%------------------------------------------------------------------------------
% File     : cvc5---1.0.5
% Problem  : SWW197+1 : TPTP v8.1.2. Released v5.2.0.
% Transfm  : none
% Format   : tptp
% Command  : do_cvc5 %s %d

% Computer : n016.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 00:19:55 EDT 2023

% Result   : Theorem 1.63s 1.84s
% Output   : Proof 1.63s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.22/0.24  % Problem    : SWW197+1 : TPTP v8.1.2. Released v5.2.0.
% 0.22/0.25  % Command    : do_cvc5 %s %d
% 0.25/0.46  % Computer : n016.cluster.edu
% 0.25/0.46  % Model    : x86_64 x86_64
% 0.25/0.46  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.25/0.46  % Memory   : 8042.1875MB
% 0.25/0.46  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.25/0.46  % CPULimit   : 300
% 0.25/0.46  % WCLimit    : 300
% 0.25/0.46  % DateTime   : Sun Aug 27 23:24:59 EDT 2023
% 0.25/0.46  % CPUTime    : 
% 0.31/0.73  %----Proving TF0_NAR, FOF, or CNF
% 1.63/1.84  ------- convert to smt2 : /export/starexec/sandbox/tmp/tmp.oFoR70JTGy/cvc5---1.0.5_27707.p...
% 1.63/1.84  ------- get file name : TPTP file name is SWW197+1
% 1.63/1.84  ------- cvc5-fof : /export/starexec/sandbox/solver/bin/cvc5---1.0.5_27707.smt2...
% 1.63/1.84  --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10...
% 1.63/1.84  % SZS status Theorem for SWW197+1
% 1.63/1.84  % SZS output start Proof for SWW197+1
% 1.63/1.84  (
% 1.63/1.84  (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))) (let ((_let_2 (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ocsqrt tptp.v_z____))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower tptp.tc_Complex_Ocomplex))) (let ((_let_4 (tptp.hAPP _let_3 _let_2))) (let ((_let_5 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Complex_Ocomplex) tptp.v_b))) (let ((_let_6 (tptp.c_Groups_Oone__class_Oone tptp.tc_Complex_Ocomplex))) (let ((_let_7 (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex _let_6 (tptp.hAPP _let_5 (tptp.hAPP _let_4 tptp.v_na____)))))) (let ((_let_8 (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_7 _let_1))) (let ((_let_9 (not _let_8))) (let ((_let_10 (tptp.class_Rings_Ocomm__semiring__1 tptp.tc_Complex_Ocomplex))) (let ((_let_11 (tptp.c_Int_OBit1 tptp.c_Int_OPls))) (let ((_let_12 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_13 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_12))) (let ((_let_14 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_15 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_16 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat _let_11))) (let ((_let_17 (tptp.c_Int_OBit0 _let_11))) (let ((_let_18 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint _let_17))) (let ((_let_19 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat _let_17))) (let ((_let_20 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (let ((_let_21 (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Complex_Ocomplex) (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ocsqrt V_z)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_z)))) (let ((_let_22 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_23 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_24 (tptp.hAPP (tptp.hAPP _let_23 _let_19) tptp.v_m____))) (let ((_let_25 (= tptp.v_na____ _let_24))) (let ((_let_26 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint tptp.c_Int_OPls))) (let ((_let_27 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat tptp.c_Int_OPls))) (let ((_let_28 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (let ((_let_29 (tptp.hAPP tptp.c_Int_Onat _let_12))) (let ((_let_30 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (let ((_let_31 (tptp.hAPP _let_30 _let_15))) (let ((_let_32 (= _let_31 _let_1))) (let ((_let_33 (= _let_30 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_RealDef_Oreal)))) (let ((_let_34 (tptp.hAPP _let_20 _let_15))) (let ((_let_35 (tptp.hAPP _let_20 _let_22))) (let ((_let_36 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Complex_Ocomplex))) (let ((_let_37 (not (= tptp.v_m____ _let_22)))) (let ((_let_38 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat tptp.v_m____ tptp.v_na____))) (let ((_let_39 (forall ((V_q $$unsorted) (V_p $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_p)) V_q) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_p) V_q))))))))) (let ((_let_40 (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex _let_6 (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_3 tptp.v_z____) tptp.v_m____)))) _let_1))) (let ((_let_41 (= _let_1 _let_31))) (let ((_let_42 (tptp.hAPP _let_4 _let_19))) (let ((_let_43 (= tptp.v_z____ _let_42))) (let ((_let_44 (= (tptp.hAPP (tptp.hAPP _let_3 _let_42) tptp.v_m____) (tptp.hAPP _let_4 _let_24)))) (let ((_let_45 (ASSUME :args (_let_40)))) (let ((_let_46 (SYMM (ASSUME :args (_let_32))))) (let ((_let_47 (ASSUME :args (_let_25)))) (let ((_let_48 (ASSUME :args (_let_9)))) (let ((_let_49 (forall ((V_z $$unsorted)) (= V_z (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Complex_Ocomplex) (tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ocsqrt V_z)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))))) (let ((_let_50 (EQ_RESOLVE (ASSUME :args (_let_21)) (MACRO_SR_EQ_INTRO :args (_let_21 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_51 (not _let_10))) (let ((_let_52 (or _let_51 _let_44))) (let ((_let_53 (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))) (or (not (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))))))))) (let ((_let_54 (EQ_RESOLVE (ASSUME :args (_let_39)) (MACRO_SR_EQ_INTRO :args (_let_39 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_55 (not _let_43))) (let ((_let_56 (not _let_44))) (let ((_let_57 (not _let_25))) (let ((_let_58 (not _let_41))) (let ((_let_59 (not _let_40))) (let ((_let_60 (APPLY_UF tptp.c_Orderings_Oord__class_Oless))) (let ((_let_61 (APPLY_UF tptp.hAPP))) (let ((_let_62 (ASSUME :args (_let_43)))) (let ((_let_63 (ASSUME :args (_let_44)))) (let ((_let_64 (REFL :args (tptp.tc_Complex_Ocomplex)))) (let ((_let_65 (REFL :args (tptp.tc_RealDef_Oreal)))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (NOT_AND (MACRO_SR_PRED_TRANSFORM (SCOPE (AND_INTRO _let_45 _let_62 _let_63 _let_47 _let_46 _let_48) :args (_let_40 _let_25 _let_9 _let_41 _let_44 _let_43)) (SCOPE (MACRO_SR_PRED_ELIM (TRANS (SYMM (FALSE_INTRO _let_48)) (CONG _let_65 (REFL :args (_let_7)) _let_46 :args _let_60) (CONG _let_65 (CONG _let_64 (CONG _let_64 (REFL :args (_let_6)) (CONG (REFL :args (_let_5)) (TRANS (CONG (REFL :args (_let_4)) (SYMM (SYMM _let_47)) :args _let_61) (SYMM _let_63) (CONG (CONG (REFL :args (_let_3)) (SYMM _let_62) :args _let_61) (REFL :args (tptp.v_m____)) :args _let_61)) :args _let_61) :args (APPLY_UF tptp.c_Groups_Oplus__class_Oplus)) :args (APPLY_UF tptp.c_RealVector_Onorm__class_Onorm)) (SYMM _let_46) :args _let_60) (TRUE_INTRO _let_45))) :args (_let_40 _let_43 _let_44 _let_25 _let_41 _let_9)) :args ((not (and _let_40 _let_25 _let_9 _let_41 _let_44 _let_43)) SB_LITERAL))) (CONG (REFL :args (_let_59)) (REFL :args (_let_57)) (MACRO_SR_PRED_INTRO :args ((= (not _let_9) _let_8))) (REFL :args (_let_58)) (REFL :args (_let_56)) (REFL :args (_let_55)) :args (or))) :args ((or _let_8 _let_59 _let_58 _let_57 _let_56 _let_55))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_52)) :args ((or _let_51 _let_44 (not _let_52)))) (ASSUME :args (_let_10)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_54 :args (tptp.v_m____ _let_19 _let_2 tptp.tc_Complex_Ocomplex QUANTIFIERS_INST_E_MATCHING ((tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) (tptp.hAPP (tptp.hAPP _let_23 V_p) V_q))))) :args (_let_53))) _let_54 :args (_let_52 false _let_53)) :args (_let_44 false _let_10 false _let_52)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_50 :args (tptp.v_z____ QUANTIFIERS_INST_E_MATCHING_SIMPLE ((tptp.c_Fundamental__Theorem__Algebra__Mirabelle_Ocsqrt V_z)))) :args (_let_49))) _let_50 :args (_let_43 false _let_49)) _let_48 _let_47 _let_46 _let_45 :args (false false _let_44 false _let_43 true _let_8 false _let_25 false _let_41 false _let_40)) :args ((forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))) (= V_f_2 V_g_2))) (not (= tptp.v_na____ _let_22)) (not (= tptp.v_b _let_36)) _let_40 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat tptp.v_na____) (not (= tptp.v_n _let_22)) (forall ((V_n $$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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1)))))) (not (forall ((B_z $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oone__class_Oone tptp.tc_Complex_Ocomplex) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Complex_Ocomplex) tptp.v_b) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Complex_Ocomplex) B_z) tptp.v_m____)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))))) (forall ((T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra__1 T_a) (= (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)))) (forall ((V_s $$unsorted) (V_y $$unsorted) (V_r $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_x) V_r) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_y) V_s) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_x V_y)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_r V_s)))))) (forall ((V_s $$unsorted) (V_y $$unsorted) (V_r $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_x) V_r) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_y) V_s) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_r) V_s)))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) V_a_2) (= (= (tptp.hAPP _let_1 V_ma_2) (tptp.hAPP _let_1 V_nb_2)) (= V_ma_2 V_nb_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_ba_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_ba_2))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) V_ba_2) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x_2 V_y_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)))))) (forall ((V_a $$unsorted) (V_N $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_N) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) V_a) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_n) (tptp.hAPP _let_1 V_N))))))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_m V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oplus__class_Oplus T_a _let_1 _let_1)) V_m))))) _let_37 _let_38 (forall ((B_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_m tptp.v_na____) (=> (not (= B_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (exists ((B_z $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oone__class_Oone tptp.tc_Complex_Ocomplex) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Complex_Ocomplex) tptp.v_b) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Complex_Ocomplex) B_z) B_m)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (= (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)))) (forall ((V_nb_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_nb_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_x_2) (= V_nb_2 _let_1))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (= (= (tptp.c_RealVector_Onorm__class_Onorm T_a V_x_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) (= V_x_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_nb_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (and (tptp.class_Power_Opower T_a) (tptp.class_Rings_Omult__zero T_a) (tptp.class_Rings_Ono__zero__divisors T_a) (tptp.class_Rings_Ozero__neq__one T_a)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) V_nb_2) _let_1) (and (= V_a_2 _let_1) (not (= V_nb_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))) (forall ((V_n $$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))))) _let_39 (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_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_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_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_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 $$unsorted) (V_ba_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (= (= V_ba_2 (tptp.c_Groups_Oplus__class_Oplus T_a V_ba_2 V_a_2)) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) V_a))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_a))) (forall ((V_n $$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_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)))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_a V_x_2)) (not (= V_x_2 (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (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.c_Groups_Oplus__class_Oplus 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.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_1 (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_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))) (=> (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.c_Groups_Oplus__class_Oplus T_a V_a (tptp.hAPP _let_1 V_c)) (tptp.c_Groups_Oplus__class_Oplus T_a V_b (tptp.hAPP _let_1 V_d))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Power_Opower T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_a $$unsorted) (V_N $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_N) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_N) (tptp.hAPP _let_1 V_n)))))))) (forall ((V_n $$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_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1) _let_1)))))) (forall ((V_ry $$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)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_d)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_d))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) V_c) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) V_b)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_c))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_d)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) V_d)))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_d)) (tptp.c_Groups_Oplus__class_Oplus T_a V_c (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_d))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_a)))) (forall ((V_za_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (let ((_let_3 (tptp.hAPP _let_1 V_w_2))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_3 V_y_2) (tptp.hAPP _let_2 V_za_2)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_3 V_za_2) (tptp.hAPP _let_2 V_y_2))) (or (= V_w_2 V_x_2) (= V_y_2 V_za_2)))))))) (forall ((V_b $$unsorted) (V_m $$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.