_ya $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_RealVector_Oreal__normed__field T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_ya) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_ya)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_ya))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_ya $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_RealVector_Oreal__normed__field T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_ya) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_ya)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_ya))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_RealVector_Oreal__normed__field T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_x)) V_y) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_b)))))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_n) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Nat_Osemiring__char__0 T_b) (= (= (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b V_m_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b V_n_2)) (= V_m_2 V_n_2)))) (forall ((V_w $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_2 V_z) V_w)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_w))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (= V_c_2 _let_1))) (=> (tptp.class_Fields_Ofield__inverse__zero T_b) (= (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) (and (=> (not _let_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2) V_b_2)) (=> _let_2 (= V_a_2 _let_1)))))))) (forall ((V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (= V_c_2 _let_1))) (=> (tptp.class_Fields_Ofield__inverse__zero T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2) V_a_2) (and (=> (not _let_2) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2))) (=> _let_2 (= V_a_2 _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_w_2 $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_b))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Fields_Ofield T_b) (=> (not (= V_y_2 _let_3)) (=> (not (= V_z_2 _let_3)) (= (= (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_y_2) (tptp.hAPP (tptp.hAPP _let_2 V_w_2) V_z_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2) (tptp.hAPP (tptp.hAPP _let_1 V_w_2) V_y_2)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_b) (=> (not (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2) V_b_2))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_b) (=> (not (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2) V_a_2) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2)))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (= V_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_c)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b) V_c) V_a))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_c) V_b) (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b) V_c)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) V_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_3 (= V_a _let_2))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (and (=> _let_3 (= _let_1 _let_2)) (=> (not _let_3) (= _let_1 (tptp.c_Groups_Oone__class_Oone T_a))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_b) (=> (not (= V_b_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_a_2) V_b_2) (tptp.c_Groups_Oone__class_Oone T_b)) (= V_a_2 V_b_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_m)) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_m)) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_n))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_n) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n)))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_n)))))) (= _let_8 _let_4) (forall ((V_n_2 $$unsorted)) (= (= (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_b))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)))) (forall ((V_z $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n)) V_z)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n))) V_z)))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n_2)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m_2)) (and (= V_n_2 _let_1) (= V_m_2 _let_1))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n)))) (= _let_6 _let_3) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_m)) V_n)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_x_2)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_y_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m)) V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m)) V_n) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_m) V_n)))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_m))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (=> (not (= V_y (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z) (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) V_y))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_z (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_x)) V_y)) V_z))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (=> (not (= V_y (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) V_y))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_z (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_z)) V_y) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) V_z))))))) (forall ((V_w $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_y _let_4)) (=> (not (= V_z _let_4)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_3 V_w) V_z)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_y))) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))))))) (forall ((V_w $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_3 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_y _let_4)) (=> (not (= V_z _let_4)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_3 V_w) V_z)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_y))) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_z (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_z)) V_y) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) V_z))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_z (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_x)) V_y)) V_z))))))) (forall ((V_h $$unsorted) (V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_4 (tptp.hAPP _let_3 V_a))) (=> (tptp.class_RealVector_Oreal__field T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_4 V_b)) (tptp.hAPP (tptp.hAPP _let_3 V_c) V_d))) V_h) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d)) V_h))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) V_h)) V_d))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))))) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint _let_9) _let_3) (forall ((V_m $$unsorted)) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m)))) (forall ((V_E_2 $$unsorted) (V_g_2 $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_g_2 V_x_2))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_3 (tptp.c_Rings_Oinverse__class_Odivide T_b))) (=> (tptp.class_RealVector_Oreal__normed__field T_b) (=> (tptp.c_Deriv_Oderiv T_b V_f_2 V_x_2 V_D_2) (=> (tptp.c_Deriv_Oderiv T_b V_g_2 V_x_2 V_E_2) (=> (not (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (tptp.c_Deriv_Oderiv T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_3) V_f_2)) V_g_2) V_x_2 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (tptp.hAPP (tptp.hAPP _let_2 V_D_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP V_f_2 V_x_2)) V_E_2))) (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_1))))))))))) (forall ((V_h $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_4 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_5 (tptp.hAPP _let_1 V_a))) (let ((_let_6 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_7 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_RealVector_Oreal__normed__field T_a) (=> (not (= V_a _let_7)) (=> (not (= V_b _let_7)) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 _let_5) _let_2)) V_h) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_6 _let_5) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_a) V_b)) V_h))) _let_2)))))))))))))) (forall ((V_n_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b V_n_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_b (tptp.hAPP (tptp.c_COMBC T_b T_b T_b (tptp.c_Groups_Oplus__class_Oplus T_b)) (tptp.c_Groups_Oone__class_Oone T_b)) V_n_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_OpCons T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (let ((_let_3 (tptp.hAPP _let_2 V_r))) (let ((_let_4 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_5 (tptp.c_Polynomial_Odegree