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_m) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_m)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b)) V_m))))) (forall ((V_ya $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_x V_y)) V_ya) (tptp.c_Groups_Oplus__class_Oplus T_a (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))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_a_H)) V_b) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_a_H) 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_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b)) V_c) (tptp.c_Groups_Oplus__class_Oplus T_a (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_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_ba_2))) (let ((_let_3 (tptp.hAPP _let_1 V_a_2))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (= (and (not (= V_a_2 V_ba_2)) (not (= V_c_2 V_d_2))) (not (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_3 V_c_2) (tptp.hAPP _let_2 V_d_2)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_3 V_d_2) (tptp.hAPP _let_2 V_c_2)))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_xa $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_xa))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_x V_y)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_1 V_x) (tptp.hAPP _let_1 V_y)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_y V_z)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_1 V_y) (tptp.hAPP _let_1 V_z)))))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_b_H)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_b_H)))))) (forall ((V_a $$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 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a) V_a))) (forall ((V_q $$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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__div__algebra T_a) (= (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_a V_x)) (tptp.c_RealVector_Onorm__class_Onorm T_a V_y))))) (forall ((V_n $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__div__algebra T_a) (= (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_a V_x)) V_n)))) (forall ((V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_m) V_m) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Oone__class_Oone T_a))) V_m))))) (forall ((V_a $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_m (tptp.hAPP (tptp.hAPP _let_1 V_a) V_m)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Oone__class_Oone T_a))) V_m))))) (=> _let_38 (=> _let_37 (exists ((B_z $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oone__class_Oone tptp.tc_Complex_Ocomplex) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Complex_Ocomplex) tptp.v_b) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Complex_Ocomplex) B_z) tptp.v_m____)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))))) (forall ((V_nb_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2) V_nb_2)) (or (= V_nb_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (and _let_2 (not (= V_x_2 _let_1))) (and (not _let_2) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_x_2)))))))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_nb_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2) V_nb_2) _let_1) (and (not (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 _let_1)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Groups_Oplus__class_Oplus T_a _let_1 _let_1))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a (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 ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) V_x) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_y)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_x_2) V_y_2)) (and (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_x_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_y_2)))) (forall ((V_n $$unsorted)) (=> (not (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) 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.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_m) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_n) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_m) V_n))))))) (forall ((T_a $$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 ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n) V_n)) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_ma_2) V_nb_2)) (and (= V_ma_2 _let_1) (= V_nb_2 _let_1))))) (forall ((V_n $$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 $$unsorted) (V_m $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_m))) (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_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_x (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y V_z)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_x V_z)))) (forall ((V_n $$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_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$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_ma_2) V_nb_2) _let_1) (and (= V_ma_2 _let_1) (= V_nb_2 _let_1))))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n) V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_k)))) (forall ((V_k $$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_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n)) V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_ma_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_nb_2)) (= V_ma_2 V_nb_2))) (forall ((V_nb_2 $$unsorted) (V_k_2 $$unsorted) (V_ma_2 $$unsorted)) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_ma_2 V_k_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_nb_2 V_k_2)) (= V_ma_2 V_nb_2))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))))) (forall ((V_y $$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 $$unsorted) (V_m $$unsorted)) (=> (= (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 $$unsorted) (V_m $$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_nb_2 $$unsorted) (V_k_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_ma_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_nb_2) V_k_2)) (or (= V_ma_2 V_nb_2) (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_nb_2 $$unsorted) (V_ma_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_ma_2) (tptp.hAPP _let_1 V_nb_2)) (or (= V_ma_2 V_nb_2) (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$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_ma_2) V_nb_2) _let_1) (or (= V_ma_2 _let_1) (= V_nb_2 _let_1))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_ma_2 V_nb_2) _let_1) (and (= V_ma_2 _let_1) (= V_nb_2 _let_1))))) (forall ((V_m $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (forall ((V_m $$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 $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) V_n)) (forall ((V_n $$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_P_2 $$unsorted) (V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_nb_2) V_ma_2)))) (=> (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2) _let_1) (=> (=> (= V_ma_2 V_nb_2) _let_1) (=> (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_nb_2 V_ma_2) _let_1) _let_1))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (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)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_k V_l) (=> (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_l) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat 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)) (=> (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.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_l))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_k)))) (forall ((V_m $$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.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_j)))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_m)))) (forall ((V_t $$unsorted) (V_s $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_s V_t) (not (= V_s V_t)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_m) (not (= V_m V_n)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_n))) (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_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_ma_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_nb_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (= (not (= V_ma_2 V_nb_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_nb_2 V_ma_2)))) (forall ((V_i $$unsorted) (V_j $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_i) V_i))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_j) V_i))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_n))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_y_2))) (let ((_let_2 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_x_2))) (= (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_x_2 V_y_2)) (or (and _let_2 _let_1) (and (not _let_2) (not _let_1))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_x_2) V_y_2)) (or (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_x_2) (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_y_2)))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (= (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_ma_2 V_nb_2)) (= (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_ma_2) (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2)))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (= (not (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_ma_2 V_nb_2))) (= (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_ma_2) (not (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2))))) (not (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat _let_15)) (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 ((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_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oring__no__zero__divisors T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_ba_2) _let_1) (or (= V_a_2 _let_1) (= V_ba_2 _let_1)))))) (forall ((V_b $$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 $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ono__zero__divisors T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) _let_1) (or (= V_a _let_1) (= V_b _let_1)))))) (forall ((T_a $$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 ((T_a $$unsorted)) (=> (tptp.class_Rings_Ozero__neq__one T_a) (not (= (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b)) V_c) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_e $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Osemiring T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_e) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_b) V_e) V_c)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b)) V_e) V_c))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not (= V_n _let_1)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n)))) (forall ((V_k $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_nb_2 $$unsorted) (V_k_2 $$unsorted) (V_ma_2 $$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_ma_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_nb_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_ma_2 V_nb_2))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_ma_2) (tptp.hAPP _let_1 V_nb_2)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2))))) (forall ((V_n $$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_nb_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_nb_2)) (or (= V_nb_2 _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_x_2))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$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_ma_2) V_nb_2)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_ma_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_nb_2))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$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.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_ma_2 V_nb_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_ma_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_nb_2))))) (forall ((V_nb_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (not (= V_nb_2 _let_1)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_nb_2)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat _let_22) (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_ba_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP _let_2 V_ba_2) V_c_2)) (or (and (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_c_2) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_ba_2)) (and (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 _let_1) (tptp.c_Orderings_Oord__class_Oless T_a V_ba_2 V_a_2)))))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_2 V_a_2) (tptp.hAPP _let_2 V_ba_2)) (or (and (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_c_2) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_ba_2)) (and (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 _let_1) (tptp.c_Orderings_Oord__class_Oless T_a V_ba_2 V_a_2)))))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c_2) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_ba_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_ba_2)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))))))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b _let_1) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a) _let_1)))))) (forall ((V_b $$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 (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_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a)) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b)))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_ba_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_ba_2 V_a_2)))))) (forall ((V_b $$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 $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a _let_1) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b _let_1) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_c $$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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__comm__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))))) (forall ((V_c $$unsorted) (V_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_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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_c) (tptp.c_Orderings_Oord__class_Oless T_a V_b (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a (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 ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_x_2) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_y_2)) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_z))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_z) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x V_y) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 V_x) (tptp.hAPP _let_1 V_y)))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_y) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_x) V_y)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_za_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_za_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_za_2) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_za_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x_2 V_y_2))))) (forall ((V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (= (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_x_2) V_x_2))) (= V_x_2 _let_1)))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_ma_2) (tptp.hAPP _let_1 V_nb_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2))))) (forall ((V_nb_2 $$unsorted) (V_ma_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_ma_2) (tptp.hAPP _let_1 V_nb_2)) (= V_ma_2 V_nb_2))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2)) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a_2))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2) _let_1) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 _let_1))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2) _let_1) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 _let_1))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Ozero T_a) (= (= _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 $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_a) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_a)) (= V_b V_c)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocancel__ab__semigroup__add T_a) (=> (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)) (= V_b V_c)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)) (= V_b V_c)))) (forall ((V_c_2 $$unsorted) (V_a_2 $$unsorted) (V_ba_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a V_ba_2 V_a_2) (tptp.c_Groups_Oplus__class_Oplus T_a V_c_2 V_a_2)) (= V_ba_2 V_c_2)))) (forall ((V_c_2 $$unsorted) (V_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_ba_2) (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_c_2)) (= V_ba_2 V_c_2)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__semigroup__add T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_c))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Groups_Oone T_a) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x))) (= (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y V_z)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z)))) (forall ((V_z3 $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (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_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_y) V_z)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (= _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2)) (= V_a_2 _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_a) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_b)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b)))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_c)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b)))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (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.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_d)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (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.c_Groups_Oplus__class_Oplus T_a V_c V_a) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_b))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (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.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_c))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_c_2 V_a_2) (tptp.c_Groups_Oplus__class_Oplus T_a V_c_2 V_ba_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_ba_2)))) (forall ((V_ba_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_c_2) (tptp.c_Groups_Oplus__class_Oplus T_a V_ba_2 V_c_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_ba_2)))) (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_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_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_ma_2) (tptp.hAPP _let_1 V_nb_2)) (or (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= V_ma_2 V_nb_2))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (=> (not (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP _let_1 V_ba_2) V_c_2)) (= V_a_2 V_ba_2))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_c_2))) (=> (not (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (= (= (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_ba_2)) (= V_a_2 V_ba_2))))) (not (= _let_28 _let_1)) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_u $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i) V_u) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_j) V_u) V_k)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_j)) V_u) V_k)))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)) V_z) V_z)) (forall ((V_w $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_z1 V_z2)) V_w) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.hAPP _let_1 V_z1) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_w))))) (forall ((V_b $$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.c_Groups_Oplus__class_Oplus T_a V_a V_b) _let_1)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Power_Opower T_a) (= (tptp.c_Power_Opower__class_Opower T_a) (tptp.c_Power_Opower_Opower T_a (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Groups_Otimes__class_Otimes T_a))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x V_y) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y V_z)))) (forall ((V_y $$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.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y V_z)))) (forall ((V_a $$unsorted) (V_n $$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_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_a) (exists ((B_r $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) B_r) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) B_r) V_n) V_a)))))) (forall ((V_a $$unsorted) (V_n $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_a) (exists ((B_x $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) B_x) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) B_x) V_n) V_a) (forall ((B_y $$unsorted)) (=> (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) B_y) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) B_y) V_n) V_a)) (= B_y B_x)))))))) (forall ((V_w_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w_2))) (let ((_let_3 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat _let_2))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2) _let_2)) (or (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (and _let_3 (not (= V_x_2 _let_1))) (and (not _let_3) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_x_2))))))))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) 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 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n)))))))) (forall ((V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Groups_Omonoid__mult T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) V_x) (tptp.hAPP _let_1 V_n)))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z) V_z)) (forall ((V_z $$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_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_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2) V_ba_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_z2 $$unsorted) (V_z1 $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w))) (= (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_z1 V_z2)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_z1) (tptp.hAPP _let_1 V_z2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_v $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.c_Int_Onumber__class_Onumber__of T_b V_v)))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Rings_Oring T_b)) (= (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_b V_b V_c)) (tptp.c_Groups_Ominus__class_Ominus T_b (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_c)))))) (forall ((V_c $$unsorted) (V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_v) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_w) V_c)) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_v V_w)) V_c)))) (forall ((V_z3 $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_z1))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_z2)) V_z3) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_z3)))))) (forall ((V_w $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_z1 V_z2)) V_w) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_z1) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_w))))) (forall ((V_v $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b V_v))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Rings_Oring T_b)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ominus__class_Ominus T_b V_a V_b)) _let_1) (tptp.c_Groups_Ominus__class_Ominus T_b (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1))))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$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_ma_2) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_ma_2) V_nb_2) _let_1) (and (= V_ma_2 _let_1) (= V_nb_2 _let_1)))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP _let_1 V_i) (tptp.hAPP _let_1 V_j)))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_v V_w)) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_v) (tptp.c_Int_Onumber__class_Onumber__of T_a V_w))))) (forall ((V_x_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a V_w_2))) (=> (tptp.class_Int_Onumber T_a) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Int_Oring__char__0 T_a)) (= (= (tptp.c_Int_Onumber__class_Onumber__of T_a V_x_2) (tptp.c_Int_Onumber__class_Onumber__of T_a V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_x_2)) (tptp.c_Int_Onumber__class_Onumber__of T_a V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_x_2) V_y_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (=> (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2) (tptp.c_Groups_Ominus__class_Ominus T_a V_c_2 V_d_2)) (= (= V_a_2 V_ba_2) (= V_c_2 V_d_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (=> (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2) (tptp.c_Groups_Ominus__class_Ominus T_a V_c_2 V_d_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2) V_ba_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c_2) V_d_2)))))) (forall ((V_w_2 $$unsorted) (V_v_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a V_v_2))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of T_a V_w_2))) (=> (and (tptp.class_Int_Onumber T_a) (tptp.class_Orderings_Olinorder T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1) _let_2)) (not (tptp.c_Orderings_Oord__class_Oless T_a _let_2 _let_1))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x V_y) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x) V_y)))) (forall ((V_y $$unsorted) (V_z $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y V_z)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y V_z)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_a) (tptp.c_RealVector_Onorm__class_Onorm T_a V_b))) (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n)) V_m))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i V_j) V_k) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i V_k) V_j))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_l V_n)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_l V_m))))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_l)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n V_l))))) (forall ((V_n $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i) V_n)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n V_i)) V_i))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_ma_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_nb_2)) (= (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_ma_2 V_k_2) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_nb_2 V_k_2)) (= V_ma_2 V_nb_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_k) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n V_k)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n)))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_ma_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_nb_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_ma_2 V_k_2)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_nb_2 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2))))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_ma_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_nb_2)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_ma_2 V_k_2) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_nb_2 V_k_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_a V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_c) V_a)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_a V_c) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_b V_c))))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_j)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_i) V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j V_k) V_i)))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_j)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j V_k) V_i) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_i) V_k)))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_j)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_j) V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j V_k))))) (forall ((V_k_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i_2) V_j_2)) (= (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j_2 V_i_2) V_k_2) (= V_j_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_i_2))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_m)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n) V_n) V_m))) (forall ((V_i_2 $$unsorted) (V_j_2 $$unsorted) (V_k_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2) V_j_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i_2) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j_2 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i_2 V_k_2)) V_j_2))))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_j)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j V_k)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_j) V_k)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_m)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n)) V_m))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted) (V_n $$unsorted) (V_k $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_m) V_k))))) (forall ((V_i_2 $$unsorted) (V_k_2 $$unsorted) (V_j_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j_2 V_k_2)) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_j_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i_2 V_k_2))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (= (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_ma_2 V_nb_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2)))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_j)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j V_k)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_k) V_j)))) (forall ((V_d_2 $$unsorted) (V_ba_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2) V_c_2)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_ba_2) V_e_2) V_d_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c_2) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_ba_2 V_a_2)) V_e_2) V_d_2))))))) (forall ((V_d_2 $$unsorted) (V_ba_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2) V_c_2)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_ba_2) V_e_2) V_d_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2)) V_e_2) V_c_2)) V_d_2)))))) (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.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_N)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oone__class_Oone T_a)) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 V_N)))))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_u_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i_2) V_j_2)) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2) V_ma_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2) V_nb_2)) (= V_ma_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j_2 V_i_2)) V_u_2) V_nb_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_u $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i) V_j)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i) V_u) V_m) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_j) V_u) V_n)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j V_i)) V_u) V_n)))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_u_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i_2) V_j_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2) V_ma_2)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2) V_nb_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j_2 V_i_2)) V_u_2) V_nb_2))))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_u_2 $$unsorted) (V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_j_2) V_i_2)) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2) V_ma_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2) V_nb_2)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i_2 V_j_2)) V_u_2) V_ma_2) V_nb_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_u $$unsorted) (V_i $$unsorted) (V_j $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_j) V_i)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i) V_u) V_m) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_j) V_u) V_n)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i V_j)) V_u) V_m) V_n))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_u_2 $$unsorted) (V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_j_2) V_i_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2) V_ma_2)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2) V_nb_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i_2 V_j_2)) V_u_2) V_ma_2)) V_nb_2)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_a) (tptp.c_RealVector_Onorm__class_Onorm T_a V_b)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (= V_a_2 V_ba_2) (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2) (tptp.c_Groups_Ozero__class_Ozero T_a)) (= V_a_2 V_ba_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_x_2) (tptp.c_Int_Onumber__class_Onumber__of T_a V_y_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_x_2 V_y_2)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (=> (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2) (tptp.c_Groups_Ominus__class_Ominus T_a V_c_2 V_d_2)) (= (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_ba_2) (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 V_d_2))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Int_Onumber__class_Onumber__of T_a V_v)) (tptp.c_Int_Onumber__class_Onumber__of T_a V_w))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Int_Onumber__class_Onumber__of T_a V_v)) (tptp.c_Int_Onumber__class_Onumber__of T_a V_w)) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_w))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a