T_a V_y))) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q V_r) (=> (not (= V_y (tptp.c_Groups_Ozero__class_Ozero _let_4))) (=> (= V_b (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a _let_3) _let_5)) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_y) _let_5))) (tptp.c_Polynomial_Opdivmod__rel T_a (tptp.hAPP _let_2 V_x) V_y (tptp.hAPP (tptp.hAPP _let_1 V_b) V_q) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_4) _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_b) V_y))))))))))))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_x)) V_n) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_x) V_n)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (= (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_x_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_y_2)) (= V_x_2 V_y_2))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (= (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n_2)) (= V_m_2 V_n_2))) (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)))) (=> (tptp.class_Fields_Ofield T_b) (= (tptp.c_Polynomial_Opdivmod__rel T_b _let_1 V_y_2 V_q_2 V_r_2) (and (= V_q_2 _let_1) (= V_r_2 _let_1)))))) (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b)))) (=> (tptp.class_Fields_Ofield T_b) (= (tptp.c_Polynomial_Opdivmod__rel T_b V_x_2 _let_1 V_q_2 V_r_2) (and (= V_q_2 _let_1) (= V_r_2 V_x_2)))))) (forall ((V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Fields_Ofield T_a) (tptp.c_Polynomial_Opdivmod__rel T_a _let_1 V_y _let_1 _let_1)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))) (=> (tptp.class_Fields_Ofield T_a) (tptp.c_Polynomial_Opdivmod__rel T_a V_x _let_1 _let_1 V_x)))) (forall ((V_a $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Polynomial_Osmult T_a) V_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q V_r) (tptp.c_Polynomial_Opdivmod__rel T_a (tptp.hAPP _let_1 V_x) V_y (tptp.hAPP _let_1 V_q) (tptp.hAPP _let_1 V_r)))))) (forall ((V_r2 $$unsorted) (V_q2 $$unsorted) (V_r1 $$unsorted) (V_q1 $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q1 V_r1) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q2 V_r2) (and (= V_q1 V_q2) (= V_r1 V_r2)))))) (forall ((V_r2 $$unsorted) (V_q2 $$unsorted) (V_r1 $$unsorted) (V_q1 $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q1 V_r1) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q2 V_r2) (= V_r1 V_r2))))) (forall ((V_r2 $$unsorted) (V_q2 $$unsorted) (V_r1 $$unsorted) (V_q1 $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q1 V_r1) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q2 V_r2) (= V_q1 V_q2))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q V_r) (= (tptp.c_Divides_Odiv__class_Omod (tptp.tc_Polynomial_Opoly T_a) V_x V_y) V_r)))) (forall ((V_i_2 $$unsorted) (V_inc_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_b V_inc_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_i_2) V_i_2))) (forall ((V_i_2 $$unsorted) (V_n_2 $$unsorted) (V_inc_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_b V_inc_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2) V_i_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_b V_inc_2 V_n_2 (tptp.hAPP V_inc_2 V_i_2))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_z $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) V_y))) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q V_r) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_q V_z V_q_H V_r_H) (tptp.c_Polynomial_Opdivmod__rel T_a V_x (tptp.hAPP _let_2 V_z) V_q_H (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus _let_1) (tptp.hAPP _let_2 V_r_H)) V_r)))))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Osmult T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y V_q V_r) (tptp.c_Polynomial_Opdivmod__rel T_a V_x (tptp.hAPP (tptp.hAPP _let_1 V_a) V_y) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a)) V_q) V_r)))))) (= _let_4 _let_8) (forall ((V_y $$unsorted) (V_x $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_x)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_y)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_x) V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_x)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_y)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_x) V_y)))) (= _let_3 _let_6) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_x_2)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_x_2) V_y_2)))) (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_b))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (let ((_let_3 (= V_y_2 _let_2))) (=> (tptp.class_Fields_Ofield T_b) (= (tptp.c_Polynomial_Opdivmod__rel T_b V_x_2 V_y_2 V_q_2 V_r_2) (and (= V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes _let_1) V_q_2) V_y_2)) V_r_2)) (=> _let_3 (= V_q_2 _let_2)) (=> (not _let_3) (or (= V_r_2 _let_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_b V_r_2) (tptp.c_Polynomial_Odegree T_b V_y_2))))))))))) (forall ((V_n_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (=> (tptp.class_Rings_Oring__1 T_b) (= (tptp.c_Transcendental_Odiffs T_b V_c_2 V_n_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b _let_1)) (tptp.hAPP V_c_2 _let_1)))))) (forall ((V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Groups_Omonoid__mult T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) V_x) (tptp.hAPP _let_1 V_n)))))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.hAPP tptp.c_Nat_OSuc V_n))) (forall ((V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n))) (forall ((V_n_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Ozero__class_Ozero T_b) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b V_m_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_m) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_m) (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__comm__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a _let_1) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b _let_1) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a _let_1) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) _let_1)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (=> (tptp.c_Orderings_Oord__class_Oless T_b V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b)) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_b_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 V_a_2)))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a)) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b _let_1) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a) _let_1)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b _let_1) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) _let_1)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (=> (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Ozero__class_Ozero T_b) V_c_2) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_b_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 V_b_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP _let_2 V_a_2) (tptp.hAPP _let_2 V_b_2)) (or (and (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_c_2) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 V_b_2)) (and (tptp.c_Orderings_Oord__class_Oless T_b V_c_2 _let_1) (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 V_a_2)))))))) (forall ((V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) (or (and (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_c_2) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 V_b_2)) (and (tptp.c_Orderings_Oord__class_Oless T_b V_c_2 _let_1) (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 V_a_2)))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__ring T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_c) (tptp.c_Orderings_Oord__class_Oless T_a V_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_c)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a _let_1) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b _let_1) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b) _let_1)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2) _let_1) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 _let_1))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2)) (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_a_2))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2) _let_1) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 _let_1))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 V_b_2) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a $$unsorted) (V_N $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_N) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) V_a) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_n) (tptp.hAPP _let_1 V_N))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_b_2))) (=> (tptp.class_Rings_Olinordered__semidom T_b) (=> (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Oone__class_Oone T_b) V_b_2) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x_2 V_y_2)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c V_d) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c))) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_c_2))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_b_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 V_b_2))))) (forall ((V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 V_b_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2) (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_d_2)) (= (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 V_b_2) (tptp.c_Orderings_Oord__class_Oless T_b V_c_2 V_d_2)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP _let_1 V_b_2) (tptp.hAPP _let_1 V_a_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 V_b_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP _let_1 V_a_2) V_b_2) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP _let_1 V_b_2) V_a_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 (tptp.hAPP _let_1 V_b_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not (= V_n _let_1)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (not (= V_n_2 _let_1)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n_2)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (not (= V_x V_y)) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)) (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_n))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (not (= V_m_2 V_n_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (not (= V_x V_y)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x V_y)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_y V_x)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_n))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_m) (not (= V_m V_n)))) (forall ((V_t $$unsorted) (V_s $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_s V_t) (not (= V_s V_t)))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_n_2) V_m_2)))) (=> (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) _let_1) (=> (=> (= V_m_2 V_n_2) _let_1) (=> (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2) _let_1) _let_1))))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_l))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_l) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_n) (tptp.hAPP _let_1 V_m)))))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_j $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_j V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j) V_n) V_k))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_j) V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_k))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_k V_l) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_l) (tptp.hAPP (tptp.hAPP _let_1 V_k) V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n))))) (forall ((V_l $$unsorted) (V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_k V_l) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_l)))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_j)))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j) V_m)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2)))) (forall ((V_i $$unsorted) (V_j $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j) V_i) V_i))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_j) V_i))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc V_n)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)) (= V_m V_n)))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_j V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_i) V_k)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (=> (not (= _let_1 V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_m)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.hAPP tptp.c_Nat_OSuc V_m)) (= V_m V_n)))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (= V_n_2 V_m_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (= V_m_2 V_n_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 (tptp.hAPP tptp.c_Nat_OSuc V_m_2)))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))))))) (forall ((V_a $$unsorted) (V_N $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_N) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_N) (tptp.hAPP _let_1 V_n)))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_x)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (not (= V_x_2 V_y_2)) (or (tptp.c_Orderings_Oord__class_Oless T_b V_x_2 V_y_2) (tptp.c_Orderings_Oord__class_Oless T_b V_y_2 V_x_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (not (tptp.c_Orderings_Oord__class_Oless T_b V_x_2 V_y_2)) (or (tptp.c_Orderings_Oord__class_Oless T_b V_y_2 V_x_2) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (= V_x V_y) (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x)))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (not (tptp.c_Orderings_Oord__class_Oless T_b V_y_2 V_x_2)) (= (not (tptp.c_Orderings_Oord__class_Oless T_b V_x_2 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (= V_x V_y)) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)) (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (not (= V_x V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (not (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (not (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (not (= V_x V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (not (= V_y V_x))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (not (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a) (not (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (=> (= V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_c) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (= V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c V_b) (tptp.c_Orderings_Oord__class_Oless T_a V_c V_a))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (= V_b V_c) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_c))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a) (=> (= V_b V_c) (tptp.c_Orderings_Oord__class_Oless T_a V_c V_a))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y V_z) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_z))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_z V_y) (tptp.c_Orderings_Oord__class_Oless T_a V_z V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (not (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)) (=> (not (= V_x V_y)) (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_P_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Owellorder T_b) (=> (tptp.c_Orderings_Oord__class_Oless T_b V_k_2 (tptp.c_Orderings_Oord__class_OLeast T_b V_P_2)) (not (tptp.hBOOL (tptp.hAPP V_P_2 V_k_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) V_m))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2) V_a_2) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Ozero__class_Ozero T_b) V_a_2)))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2) _let_1) (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_a_2))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 _let_1))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_m) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_n) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_m) V_n))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.c_Orderings_Oord__class_Oless T_a V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_a_2) V_b_2)) (or (and (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_a_2) (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_b_2)) (and (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 _let_1) (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 _let_1))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_a_2) V_b_2) _let_1) (or (and (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_a_2) (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 _let_1)) (and (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 _let_1) (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_b_2))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_x) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_y) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_x) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y _let_1) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y) _let_1)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x _let_1) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_y) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y) _let_1)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x _let_1) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y _let_1) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_a_2))) (=> (tptp.class_Rings_Olinordered__semidom T_b) (=> (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Oone__class_Oone T_b) V_a_2) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2)))))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a V_m) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b _let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_b) V_a_2)) (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_a_2))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_b) V_a_2) _let_1) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a)) (=> (not (= V_a _let_1)) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a _let_1) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a) _let_1))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_a))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_a))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a) _let_1) (=> (not (= V_a _let_1)) (tptp.c_Orderings_Oord__class_Oless T_a V_a _let_1)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Oinverse T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (or (= V_m_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (exists ((B_j $$unsorted)) (and (= V_m_2 (tptp.hAPP tptp.c_Nat_OSuc B_j)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_j V_n_2)))))) (forall ((V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1)) (= V_n_2 _let_1)))) (forall ((V_n_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2) (exists ((B_m $$unsorted)) (= V_n_2 (tptp.hAPP tptp.c_Nat_OSuc B_m))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_m) V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_m) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_n) V_m)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (exists ((B_k $$unsorted)) (= V_n_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) B_k)))))) (forall ((V_m $$unsorted) (V_i $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_i)))) (forall ((V_m $$unsorted) (V_i $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_m)))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_i) (tptp.hAPP _let_1 V_j)))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k)))))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_k_2)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n) V_m))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n_2) V_m_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n) (tptp.hAPP tptp.c_Nat_OSuc V_m))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_m))) (forall ((V_k_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_k_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i_2) V_k_2) V_j_2))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n))) (and (=> _let_2 (= _let_1 V_m)) (=> (not _let_2) (= _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n) V_n))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n) V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_i))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_i) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n))))) (forall ((V_n_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_x_2) V_n_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_x_2) (= V_n_2 _let_1))))) (forall ((V_p $$unsorted) (V_q $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Odegree T_a V_p))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a V_q) _let_1) (= (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) _let_1))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Odegree T_a V_q))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a V_p) _let_1) (= (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) _let_1))))) (forall ((V_q $$unsorted) (V_n $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a V_p) V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a V_q) V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) V_n))))) (forall ((V_n_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Rings_Oring__1 T_b) (= (tptp.c_Transcendental_Odiffs T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b tptp.tc_Nat_Onat) _let_1) V_c_2) V_n_2) (tptp.hAPP _let_1 (tptp.c_Transcendental_Odiffs T_b V_c_2 V_n_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a V_x) (tptp.c_Polynomial_Odegree T_a V_y)) (= (tptp.c_Divides_Odiv__class_Omod (tptp.tc_Polynomial_Opoly T_a) V_x V_y) V_x)))) (forall ((V_q $$unsorted) (V_n $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a V_p) V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a V_q) V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)) V_n))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_1 V_x) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_y)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_x_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_y_2))) (or (not (= V_x_2 _let_1)) (not (= V_y_2 _let_1)))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) _let_1) _let_1))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Rings_Oordered__ring T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_c_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_b_2) V_a_2)) V_e_2)) V_d_2))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Rings_Oordered__ring T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2)) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_a_2) V_b_2)) V_e_2)) V_c_2) V_d_2)))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b V_c_2 _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_c_2))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) (and (=> _let_4 (tptp.c_Orderings_Oord__class_Oless T_b _let_3 V_b_2)) (=> (not _let_4) (and (=> _let_2 (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 _let_3)) (=> (not _let_2) (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 _let_1)))))))))))) (forall ((V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b V_c_2 _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_c_2))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2) V_a_2) (and (=> _let_4 (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 _let_3)) (=> (not _let_4) (and (=> _let_2 (tptp.c_Orderings_Oord__class_Oless T_b _let_3 V_b_2)) (=> (not _let_2) (tptp.c_Orderings_Oord__class_Oless T_b _let_1 V_a_2)))))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Ozero__class_Ozero T_b) V_c_2) (= (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2) V_b_2))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Ozero__class_Ozero T_b) V_c_2) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2) V_a_2) (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2)))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_y) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y) V_z))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_y) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y) V_x) (tptp.c_Orderings_Oord__class_Oless T_a V_z (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.c_Orderings_Oord__class_Oless T_b V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b)) (= (tptp.c_Orderings_Oord__class_Oless T_b V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) (tptp.c_Orderings_Oord__class_Oless T_b V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.c_Orderings_Oord__class_Oless T_b V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b)) (= (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2) V_a_2) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2) V_b_2))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_c))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_2 V_c) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_c))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c _let_2) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_1)) V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (tptp.c_Orderings_Oord__class_Oless T_a V_a (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_1)))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.c_Orderings_Oord__class_Oless T_b _let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_b) V_x_2)) (and (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Ozero__class_Ozero T_b) V_x_2) (tptp.c_Orderings_Oord__class_Oless T_b V_x_2 _let_1)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a _let_1) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Oinverse T_a) V_a))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP tptp.c_Nat_OSuc V_n))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) V_m)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)))))) (forall ((V_i $$unsorted) (V_n $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.hAPP tptp.c_Nat_OSuc V_i)) V_n))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.hAPP tptp.c_Nat_OSuc _let_1))) V_n)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) (tptp.hAPP _let_2 V_n))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_a_2) V_b_2))) (not (or (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_a_2 V_b_2) (not (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (exists ((B_d $$unsorted)) (and (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_b_2) B_d)) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_d))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_a_2) V_b_2))) (and (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_a_2 V_b_2) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((B_d $$unsorted)) (=> (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_b_2) B_d)) (tptp.hBOOL (tptp.hAPP V_P_2 B_d))))))) (forall ((V_n $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ozero T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a V_p) V_n) (= (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_a V_p) V_n) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Omod _let_1 V_x V_y))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_y _let_3)) (or (= _let_2 _let_3) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a _let_2) (tptp.c_Polynomial_Odegree T_a V_y))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1) _let_1)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a _let_1) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP tptp.c_Nat_OSuc V_n)) _let_1)))))) (forall ((V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (= V_x (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) V_n)))))) (forall ((V_n $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n))) (forall ((V_n $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (= V_n (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_m_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2)) V_m_2)) (= V_n_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_m_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n_2) V_m_2)) V_m_2)) (= V_n_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_k_2 $$unsorted) (V_n_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_n_2 V_k_2))) (and (=> _let_1 (tptp.hBOOL (tptp.hAPP V_P_2 V_n_2))) (=> (not _let_1) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_j V_k_2) (=> (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2) B_i)) B_j)) (tptp.hBOOL (tptp.hAPP V_P_2 B_j)))))))))) (forall ((V_q $$unsorted) (V_b $$unsorted) (V_r $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_b))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_c) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_r V_b) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_q V_c))) V_r) (tptp.hAPP _let_1 V_c)))))) (forall ((V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_k) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k) V_m)) V_k) _let_1)))) (forall ((V_n_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_x_2) V_n_2)) (or (= V_n_2 _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_x_2))))) (forall ((V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x (tptp.hAPP tptp.c_Nat_OSuc _let_2)) (or (= V_x _let_1) (= V_x _let_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (=> (tptp.class_Orderings_Oorder T_b) (=> (tptp.c_Orderings_Oorder_Ostrict__mono tptp.tc_Nat_Onat T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_2) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1))) V_f_2) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)))) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)))))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP tptp.c_Nat_OSuc V_n))) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_i V_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))))) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_4 _let_3) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP _let_1 V_i) (tptp.hAPP _let_1 V_j)))))) (forall ((V_k $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_k) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_x_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_y_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x_2 V_y_2))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 V_z_2) (= V_w_2 V_z_2)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k_2 V_l_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_k_2) V_l_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_b) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) V_b))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_b (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_m) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_m V_n) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_n) V_m)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m))))) (forall ((V_z_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z_2)) V_z_2) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_z_2 _let_1))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_m_2) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_m_2) V_n_2) _let_1) (and (= V_m_2 _let_1) (= V_n_2 _let_1)))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 V_z_2) (exists ((B_n $$unsorted)) (= V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_w_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP tptp.c_Nat_OSuc B_n))))))) (forall ((V_n_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_k)))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP _let_1 V_i) (tptp.hAPP _let_1 V_j)))))) (forall ((V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP tptp.c_Nat_OSuc V_n))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (=> (tptp.class_Orderings_Oorder T_b) (=> (tptp.c_Orderings_Oorder_Ostrict__mono tptp.tc_Nat_Onat T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_2) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1))) V_f_2) (tptp.c_Orderings_Oorder_Omono tptp.tc_Nat_Onat T_b _let_1 V_f_2)))))))) (forall ((V_y $$unsorted) (V_z $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x V_z) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) V_z)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_x V_y) (= V_x V_y) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_y V_x))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x V_y) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) V_z)))) (forall ((V_pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.c_Polynomial_Opos__poly T_b V_pa_2) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Ozero__class_Ozero T_b) (tptp.hAPP (tptp.c_Polynomial_Ocoeff T_b V_pa_2) (tptp.c_Polynomial_Odegree T_b V_pa_2)))))) (forall ((V_pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.c_Polynomial_Opos__poly T_b (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons T_b) V_a_2) V_pa_2)) (or (tptp.c_Polynomial_Opos__poly T_b V_pa_2) (and (= V_pa_2 (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_b))) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Groups_Ozero__class_Ozero T_b) V_a_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.c_Orderings_Oord__class_Oless _let_1 V_x_2 V_y_2) (tptp.c_Polynomial_Opos__poly T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_y_2) V_x_2)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (not (tptp.c_Polynomial_Opos__poly T_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly T_a)))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.c_Polynomial_Opos__poly T_a V_p) (=> (tptp.c_Polynomial_Opos__poly T_a V_q) (tptp.c_Polynomial_Opos__poly T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.c_Polynomial_Opos__poly T_a V_p) (=> (tptp.c_Polynomial_Opos__poly T_a V_q) (tptp.c_Polynomial_Opos__poly T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus (tptp.tc_Polynomial_Opoly T_a)) V_p) V_q)))))) (forall ((V_p $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (or (= V_p (tptp.c_Groups_Ozero__class_Ozero _let_1)) (tptp.c_Polynomial_Opos__poly T_a V_p) (tptp.c_Polynomial_Opos__poly T_a (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_p)))))) (forall ((V_k $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_k))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_k) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.hAPP _let_2 V_m) V_k) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.hAPP tptp.c_Nat_OSuc _let_1))))))))) (forall ((V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_x (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (exists ((B_n $$unsorted)) (= V_x (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint (tptp.hAPP tptp.c_Nat_OSuc B_n))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_b) (=> (not (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_a_2) V_b_2)) (= (= (tptp.c_Divides_Odiv__class_Odiv T_b V_b_2 V_a_2) V_c_2) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2) V_a_2))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd T_b))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Divides_Osemiring__div T_b) (=> (not (= V_a_2 _let_3)) (=> (not (= V_c_2 _let_3)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c_2) V_d_2)) (= (= (tptp.c_Divides_Odiv__class_Odiv T_b V_b_2 V_a_2) (tptp.c_Divides_Odiv__class_Odiv T_b V_d_2 V_c_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_d_2)))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_4 (= V_c _let_3))) (=> (tptp.class_Divides_Osemiring__div T_a) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b)))))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) V_b) V_a)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a) V_b) V_a)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) V_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.c_Divides_Odiv__class_Odiv T_a V_b V_a)) V_b)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_c) V_b)) (= (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_a V_b V_c)) (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP _let_1 V_b) V_c)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) V_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Divides_Odiv__class_Odiv T_a V_b V_a)) V_a) V_b)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) V_b)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_a V_b V_a)) V_c) (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c) V_a)))))) (forall ((V_w $$unsorted) (V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_z) V_w)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_a V_x V_y)) (tptp.c_Divides_Odiv__class_Odiv T_a V_w V_z)) (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_z))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) V_z) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_a V_x V_z)) (tptp.c_Divides_Odiv__class_Odiv T_a V_y V_z))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b) V_b) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_y) V_x)) (= (tptp.c_Divides_Odiv__class_Odiv T_a V_x (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_a V_x V_y))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_y) V_x)) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP _let_1 V_x) V_y) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_a V_x V_y))))))) (forall ((V_n $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_y) V_x)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_a V_x V_y)) V_n) (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_n))))))) (forall ((V_k $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_k)))) (forall ((V_k $$unsorted) (V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_b))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_k)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a))) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.hAPP _let_1 V_b) V_c) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_b V_c))) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c)) V_c))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_b))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_m V_n)) V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_n) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_m V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m V_n)))) (forall ((V_a $$unsorted)) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Odiv T_a V_a (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Odiv T_a V_a _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Odiv T_a _let_1 V_a) _let_1)))) (forall ((V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint _let_1 V_b) _let_1))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a (tptp.hAPP _let_1 V_b)) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) V_b)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_b)))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_a_2))) (=> (tptp.class_Divides_Osemiring__div T_b) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_b_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_b V_b_2 V_a_2)) (tptp.c_Divides_Odiv__class_Odiv T_b V_c_2 V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2))))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_n) V_m)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_n) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_m V_n)) V_m))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_c)) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_b V_c))) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_c)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c)) V_c))))) (forall ((V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (and (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a _let_1) _let_1) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a _let_1) V_a)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) V_b) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b)) (tptp.c_Groups_Oone__class_Oone T_a))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_b) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b)) (tptp.c_Groups_Oone__class_Oone T_a))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c) V_b)) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_c) (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Divides_Odiv__class_Odiv T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_c)) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_c) (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b))))))) (forall ((V_k $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_x) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_x V_k) V_x)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_a _let_1) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_b) _let_1))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_b_2 _let_1) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a_2 V_b_2) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_a_2))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_b_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a_2 V_b_2) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_a_2 _let_1))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_c) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) V_c)) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_b) V_c)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b))) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b))) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b))) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_a))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b))) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Divides_Odiv__class_Odiv T_a V_a V_b)) V_b)) V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_c) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_b) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_b)))) (let ((_let_3 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a (tptp.hAPP _let_1 V_b)))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_5 (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) _let_4))) (=> (not (= V_b _let_4)) (and (=> _let_5 (= _let_3 _let_2)) (=> (not _let_5) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) _let_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))))))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_a V_b)))) (let ((_let_3 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) V_b))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_5 (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) _let_4))) (=> (not (= V_b _let_4)) (and (=> _let_5 (= _let_3 _let_2)) (=> (not _let_5) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) _let_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone _let_1))) (let ((_let_3 (tptp.c_Groups_Osgn__class_Osgn _let_1 V_x))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero _let_1))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless _let_1 _let_4 V_x))) (let ((_let_6 (= V_x _let_4))) (=> (tptp.class_Rings_Olinordered__idom T_a) (and (=> _let_6 (= _let_3 _let_4)) (=> (not _let_6) (and (=> _let_5 (= _let_3 _let_2)) (=> (not _let_5) (= _let_3 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) _let_2)))))))))))))) (forall ((V_n_2 $$unsorted) (V_P_2 $$unsorted) (V_k_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_k_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint V_n_2 V_k_2)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_n_2 V_k_2))) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (and (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) B_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint B_j V_k_2) (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_i) B_j))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_x))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_n)) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint V_m))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Polynomial_Odegree T_a V_x) (tptp.c_Polynomial_Odegree T_a V_y)) (= (tptp.c_Divides_Odiv__class_Odiv _let_1 V_x V_y) (tptp.c_Groups_Ozero__class_Ozero _let_1)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Polynomial_Opoly T_a))) (=> (tptp.class_Fields_Ofield T_a) (tptp.c_Polynomial_Opdivmod__rel T_a V_x V_y (tptp.c_Divides_Odiv__class_Odiv _let_1 V_x V_y) (tptp.c_Divides_Odiv__class_Omod _let_1 V_x V_y))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_b))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_a (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat V_a V_b) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_a V_b))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_a))) (= (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_b) V_c) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat V_b V_c))) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_b V_c)) V_c))))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Ocancel__ab__semigroup__add T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Oring__1__no__zero__divisors T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Oring__no__zero__divisors T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Ocancel__semigroup__add T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Oab__semigroup__mult T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Ocomm__monoid__mult T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Oab__semigroup__add T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Ono__zero__divisors T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Ocomm__monoid__add T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Ocomm__semiring__1 T))) _let_2 (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Ocomm__semiring T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Oab__group__add T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Ozero__neq__one T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Omonoid__mult T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Ocomm__ring__1 T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Omonoid__add T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Osemiring__1 T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Osemiring__0 T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Ogroup__add T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Omult__zero T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Ocomm__ring T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Osemiring T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Ouminus T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Oring__1 T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Ominus T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Power_Opower T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Ozero T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Oring T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Oone T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Rings_Odvd T))) (forall ((T $$unsorted)) (=> (tptp.class_Rings_Oidom T) (tptp.class_Groups_Ocancel__comm__monoid__add T))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Ocancel__comm__monoid__add T_1) (tptp.class_Groups_Ocancel__comm__monoid__add (tptp.tc_Polynomial_Opoly T_1)))) (tptp.class_Groups_Ocancel__comm__monoid__add tptp.tc_Nat_Onat) (tptp.class_Groups_Ocancel__comm__monoid__add tptp.tc_Int_Oint) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_1) (tptp.class_Lattices_Oboolean__algebra (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_1) (tptp.class_Orderings_Opreorder (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Oorder T_1) (tptp.class_Orderings_Oorder (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Oord T_1) (tptp.class_Orderings_Oord (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Groups_Ouminus T_1) (tptp.class_Groups_Ouminus (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Groups_Ominus T_1) (tptp.class_Groups_Ominus (tptp.tc_fun T_2 T_1)))) (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__comm__semiring__strict tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__semiring__strict tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__comm__monoid__add tptp.tc_Int_Oint) (tptp.class_Groups_Olinordered__ab__group__add tptp.tc_Int_Oint) (tptp.class_Groups_Ocancel__ab__semigroup__add tptp.tc_Int_Oint) (tptp.class_Rings_Oring__1__no__zero__divisors tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__ring__strict tptp.tc_Int_Oint) (tptp.class_Rings_Oring__no__zero__divisors tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__ab__group__add tptp.tc_Int_Oint) (tptp.class_Groups_Ocancel__semigroup__add tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__semidom tptp.tc_Int_Oint) (tptp.class_Groups_Oab__semigroup__mult tptp.tc_Int_Oint) (tptp.class_Groups_Ocomm__monoid__mult tptp.tc_Int_Oint) (tptp.class_Groups_Oab__semigroup__add tptp.tc_Int_Oint) (tptp.class_Rings_Ono__zero__divisors tptp.tc_Int_Oint) (tptp.class_Groups_Ocomm__monoid__add tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__ring tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__idom tptp.tc_Int_Oint) (tptp.class_Rings_Ocomm__semiring__1 tptp.tc_Int_Oint) (tptp.class_Rings_Ocomm__semiring__0 tptp.tc_Int_Oint) (tptp.class_Divides_Osemiring__div tptp.tc_Int_Oint) (tptp.class_Rings_Ocomm__semiring tptp.tc_Int_Oint) (tptp.class_Nat_Osemiring__char__0 tptp.tc_Int_Oint) (tptp.class_Groups_Oab__group__add tptp.tc_Int_Oint) (tptp.class_Rings_Ozero__neq__one tptp.tc_Int_Oint) (tptp.class_Rings_Oordered__ring tptp.tc_Int_Oint) (tptp.class_Orderings_Opreorder tptp.tc_Int_Oint) (tptp.class_Orderings_Olinorder tptp.tc_Int_Oint) (tptp.class_Groups_Omonoid__mult tptp.tc_Int_Oint) (tptp.class_Rings_Ocomm__ring__1 tptp.tc_Int_Oint) (tptp.class_Groups_Omonoid__add tptp.tc_Int_Oint) (tptp.class_Rings_Osemiring__1 tptp.tc_Int_Oint) (tptp.class_Rings_Osemiring__0 tptp.tc_Int_Oint) (tptp.class_Groups_Ogroup__add tptp.tc_Int_Oint) (tptp.class_Divides_Oring__div tptp.tc_Int_Oint) (tptp.class_Rings_Omult__zero tptp.tc_Int_Oint) (tptp.class_Rings_Ocomm__ring tptp.tc_Int_Oint) (tptp.class_Orderings_Oorder tptp.tc_Int_Oint) (tptp.class_Int_Oring__char__0 tptp.tc_Int_Oint) (tptp.class_Rings_Osemiring tptp.tc_Int_Oint) (tptp.class_Orderings_Oord tptp.tc_Int_Oint) (tptp.class_Groups_Ouminus tptp.tc_Int_Oint) (tptp.class_Rings_Oring__1 tptp.tc_Int_Oint) (tptp.class_Groups_Ominus tptp.tc_Int_Oint) (tptp.class_Power_Opower tptp.tc_Int_Oint) (tptp.class_Groups_Ozero tptp.tc_Int_Oint) (tptp.class_Rings_Oring tptp.tc_Int_Oint) (tptp.class_Rings_Oidom tptp.tc_Int_Oint) (tptp.class_Groups_Oone tptp.tc_Int_Oint) (tptp.class_Rings_Odvd tptp.tc_Int_Oint) (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le tptp.tc_Nat_Onat) (tptp.class_Rings_Olinordered__comm__semiring__strict tptp.tc_Nat_Onat) (tptp.class_Rings_Olinordered__semiring__strict tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__comm__monoid__add tptp.tc_Nat_Onat) (tptp.class_Groups_Ocancel__ab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Groups_Ocancel__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Rings_Olinordered__semidom tptp.tc_Nat_Onat) (tptp.class_Groups_Oab__semigroup__mult tptp.tc_Nat_Onat) (tptp.class_Groups_Ocomm__monoid__mult tptp.tc_Nat_Onat) (tptp.class_Groups_Oab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Rings_Ono__zero__divisors tptp.tc_Nat_Onat) (tptp.class_Groups_Ocomm__monoid__add tptp.tc_Nat_Onat) (tptp.class_Rings_Ocomm__semiring__1 tptp.tc_Nat_Onat) (tptp.class_Rings_Ocomm__semiring__0 tptp.tc_Nat_Onat) (tptp.class_Divides_Osemiring__div