V_v)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a V_w)) V_z)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_w))) V_z))))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_b V_b_H)) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_b_H)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_xa $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_xa))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_x V_y)) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.hAPP _let_1 V_x) (tptp.hAPP _let_1 V_y)))))) (forall ((V_b $$unsorted) (V_a_H $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_a_H)) V_b) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b)))))) (forall ((V_ya $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_x V_y)) V_ya) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) V_ya) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_ya)))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_v V_w)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_v) (tptp.c_Int_Onumber__class_Onumber__of T_a V_w))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_v) (tptp.c_Int_Onumber__class_Onumber__of T_a V_w)) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_v V_w))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_v) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_w) V_z)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_v V_w)) V_z)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b) V_b) V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) V_b) V_a))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n) _let_1) (=> (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n V_m) _let_1) (= V_m V_n))))) (forall ((V_m $$unsorted)) (= (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 $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat _let_1 V_n) _let_1))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (= (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b)) (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_b V_a))))) (forall ((V_ba_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_c_2)) (tptp.c_Groups_Oplus__class_Oplus T_a V_ba_2 V_c_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2) V_ba_2))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_c_2 V_a_2)) (tptp.c_Groups_Oplus__class_Oplus T_a V_c_2 V_ba_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2) V_ba_2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_a)) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_b)))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_d))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_c))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_a)) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_b))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b))))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_l) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_l V_n) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_l V_m))))) (forall ((V_n $$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.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j V_n) V_k))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n) V_n) V_m)) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_m) V_n) V_m)) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i V_j) V_k) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_k)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k V_m) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k V_n)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_k)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n $$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_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n)) (tptp.c_Groups_Ominus__class_Ominus 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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n)) V_k) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k))))) (forall ((V_nb_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_nb_2) _let_1)) (= V_nb_2 _let_1)))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (= V_m V_n)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)) (=> (not (= V_m V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2) (= V_ma_2 V_nb_2)))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2)) (not (= V_ma_2 V_nb_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x_2 V_y_2) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x_2) V_y_2)) (not (= V_x_2 V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x_2) V_y_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x_2 V_y_2) (= V_x_2 V_y_2)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_m)))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2)) (exists ((B_k $$unsorted)) (= V_nb_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_ma_2 B_k))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_ma_2)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_nb_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2)))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_m)))))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_j)))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_k)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_k))))) (forall ((V_l $$unsorted) (V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_l)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_k)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_l)))))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_k)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_n)))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_k)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_k)) V_n)) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_n))))))) (forall ((V_m $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_m)))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_m)))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_i)) (tptp.hAPP _let_1 V_j)))))) (forall ((V_l $$unsorted) (V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k) V_l)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_l))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_z V_x)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_z V_y))))) (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.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_N)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP _let_1 V_N)) (tptp.hAPP _let_1 V_n))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_ba_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_ba_2))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) V_ba_2) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x_2) V_y_2))))))) (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.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_d)))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_c)) (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_b V_d))))))) (forall ((V_w_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w_2))) (let ((_let_3 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat _let_2))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2) _let_2))) (or _let_3 (and (not _let_3) (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)))))))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_u_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i_2) V_j_2)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2) V_ma_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2) V_nb_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j_2 V_i_2)) V_u_2) V_nb_2)))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_u_2 $$unsorted) (V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_j_2) V_i_2)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2) V_ma_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2) V_nb_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i_2 V_j_2)) V_u_2) V_ma_2) V_nb_2))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_ba_2) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_w_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w_2))) (let ((_let_3 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat _let_2))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2) _let_2)) _let_1)) (and (not (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (or (and (not _let_3) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) _let_1))) (and _let_3 (= V_x_2 _let_1)))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_v $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.c_Int_Onumber__class_Onumber__of T_b V_v)))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Rings_Osemiring T_b)) (= (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_b V_b V_c)) (tptp.c_Groups_Oplus__class_Oplus T_b (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_c)))))) (forall ((V_v $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b V_v))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Rings_Osemiring T_b)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oplus__class_Oplus T_b V_a V_b)) _let_1) (tptp.c_Groups_Oplus__class_Oplus T_b (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1))))))) (forall ((V_d_2 $$unsorted) (V_ba_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oring T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2) V_c_2) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_ba_2) V_e_2) V_d_2)) (= V_c_2 (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_ba_2 V_a_2)) V_e_2) V_d_2)))))) (forall ((V_d_2 $$unsorted) (V_ba_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oring T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2) V_c_2) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_ba_2) V_e_2) V_d_2)) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2)) V_e_2) V_c_2) V_d_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_y V_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.c_Groups_Ominus__class_Ominus T_a V_x V_a)))) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (= (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y) (tptp.hAPP _let_3 V_b)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_4 _let_1) (tptp.hAPP _let_4 V_b)) (tptp.hAPP _let_3 _let_1))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.hAPP _let_2 V_y) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_2 (tptp.c_Groups_Ominus__class_Ominus T_a V_y V_b)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_x V_a)) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (or (and (tptp.hBOOL (tptp.hAPP _let_2 V_a)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) _let_1)) (tptp.hBOOL (tptp.hAPP _let_2 V_b)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) _let_1))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1))) (=> (tptp.class_Rings_Oordered__ring T_a) (=> (or (and (tptp.hBOOL (tptp.hAPP _let_2 V_a)) (tptp.hBOOL (tptp.hAPP _let_2 V_b))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) _let_1)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Oordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c) V_d)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_b)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))))))) (forall ((V_d $$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_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Oordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c) V_d)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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 $$unsorted) (V_a $$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_Oordered__ring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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_Oordered__ring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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_Oordered__comm__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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_Oordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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_Oordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a)) _let_1))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)))))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_ba_2)) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP _let_2 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_ba_2) _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP _let_2 V_ba_2))))))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_ba_2))) (or (and (tptp.hBOOL (tptp.hAPP _let_2 V_a_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_ba_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_ba_2) _let_1))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__ring T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a))))) (forall ((V_b $$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.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b)) _let_1))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_c)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y_2)) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a V_x_2 V_y_2) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b)))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2) _let_1)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c) V_d)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_d)))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq 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.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_d)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n) V_m))))) (forall ((V_ma_2 $$unsorted) (V_nb_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_nb_2 V_ma_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_m)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n)) V_m))) (forall ((V_k_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i_2 (tptp.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.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i_2 V_k_2) V_j_2))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oone__class_Oone T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_y_2))) (let ((_let_2 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_x_2))) (= (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_x_2 V_y_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x_2 V_y_2) (and _let_2 _let_1) (and (not _let_2) (not _let_1))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) (tptp.c_RealVector_Onorm__class_Onorm T_a V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (not (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_n)))))))) (forall ((V_d_2 $$unsorted) (V_ba_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2) V_c_2) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_ba_2) V_e_2) V_d_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_ba_2 V_a_2)) V_e_2) V_d_2)))))) (forall ((V_d_2 $$unsorted) (V_ba_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2) V_c_2) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_ba_2) V_e_2) V_d_2)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_ba_2)) V_e_2) V_c_2) V_d_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.hAPP (tptp.hAPP _let_2 V_x) V_x) _let_1) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oplus__class_Oplus T_a V_x _let_1)) (tptp.c_Groups_Ominus__class_Ominus T_a V_x _let_1))))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c_2) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP _let_1 V_a_2)) (tptp.hAPP _let_1 V_ba_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2) V_ba_2))))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP _let_1 V_a_2)) (tptp.hAPP _let_1 V_ba_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_ba_2) V_a_2))))))) (forall ((V_d $$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_Groups_Ozero__class_Ozero 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 V_c V_d) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_2 V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_2) 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_d)))))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c V_d) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_c)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d)))))))))) (forall ((V_d $$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_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c) V_d)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_2) V_a)) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_2 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_d)))))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c V_d) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_2 V_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_2) V_c)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d)))))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__semiring T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b)))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__semiring T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b)))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1) V_b)) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1) V_a)) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) V_c)) (tptp.c_Orderings_Oord__class_Oless T_a V_b (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_c) (tptp.c_Orderings_Oord__class_Oless T_a V_b (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)))))) (forall ((V_b $$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.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) _let_1)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) _let_1)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) _let_1)) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b _let_1) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) _let_1)))))) (forall ((V_y $$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) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (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))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_x_2) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_y_2))) _let_1)) (and (= V_x_2 _let_1) (= V_y_2 _let_1))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) V_y)) V_x)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_y) V_x)) V_x)))))))) (forall ((V_w_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w_2))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (and (tptp.class_Power_Opower T_a) (tptp.class_Rings_Omult__zero T_a) (tptp.class_Rings_Ono__zero__divisors T_a) (tptp.class_Rings_Ozero__neq__one T_a)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) _let_1) _let_2) (and (= V_a_2 _let_2) (not (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_b)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b)))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_a_2 V_ba_2))) (and (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_a_2 V_ba_2) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((B_d $$unsorted)) (=> (= V_a_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_ba_2 B_d)) (tptp.hBOOL (tptp.hAPP V_P_2 B_d))))))) (forall ((V_ba_2 $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_a_2 V_ba_2))) (not (or (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_a_2 V_ba_2) (not (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (exists ((B_d $$unsorted)) (and (= V_a_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_ba_2 B_d)) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_d))))))))) (forall ((V_a_2 $$unsorted) (V_times_2 $$unsorted) (V_one_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower_Opower T_a 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 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_x_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (= V_x_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_x_2 $$unsorted) (V_nb_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (not (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2)) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2) V_nb_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_x_2))))))) (forall ((V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Groups_Omonoid__mult T_a) (tptp.class_Rings_Olinordered__ring T_a)) (=> (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_n) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) V_n)))))) (forall ((V_nb_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) V_nb_2))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_a_2)) (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2)))))) (forall ((V_nb_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (let ((_let_2 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2) V_nb_2))) (or _let_2 (and (not _let_2) (tptp.hBOOL (tptp.hAPP _let_1 V_x_2))))))))) (forall ((V_nb_2 $$unsorted) (V_k_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_ma_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_1 V_nb_2) V_k_2))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2)))))) (forall ((V_nb_2 $$unsorted) (V_ma_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.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_ma_2)) (tptp.hAPP _let_1 V_nb_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.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2)))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_ma_2)) (tptp.hAPP _let_1 V_nb_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) V_y))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_a V_x)) (tptp.c_RealVector_Onorm__class_Onorm T_a V_y)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_za_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_za_2))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_za_2) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x_2) V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_za_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_za_2) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_za_2)) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_za_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x_2) V_y_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_x V_y))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_x) (tptp.c_RealVector_Onorm__class_Onorm T_a V_y)))))) (forall ((V_n $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) V_n))) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_a V_x)) V_n))))) (forall ((V_w_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w_2))) (= (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) _let_2)) (or (= _let_2 _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_x_2)))))) (forall ((V_z $$unsorted) (V_w $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex V_w)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex V_w V_z)) (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex V_z))))) (forall ((V_v $$unsorted) (V_u $$unsorted) (V_y $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_u)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_v)) (=> (= (tptp.c_Groups_Oplus__class_Oplus T_a V_u V_v) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_u) V_x) (tptp.hAPP (tptp.hAPP _let_1 V_v) V_y))) V_a))))))))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) _let_2))) (let ((_let_4 (= _let_2 (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_4 (= _let_3 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (not _let_4) (= _let_3 _let_1))))))))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_n)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n) (= V_a V_b))))))))) (forall ((V_w_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w_2))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2) _let_2) _let_1) (and (not (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat _let_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 _let_1))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_4 (= V_m _let_3))) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n)))))))))) (forall ((V_nb_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2) V_nb_2)) _let_1)) (and (not (= V_nb_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (or (and (not _let_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) _let_1))) (and _let_2 (= V_x_2 _let_1))))))))) (forall ((V_nb_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_nb_2) (= (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2) (not (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_nb_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))))) (forall ((V_p $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_p))) (let ((_let_3 (tptp.hAPP _let_2 V_m))) (let ((_let_4 (= V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_4 (= _let_3 _let_1)) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_p) (tptp.hAPP _let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m _let_1))))))))))) (forall ((V_v $$unsorted) (V_u $$unsorted) (V_y $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semiring__1__strict T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y V_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_u)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_v)) (=> (= (tptp.c_Groups_Oplus__class_Oplus T_a V_u V_v) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_1 V_u) V_x) (tptp.hAPP (tptp.hAPP _let_1 V_v) V_y)) V_a)))))))))) (forall ((V_m $$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.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_c) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_x)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_c) (tptp.c_RealVector_Onorm__class_Onorm T_a V_y)))) (= V_x (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_x)))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_d)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b) (tptp.c_Groups_Ominus__class_Ominus T_a V_c V_d))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_x)) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_z V_y) (tptp.c_Orderings_Oord__class_Oless T_a V_z V_x))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y)) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y V_z) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_z))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_z) V_y)) (tptp.c_Orderings_Oord__class_Oless T_a V_z V_x))))) (forall ((V_za_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_za_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_za_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.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_za_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_za_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_w_2) V_za_2)))) (forall ((V_za_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_w_2) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_za_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 V_za_2))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_w)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_v V_w)))) (forall ((V_za_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_w_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_za_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 V_za_2))) (forall ((V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) V_z)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_z))))) (forall ((V_z $$unsorted) (V_w $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w V_z) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_w (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_z)))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_k V_i)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_k V_j))))) (forall ((V_z $$unsorted) (V_z_H $$unsorted) (V_w $$unsorted) (V_w_H $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_H V_w) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_z_H) V_z)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_w_H V_z_H) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_w V_z))))) (forall ((V_w $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_w) V_w))) (forall ((V_w $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_w) V_w))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (or (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_z) V_w)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_w) V_z)))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (or (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_z) V_w)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_w) V_z)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_k_2)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k_2) V_l_2)))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_j) V_k)) (tptp.hBOOL (tptp.hAPP _let_1 V_k)))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_j) V_k)) (tptp.hBOOL (tptp.hAPP _let_1 V_k)))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_z) V_w)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_w) V_z)) (= V_z V_w)))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_z) V_w)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_w) V_z)) (= V_z V_w)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_k_2) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_l_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k_2 V_l_2))) (forall ((V_w_2 $$unsorted) (V_za_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_za_2 V_w_2) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_za_2) V_w_2)) (not (= V_za_2 V_w_2))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (or (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_m)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (= V_m V_n) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_j) V_k)) (tptp.hBOOL (tptp.hAPP _let_1 V_k)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_m)) (= V_m V_n)))) (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_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_i V_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_i V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_j V_k)))) (forall ((V_za_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_za_2) V_za_2) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_za_2 _let_1)))) (forall ((V_z3 $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_z1 V_z2) V_z3) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_z1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_z2 V_z3)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_x (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_y V_z)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_y (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_x V_z)))) (forall ((V_za_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_za_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_za_2) (= V_w_2 V_za_2)))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_z V_w) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_w V_z))) (forall ((V_z $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_z (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (forall ((V_z $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_z) V_z)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_12 _let_14) (forall ((V_z $$unsorted)) (not (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_z) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (not (= _let_12 _let_14)) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal V_x_2 V_y_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))))) (forall ((V_w $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_z1 V_z2)) V_w) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_z1) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_w))))) (forall ((V_z2 $$unsorted) (V_z1 $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w))) (= (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_z1 V_z2)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_z1) (tptp.hAPP _let_1 V_z2))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_w)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_v) V_w))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_a) (tptp.c_RealVector_Onorm__class_Onorm T_a V_b))) (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b) V_f_2) V_g_2)) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_x))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_x))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= V_x_2 