tptp.tc_Nat_Onat) (tptp.class_Rings_Ocomm__semiring tptp.tc_Nat_Onat) (tptp.class_Orderings_Owellorder tptp.tc_Nat_Onat) (tptp.class_Nat_Osemiring__char__0 tptp.tc_Nat_Onat) (tptp.class_Rings_Ozero__neq__one tptp.tc_Nat_Onat) (tptp.class_Orderings_Opreorder tptp.tc_Nat_Onat) (tptp.class_Orderings_Olinorder tptp.tc_Nat_Onat) (tptp.class_Groups_Omonoid__mult tptp.tc_Nat_Onat) (tptp.class_Groups_Omonoid__add tptp.tc_Nat_Onat) (tptp.class_Rings_Osemiring__1 tptp.tc_Nat_Onat) (tptp.class_Rings_Osemiring__0 tptp.tc_Nat_Onat) (tptp.class_Rings_Omult__zero tptp.tc_Nat_Onat) (tptp.class_Orderings_Oorder tptp.tc_Nat_Onat) (tptp.class_Rings_Osemiring tptp.tc_Nat_Onat) (tptp.class_Orderings_Oord tptp.tc_Nat_Onat) (tptp.class_Groups_Ominus tptp.tc_Nat_Onat) (tptp.class_Power_Opower tptp.tc_Nat_Onat) (tptp.class_Groups_Ozero tptp.tc_Nat_Onat) (tptp.class_Groups_Oone tptp.tc_Nat_Onat) (tptp.class_Rings_Odvd tptp.tc_Nat_Onat) (tptp.class_Lattices_Oboolean__algebra tptp.tc_HOL_Obool) (tptp.class_Orderings_Opreorder tptp.tc_HOL_Obool) (tptp.class_Orderings_Oorder tptp.tc_HOL_Obool) (tptp.class_Orderings_Oord tptp.tc_HOL_Obool) (tptp.class_Groups_Ouminus tptp.tc_HOL_Obool) (tptp.class_Groups_Ominus tptp.tc_HOL_Obool) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Oidom T_1) (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Rings_Olinordered__comm__semiring__strict (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Rings_Olinordered__semiring__strict (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Groups_Oordered__comm__monoid__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Groups_Olinordered__ab__group__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Ocancel__comm__monoid__add T_1) (tptp.class_Groups_Ocancel__ab__semigroup__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Oidom T_1) (tptp.class_Rings_Oring__1__no__zero__divisors (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Rings_Olinordered__ring__strict (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Oidom T_1) (tptp.class_Rings_Oring__no__zero__divisors (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Groups_Oordered__ab__group__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Ocancel__comm__monoid__add T_1) (tptp.class_Groups_Ocancel__semigroup__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Rings_Olinordered__semidom (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_1) (tptp.class_Groups_Oab__semigroup__mult (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_1) (tptp.class_Groups_Ocomm__monoid__mult (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_1) (tptp.class_Groups_Oab__semigroup__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Oidom T_1) (tptp.class_Rings_Ono__zero__divisors (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_1) (tptp.class_Groups_Ocomm__monoid__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Rings_Olinordered__ring (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Rings_Olinordered__idom (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_1) (tptp.class_Rings_Ocomm__semiring__1 (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_1) (tptp.class_Rings_Ocomm__semiring__0 (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Fields_Ofield T_1) (tptp.class_Divides_Osemiring__div (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_1) (tptp.class_Rings_Ocomm__semiring (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Nat_Osemiring__char__0 (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_1) (tptp.class_Groups_Oab__group__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_1) (tptp.class_Rings_Ozero__neq__one (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Rings_Oordered__ring (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Orderings_Opreorder (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Orderings_Olinorder (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_1) (tptp.class_Groups_Omonoid__mult (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__ring__1 T_1) (tptp.class_Rings_Ocomm__ring__1 (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_1) (tptp.class_Groups_Omonoid__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_1) (tptp.class_Rings_Osemiring__1 (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_1) (tptp.class_Rings_Osemiring__0 (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_1) (tptp.class_Groups_Ogroup__add (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Fields_Ofield T_1) (tptp.class_Divides_Oring__div (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_1) (tptp.class_Rings_Omult__zero (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__ring T_1) (tptp.class_Rings_Ocomm__ring (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Orderings_Oorder (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Int_Oring__char__0 (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__0 T_1) (tptp.class_Rings_Osemiring (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_1) (tptp.class_Orderings_Oord (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_1) (tptp.class_Groups_Ouminus (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__ring__1 T_1) (tptp.class_Rings_Oring__1 (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_1) (tptp.class_Groups_Ominus (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_1) (tptp.class_Power_Opower (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Groups_Ozero T_1) (tptp.class_Groups_Ozero (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__ring T_1) (tptp.class_Rings_Oring (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Oidom T_1) (tptp.class_Rings_Oidom (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_1) (tptp.class_Groups_Oone (tptp.tc_Polynomial_Opoly T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_1) (tptp.class_Rings_Odvd (tptp.tc_Polynomial_Opoly T_1)))) (forall ((V_P $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_COMBI T_a) V_P) V_P)) (forall ((V_Q $$unsorted) (V_P $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_COMBK T_a T_b) V_P) V_Q) V_P)) (forall ((V_R_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c T_b T_d) V_P_2) V_Q_2) V_R_2) (tptp.hAPP V_P_2 (tptp.hAPP V_Q_2 V_R_2)))) (forall ((V_R_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_d T_b V_P_2) V_Q_2) V_R_2) (tptp.hAPP (tptp.hAPP V_P_2 V_R_2) V_Q_2))) (forall ((V_R_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_d T_b) V_P_2) V_Q_2) V_R_2) (tptp.hAPP (tptp.hAPP V_P_2 V_R_2) (tptp.hAPP V_Q_2 V_R_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.c_fequal V_x_2) V_y_2))) (= V_x_2 V_y_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (or (not (= V_x_2 V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_fequal V_x_2) V_y_2)))) (forall ((V_P_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP tptp.c_fNot V_P_2))) (not (tptp.hBOOL V_P_2)))) (forall ((V_P_2 $$unsorted)) (or (not (not (tptp.hBOOL V_P_2))) (tptp.hBOOL (tptp.hAPP tptp.c_fNot V_P_2)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (not (tptp.hBOOL V_P_2)) (not (tptp.hBOOL V_Q_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fconj V_P_2) V_Q_2)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fconj V_P_2) V_Q_2))) (tptp.hBOOL V_P_2))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fconj V_P_2) V_Q_2))) (tptp.hBOOL V_Q_2))) (not (exists ((B_k $$unsorted) (B_a $$unsorted)) (and (not (= B_a (tptp.c_Groups_Ozero__class_Ozero tptp.t_a))) (not (= B_k (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (exists ((B_q $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (and (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Opsize tptp.t_a B_q)) B_k)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Opsize tptp.t_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly tptp.t_a)))) (forall ((B_z $$unsorted)) (let ((_let_1 (tptp.c_Polynomial_Opoly tptp.t_a (tptp.c_Groups_Ozero__class_Ozero (tptp.tc_Polynomial_Opoly tptp.t_a))))) (= (tptp.hAPP _let_1 B_z) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.t_a) (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.t_a))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.t_a) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.t_a) B_z) B_k)) (tptp.hAPP (tptp.c_Polynomial_Opoly tptp.t_a (tptp.hAPP (tptp.hAPP (tptp.c_Polynomial_OpCons tptp.t_a) B_a) B_q)) B_z)))))))))))) _let_1 true))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 1.69/1.87  )
% 1.69/1.87  % SZS output end Proof for SWW246+1
% 1.69/1.87  % cvc5---1.0.5 exiting
% 1.69/1.87  % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------