V_y_2) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2) V_x_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (= V_x V_y) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b) V_f_2) V_g_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2)))))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2) V_x_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (=> (= V_a V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c))) (=> (tptp.class_Orderings_Oorder T_a) (=> (= V_a V_b) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 V_a))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a))) (=> (tptp.class_Orderings_Oord T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP _let_1 V_c))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) V_a)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c) V_a)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_x)) (= V_x V_y))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_z)) (tptp.hBOOL (tptp.hAPP _let_1 V_z))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y)) (= V_x V_y))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a V_z))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 V_x))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b) V_f_2) V_g_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)) (=> (not (= V_x V_y)) (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_y $$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_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_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_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_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 $$unsorted) (V_b $$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) (=> (= 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_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a) (not (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b))))) (forall ((V_b $$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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (not (= V_y V_x))))) (forall ((V_y $$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_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_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_y_2 V_x_2)) (= (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (or (tptp.c_Orderings_Oord__class_Oless T_a V_y_2 V_x_2) (= V_x_2 V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (= V_x_2 V_y_2)) (or (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (tptp.c_Orderings_Oord__class_Oless T_a V_y_2 V_x_2))))) (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_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2) V_l_2)) (exists ((B_n $$unsorted)) (= V_l_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 B_n))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2) V_x_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) V_y_2))) (tptp.c_Orderings_Oord__class_Oless T_a V_y_2 V_x_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y)) (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) V_y_2)) (not (= V_x_2 V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) V_y_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2) V_x_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) V_y_2)) (or (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_x))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_x))) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (not (= V_a V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (not (= V_a V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) V_a)) (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_x)) (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y))))) (forall ((V_y $$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) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2) V_y_2)) (= (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y)) (or (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (= V_x V_y))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b)) (=> (not (= V_a V_b)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) V_a)) (=> (not (= V_a V_b)) (tptp.c_Orderings_Oord__class_Oless T_a V_b 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.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y) V_z)) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_z))))) (forall ((V_q $$unsorted) (V_r $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_a) (=> (= V_a (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_r (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a) V_q))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) V_r)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_q) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))))) (forall ((V_q $$unsorted) (V_r $$unsorted) (V_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_a) (=> (= V_a (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_r (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a) V_q))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_r V_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_q)))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_b_H $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b_H) V_q_H) V_r_H))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_r_H V_b_H) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_b_H) (tptp.hBOOL (tptp.hAPP _let_2 V_q_H)))))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_b_H $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (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_H) V_q_H) V_r_H) _let_1) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) V_r_H)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_b_H) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_q_H) _let_1))))))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_k) V_k)) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.c_Orderings_Oord__class_Oless _let_1 V_f_2 V_g_2) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_f_2) V_g_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_g_2) V_f_2)))))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_b_H $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_2 V_b_H) V_q_H) V_r_H))) (=> (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_2 V_b) V_q) V_r) _let_3) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_3 _let_1) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_r V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) V_r_H)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_b_H) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_b_H) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_q_H) V_q)))))))))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_r_H $$unsorted) (V_q_H $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_q_H) V_r_H)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_q) V_r))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_r) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_b V_r) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_b V_r_H) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_q) V_q_H)))))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_b_H $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_3 V_b_H) V_q_H) V_r_H))) (=> (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_3 V_b) V_q) V_r) _let_4) (=> (tptp.hBOOL (tptp.hAPP _let_2 _let_4)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_r_H V_b_H) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_r)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_b_H) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_b_H) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_q) V_q_H))))))))))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_r_H $$unsorted) (V_q_H $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_q_H) V_r_H)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_q) V_r))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_r_H)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_r_H V_b) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_r V_b) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_q_H) V_q)))))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex V_b V_a)) (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex V_b))) (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex V_a)))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x) V_n)))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_x) V_y))))))) (tptp.hBOOL (tptp.hAPP _let_13 _let_12)) (tptp.hBOOL (tptp.hAPP _let_13 _let_14)) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_x V_y))))))) (forall ((V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Complex_Ocomplex))) (=> (= (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex V_z) _let_1) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex V_z _let_2)) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Complex_Ocomplex V_z _let_2)) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex V_z tptp.c_Complex_Oii)) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Complex_Ocomplex V_z tptp.c_Complex_Oii)) _let_1)))))) (forall ((V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (exists ((B_K $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) B_K) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) B_x) V_y))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_a B_x)) B_K)))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (exists ((B_K $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) B_K) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) B_x))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_a B_x)) B_K)))))))) (forall ((V_w $$unsorted)) (not (= tptp.c_Complex_Oii (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Complex_Ocomplex V_w)))) (not (= tptp.c_Complex_Oii _let_6)) (not (= tptp.c_Complex_Oii _let_36)) (forall ((T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (exists ((B_K $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) B_K) (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) B_a) B_b))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_RealVector_Onorm__class_Onorm T_a B_a)) (tptp.c_RealVector_Onorm__class_Onorm T_a B_b))) B_K))))))))) (forall ((V_x $$unsorted) (V_y $$unsorted)) (let ((_let_1 (tptp.c_Nat__Transfer_Otsub V_x V_y))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_y) V_x)))) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_x V_y))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))))) (forall ((V_nb_2 $$unsorted) (V_P_2 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_nb_2)) (exists ((B_k $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_k V_nb_2) (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat B_i) B_k)) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_i))))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat B_k (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Nat__Transfer_Otsub V_x V_y))))))) (forall ((V_x $$unsorted) (V_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_y) V_x)) (= (tptp.c_Nat__Transfer_Otsub V_x V_y) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_x V_y)))) (forall ((V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (exists ((B_K $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) B_K)) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) B_x) V_y))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_a B_x)) B_K)))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (exists ((B_K $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) B_K)) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) B_x))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_a B_x)) B_K)))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra T_a) (exists ((B_K $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) B_K)) (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) B_a) B_b))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_RealVector_Onorm__class_Onorm T_a B_a)) (tptp.c_RealVector_Onorm__class_Onorm T_a B_b))) B_K))))))))) (= (tptp.c_Complex_Oexpi _let_36) _let_6) (forall ((V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (= (tptp.c_RComplete_Onatceiling (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_x (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_RComplete_Onatceiling V_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v)))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v))))) (=> (tptp.class_Int_Onumber__ring T_a) (and (=> _let_2 (= _let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a V_v))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)))))))) (= (tptp.c_RComplete_Onatceiling _let_28) _let_22) (forall ((V_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_RComplete_Onatceiling V_x)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_RComplete_Onatceiling V_x)) (tptp.c_RComplete_Onatceiling V_y))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Nat_Osemiring__char__0 T_a) (= (= (tptp.hAPP _let_1 V_ma_2) (tptp.hAPP _let_1 V_nb_2)) (= V_ma_2 V_nb_2))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__algebra__1 T_a) (= (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_RealDef_Oreal) V_n)))) (= (tptp.c_RComplete_Onatceiling _let_1) _let_15) (forall ((V_n $$unsorted)) (= (tptp.c_RComplete_Onatceiling (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal V_n)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_n))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_m))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n))))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_m) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (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_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (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.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n)))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP _let_1 V_ma_2) (tptp.hAPP _let_1 V_nb_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP _let_1 V_ma_2)) (tptp.hAPP _let_1 V_nb_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (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.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.c_Groups_Oone__class_Oone T_a)))) (= _let_12 _let_35) (forall ((V_nb_2 $$unsorted)) (= (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_nb_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (= V_nb_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= _let_35 _let_12) (forall ((V_P_2 $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_x)))))) (forall ((V_P_2 $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_x)))))) (forall ((V_z $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_z)))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n)))) (forall ((V_k $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_k) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x_2 V_y_2)))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP _let_1 V_ma_2) (tptp.hAPP _let_1 V_nb_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 V_nb_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x_2) V_y_2))))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.hAPP _let_1 V_ma_2)) (tptp.hAPP _let_1 V_nb_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_ma_2) V_nb_2))))) (forall ((V_za_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_w_2) V_za_2)) (exists ((B_n $$unsorted)) (= V_za_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_w_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_n)))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_x) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_x V_y))))) (forall ((V_z $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_m) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_n) V_z)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n)) V_z)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_x) V_y))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))))) (= _let_14 _let_34) (= _let_34 _let_14) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (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.hAPP _let_1 V_m)) V_n))))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x)) V_n) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_x) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) V_n) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (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.hAPP _let_1 V_m)) V_n)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Nat__Transfer_Otsub (tptp.hAPP _let_1 V_x) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_x V_y))))) (forall ((V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (= (tptp.c_RComplete_Onatceiling V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_m)) (= (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n)) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n))))))) (forall ((V_nb_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_nb_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_nb_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_m)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n)))))) (forall ((V_x_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_RComplete_Onatceiling V_x_2)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))))) (forall ((V_nb_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_nb_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_nb_2)))) (forall ((V_nb_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_nb_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_nb_2))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (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_b $$unsorted) (V_a $$unsorted)) (= (tptp.c_Complex_Oexpi (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Complex_Ocomplex V_a V_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Complex_Ocomplex) (tptp.c_Complex_Oexpi V_a)) (tptp.c_Complex_Oexpi V_b)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_x_2 V_y_2)))) (and (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_y_2) V_x_2)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2))))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x_2 V_y_2) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))))) (forall ((V_x $$unsorted) (V_n $$unsorted)) (=> (not (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_n))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (= (tptp.c_RComplete_Onatceiling (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_x (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal V_n))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_RComplete_Onatceiling V_x) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_n)))))) (forall ((V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (= (tptp.c_RComplete_Onatfloor (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_x (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_RComplete_Onatfloor V_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted)) (not (tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n)))) (forall ((V_nb_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (= (tptp.hAPP _let_1 V_ma_2) (tptp.hAPP _let_1 V_nb_2)) (= V_ma_2 V_nb_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_n $$unsorted)) (= (tptp.c_RComplete_Onatfloor (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal V_n)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_n))) (not (tptp.c_Nat__Numeral_Oneg _let_14)) (not (tptp.c_Nat__Numeral_Oneg _let_12)) (forall ((V_x_2 $$unsorted)) (= (not (tptp.c_Nat__Numeral_Oneg V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x_2)))) (forall ((V_Z_2 $$unsorted)) (= (tptp.c_Nat__Numeral_Oneg V_Z_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_Z_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (= (tptp.c_RComplete_Onatfloor _let_28) _let_22) (forall ((V_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_RComplete_Onatfloor V_x)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_RComplete_Onatfloor V_x)) (tptp.c_RComplete_Onatfloor V_y))))) (= (tptp.c_RComplete_Onatfloor _let_1) _let_15) (forall ((V_x_2 $$unsorted) (V_nb_2 $$unsorted)) (=> (not (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_nb_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_nb_2)) (tptp.c_RComplete_Onatfloor V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal V_nb_2)) V_x_2)))))) (forall ((V_x $$unsorted) (V_n $$unsorted)) (=> (not (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_n))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (= (tptp.c_RComplete_Onatfloor (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_x (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal V_n))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_RComplete_Onatfloor V_x) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_n)))))) (forall ((V_v $$unsorted)) (=> (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v)) (= (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v))) (let ((_let_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v)))) (let ((_let_3 (tptp.c_Nat__Numeral_Oneg _let_1))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (= (tptp.c_RComplete_Onatfloor V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_v_H_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v_H_2))) (let ((_let_2 (tptp.c_Nat__Numeral_Oneg _let_1))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v_2))) (let ((_let_5 (tptp.c_Nat__Numeral_Oneg _let_4))) (= (= (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_2) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_H_2)) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) _let_3))) (=> (not _let_5) (and (=> _let_2 (= _let_4 _let_3)) (=> (not _let_2) (= V_v_2 V_v_H_2)))))))))))) (forall ((V_k $$unsorted) (V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_H) V_k))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat _let_2 _let_1))) (let ((_let_4 (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v_H)))) (let ((_let_5 (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v)))) (and (=> _let_5 (= _let_3 _let_1)) (=> (not _let_5) (and (=> _let_4 (= _let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat _let_2 V_k))) (=> (not _let_4) (= _let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_v V_v_H)) V_k)))))))))))) (forall ((V_x_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.c_RComplete_Onatfloor V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)) V_x_2)))) (forall ((V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v)))) (let ((_let_2 (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v)))) (=> (tptp.class_Int_Onumber__ring T_a) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_2) (= _let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a V_v)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_RComplete_Onatfloor V_a)) (tptp.c_RComplete_Onatfloor V_b))) (tptp.c_RComplete_Onatfloor (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_a) V_b)))))))) (forall ((V_x_2 $$unsorted) (V_nb_2 $$unsorted)) (=> (not (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_nb_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_RComplete_Onatceiling V_x_2)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_nb_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x_2) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal V_nb_2))))))) (forall ((V_x $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)))) (= (tptp.c_RComplete_Onatceiling V_x) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))))) (forall ((V_k_2 $$unsorted) (V_P_2 $$unsorted) (V_d_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_d_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint B_x V_d_2))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) V_k_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) V_d_2))))))))))) (forall ((V_k_2 $$unsorted) (V_P_2 $$unsorted) (V_d_2 $$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_d_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint B_x V_d_2))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) V_k_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) V_d_2))))))))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n)))) (forall ((V_n $$unsorted)) (= (tptp.c_RComplete_Onatfloor (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n)) V_n)) _let_33 _let_33 (= (tptp.hAPP _let_30 _let_22) _let_28) (forall ((V_nb_2 $$unsorted)) (= (= (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_nb_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) (= V_nb_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)))) (forall ((V_ma_2 $$unsorted) (V_nb_2 $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 V_nb_2) (tptp.hAPP _let_1 V_ma_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_nb_2 V_ma_2)))) (forall ((V_ma_2 $$unsorted) (V_nb_2 $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 V_nb_2)) (tptp.hAPP _let_1 V_ma_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_nb_2) V_ma_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (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_RealDef_Oreal) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))))) (forall ((V_ma_2 $$unsorted) (V_nb_2 $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (= (tptp.hAPP _let_1 V_nb_2) (tptp.hAPP _let_1 V_ma_2)) (= V_nb_2 V_ma_2)))) (forall ((V_n $$unsorted)) (= (tptp.c_RComplete_Onatceiling (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n)) V_n)) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n))))) _let_32 (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) (tptp.hAPP _let_1 V_m)) V_n) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) (tptp.hAPP _let_1 V_m)) V_n)))) (forall ((V_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x) (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) (tptp.c_RComplete_Onatceiling V_x))))) (forall ((V_nb_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_nb_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (= V_nb_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n) V_m)) (= (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n)))))) (forall ((V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) (tptp.c_RComplete_Onatfloor V_x))) V_x)))) (forall ((V_a $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_x)) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x) (tptp.c_RComplete_Onatfloor V_a))))) (forall ((V_a $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x) (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_RComplete_Onatceiling V_x)) V_a)))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_RComplete_Onatfloor V_x))) (=> (= V_x (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) _let_1)) (= (tptp.c_RComplete_Onatfloor (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) V_x) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) _let_1) V_n))))) (forall ((V_nb_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_nb_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_nb_2))) (forall ((V_ma_2 $$unsorted) (V_nb_2 $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_nb_2 V_ma_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 V_nb_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))) (tptp.hAPP _let_1 V_ma_2)))))) (forall ((V_ma_2 $$unsorted) (V_nb_2 $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_nb_2) V_ma_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 V_nb_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 V_ma_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)))))) (forall ((V_a_2 $$unsorted) (V_x_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_a_2) (tptp.c_RComplete_Onatfloor V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_a_2)) V_x_2))))) (forall ((V_a_2 $$unsorted) (V_x_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_RComplete_Onatceiling V_x_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x_2) (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_a_2)))))) (forall ((V_x $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) (tptp.c_RComplete_Onatfloor V_x)) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)))) (forall ((V_x $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal _let_1) V_x)) (= (tptp.c_RComplete_Onatfloor (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal V_x _let_1)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_RComplete_Onatfloor V_x) V_a))))) (forall ((V_x $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal V_x (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)) (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) (tptp.c_RComplete_Onatfloor V_x)))) (forall ((V_x $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal _let_1) V_x)) (= (tptp.c_RComplete_Onatceiling (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal V_x _let_1)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_RComplete_Onatceiling V_x) V_a))))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_RComplete_Onatfloor V_x) V_n)))) (forall ((V_v $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v)))) (let ((_let_2 (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v)))) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (not _let_2) (= _let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal V_v))))))) (forall ((V_a $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (= (tptp.c_RComplete_Onatfloor (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_x (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_a))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_RComplete_Onatfloor V_x) V_a)))) (forall ((V_n $$unsorted) (V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_RComplete_Onatfloor V_z) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_z (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n)))) (forall ((V_x $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal _let_1) V_x)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))) (= (tptp.c_RComplete_Onatfloor V_x) V_n))))) (forall ((V_a $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (= (tptp.c_RComplete_Onatceiling (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_x (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_a))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_RComplete_Onatceiling V_x) V_a)))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n)) V_x) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_x _let_1)) V_n)))))) (forall ((V_r $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_r)) (exists ((B_n $$unsorted)) (let ((_let_1 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.hAPP _let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat B_n (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) V_r)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_r (tptp.hAPP _let_1 B_n))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (V_r $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Complex_Ocomplex) (tptp.c_Complex_Orcis V_r V_a)) V_n) (tptp.c_Complex_Orcis (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) V_r) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n)) V_a)))) (forall ((V_a $$unsorted)) (= (tptp.c_Complex_Orcis (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Complex_Ocomplex))) (forall ((V_b $$unsorted) (V_r2 $$unsorted) (V_a $$unsorted) (V_r1 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Complex_Ocomplex) (tptp.c_Complex_Orcis V_r1 V_a)) (tptp.c_Complex_Orcis V_r2 V_b)) (tptp.c_Complex_Orcis (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_r1) V_r2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_a V_b)))) (forall ((V_X_2 $$unsorted) (V_M_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_RealVector_Oreal__normed__vector T_b) (tptp.class_Orderings_Oord T_a)) (=> (forall ((B_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_M_2) B_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_b (tptp.c_Groups_Ominus__class_Ominus T_b (tptp.hAPP V_X_2 V_M_2) (tptp.hAPP V_X_2 B_n))) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)))) (forall ((B_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_M_2) B_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_b (tptp.hAPP V_X_2 B_n)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal) (tptp.c_RealVector_Onorm__class_Onorm T_b (tptp.hAPP V_X_2 V_M_2))))))))) (forall ((V_X_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_SEQ_Odecseq T_a V_X_2) (forall ((B_m $$unsorted) (B_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat B_m) B_n)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP V_X_2 B_n)) (tptp.hAPP V_X_2 B_m)))))))) (forall ((V_k $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_k) (exists ((B_n $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) B_n) (= V_k (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_n)))))) (forall ((V_n $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v)) V_n))) (let ((_let_4 (tptp.c_Nat__Numeral_Oneg _let_1))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n))) (=> (not _let_4) (= _let_3 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) _let_1) V_n)))))))))) (= _let_15 (tptp.hAPP tptp.c_Int_Onat _let_14)) (forall ((V_z $$unsorted)) (=> (tptp.c_Nat__Numeral_Oneg V_z) (= (tptp.hAPP tptp.c_Int_Onat V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_w $$unsorted)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_w)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w))) (forall ((V_v $$unsorted)) (= (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v) (tptp.hAPP tptp.c_Int_Onat (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (= (= (tptp.hAPP tptp.c_Int_Onat V_x_2) (tptp.hAPP tptp.c_Int_Onat V_y_2)) (= V_x_2 V_y_2)))))) (forall ((V_P_2 $$unsorted)) (= (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP tptp.c_Int_Onat B_x))))))) (forall ((V_P_2 $$unsorted)) (= (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP tptp.c_Int_Onat B_x))))))) (forall ((V_z_H_2 $$unsorted) (V_za_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_za_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z_H_2)) (= (= (tptp.hAPP tptp.c_Int_Onat V_za_2) (tptp.hAPP tptp.c_Int_Onat V_z_H_2)) (= V_za_2 V_z_H_2)))))) (= _let_22 _let_29) (= _let_29 _let_22) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n)) V_n)) (forall ((V_z $$unsorted)) (=> (not (tptp.c_Nat__Numeral_Oneg V_z)) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_Onat V_z)) V_z))) (forall ((V_i_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Int_Onat V_i_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_i_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP tptp.c_Int_Onat V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_za_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_w_2) (tptp.hAPP tptp.c_Int_Onat V_za_2)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_za_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 V_za_2)))) (forall ((V_w_2 $$unsorted) (V_za_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_za_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_w_2) (tptp.hAPP tptp.c_Int_Onat V_za_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 V_za_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_x_2)) (tptp.hAPP tptp.c_Int_Onat V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_x_2) V_y_2))))))) (forall ((V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_Onat V_z)) V_z))) (forall ((V_za_2 $$unsorted) (V_ma_2 $$unsorted)) (= (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_ma_2) V_za_2) (and (= V_ma_2 (tptp.hAPP tptp.c_Int_Onat V_za_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_za_2))))) (forall ((V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_Onat V_z)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) V_z)))) (and (=> _let_3 (= _let_2 V_z)) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_za_2 $$unsorted) (V_ma_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_ma_2 (tptp.hAPP tptp.c_Int_Onat V_za_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_ma_2) V_za_2))) (forall ((V_za_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_za_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_za_2))) (forall ((V_za_2 $$unsorted) (V_w_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_w_2)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_w_2) (tptp.hAPP tptp.c_Int_Onat V_za_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 V_za_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_x_2) (tptp.hAPP tptp.c_Int_Onat V_y_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_x_2 V_y_2)))))) (forall ((V_za_2 $$unsorted) (V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 V_w_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) V_za_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_w_2)) (tptp.hAPP tptp.c_Int_Onat V_za_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_w_2) V_za_2)))))) (forall ((V_ma_2 $$unsorted) (V_w_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_w_2)))) (= (= (tptp.hAPP tptp.c_Int_Onat V_w_2) V_ma_2) (and (=> _let_1 (= V_w_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_ma_2))) (=> (not _let_1) (= V_ma_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_w_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_w_2)))) (= (= V_ma_2 (tptp.hAPP tptp.c_Int_Onat V_w_2)) (and (=> _let_1 (= V_w_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_ma_2))) (=> (not _let_1) (= V_ma_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_i_2 $$unsorted) (V_P_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP tptp.c_Int_Onat V_i_2))) (and (forall ((B_n $$unsorted)) (=> (= V_i_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_n)) (tptp.hBOOL (tptp.hAPP V_P_2 B_n)))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_i_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_z_H $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z_H)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_z V_z_H)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_z) (tptp.hAPP tptp.c_Int_Onat V_z_H))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_x) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_x V_y))))))) (forall ((V_z_H $$unsorted) (V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_z) V_z_H)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_z)) (tptp.hAPP tptp.c_Int_Onat V_z_H))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_x) V_y))))))) (forall ((V_v_2 $$unsorted) (V_ma_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v_2))) (= (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_ma_2) _let_1) (and (= V_ma_2 (tptp.hAPP tptp.c_Int_Onat _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) _let_1)))))) (forall ((V_z $$unsorted) (V_z_H $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_z_H) V_z)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_z V_z_H)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_z) (tptp.hAPP tptp.c_Int_Onat V_z_H)))))) (forall ((V_n $$unsorted) (V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_z) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_z)) V_n)))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) V_n) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x) V_n))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_x) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.c_Nat__Transfer_Otsub V_x V_y))))))) (forall ((V_ma_2 $$unsorted) (V_w_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_w_2)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_w_2) V_ma_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_ma_2))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_v))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v)) _let_1))) (let ((_let_5 (tptp.c_Nat__Numeral_Oneg _let_2))) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) _let_2) _let_1))))))))))) (forall ((V_n $$unsorted) (V_i $$unsorted)) (= (tptp.c_Int_Onat__aux V_i V_n) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_i) V_n))) (tptp.c_Nat__Transfer_Otransfer__morphism tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat _let_13) (forall ((V_n $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_x) (= (tptp.c_Transcendental_Oln (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) V_x) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n)) (tptp.c_Transcendental_Oln V_x))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_d $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_d) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_x V_z)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_d)) V_z))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x_2) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_y_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Transcendental_Oln V_x_2) (tptp.c_Transcendental_Oln V_y_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x_2 V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x_2) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_y_2) (= (= (tptp.c_Transcendental_Oln V_x_2) (tptp.c_Transcendental_Oln V_y_2)) (= V_x_2 V_y_2)))))) (forall ((V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_x) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Transcendental_Oln V_x) V_x))) (= (tptp.c_Transcendental_Oln _let_1) _let_28) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (= (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_m) V_n)) _let_1) (= (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint V_m) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (= (tptp.c_Groups_Oabs__class_Oabs T_a V_a) V_a)))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_a_2)) _let_1)) (= V_a_2 _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Groups_Oabs__class_Oabs T_a V_a))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (= (tptp.c_Groups_Oabs__class_Oabs T_a V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a V_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.hAPP _let_1 _let_2) _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_a)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oabs__class_Oabs T_a V_a)) (tptp.c_Groups_Oabs__class_Oabs T_a V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_a) (tptp.c_Groups_Oabs__class_Oabs T_a V_b)))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a _let_1) _let_1)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a _let_1) _let_1)))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oabs__class_Oabs T_a V_a)) V_n))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a V_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) (tptp.c_Groups_Oabs__class_Oabs T_a V_a))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_a)) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b)) (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_b V_a))))) (forall ((V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_n) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oabs__class_Oabs T_a V_x)) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_x) V_n)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_a) (tptp.c_Groups_Oabs__class_Oabs T_a V_b))) (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_b V_a)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_a) (tptp.c_Groups_Oabs__class_Oabs T_a V_b))) (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_a) (tptp.c_Groups_Oabs__class_Oabs T_a V_b)))) (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b))) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_a) (tptp.c_Groups_Oabs__class_Oabs T_a V_b)))))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a V_b))) (let ((_let_3 (tptp.c_Groups_Oabs__class_Oabs T_a V_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_3 V_c) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_2 V_d) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_1 _let_3) _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oabs__class_Oabs T_a V_a_2)) (not (= V_a_2 _let_1)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a V_a) V_a)))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m))) (= (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint _let_1) _let_1))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a _let_1) _let_1)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_2))) (=> (tptp.class_Rings_Oordered__ring__abs T_a) (=> (and (or (tptp.hBOOL (tptp.hAPP _let_3 V_a)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a) _let_2))) (or (tptp.hBOOL (tptp.hAPP _let_3 V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b) _let_2)))) (= (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oabs__class_Oabs T_a V_a)) (tptp.c_Groups_Oabs__class_Oabs T_a V_b))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_x)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oabs__class_Oabs T_a V_y)) V_x) (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.c_Groups_Oabs__class_Oabs T_a V_a)) V_n))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b))) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_a) (tptp.c_Groups_Oabs__class_Oabs T_a V_b)))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_d)))) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_c)) (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_b V_d))))))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oabs__class_Oabs T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_x_2 V_a_2)) V_r_2) (and (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_r_2) V_x_2) (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_r_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_n) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oabs__class_Oabs T_a V_x)) (tptp.c_Groups_Oabs__class_Oabs T_a V_y))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_n)))))))) (forall ((V_x $$unsorted) (V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_w))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat _let_1) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oabs__class_Oabs T_a V_x)) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1))))))) (forall ((V_za_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint V_za_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) (= V_za_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (forall ((V_z $$unsorted) (V_w $$unsorted)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w) V_z))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint V_w))) (tptp.hAPP tptp.c_Int_Onat (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint V_z))))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint V_x)) V_n)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x_2) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_y_2) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Transcendental_Oln V_x_2)) (tptp.c_Transcendental_Oln V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_x_2) V_y_2))))))) (forall ((V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) (tptp.c_Transcendental_Oln V_x))))) (forall ((V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal) V_x) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) (tptp.c_Transcendental_Oln V_x)))) (forall ((V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 (tptp.c_Transcendental_Oln V_x_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal) V_x_2))))) (forall ((V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x_2) (= (= (tptp.c_Transcendental_Oln V_x_2) _let_1) (= V_x_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)))))) (forall ((V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Transcendental_Oln V_x_2) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)))))) (forall ((V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Transcendental_Oln V_x) _let_1))))) (forall ((V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 (tptp.c_Transcendental_Oln V_x)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal) V_x))))) (forall ((V_nb_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint V_x_2)) V_nb_2)) (or (not (= V_x_2 _let_1)) (= V_nb_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x_2) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal _let_1) (tptp.c_Transcendental_Oln V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)) V_x_2)))))) (forall ((V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal _let_1) (tptp.c_Transcendental_Oln V_x))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)) V_x)))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_x) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal _let_1 V_y) (= (tptp.c_Transcendental_Oln (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_x) V_y)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_Transcendental_Oln V_x) (tptp.c_Transcendental_Oln V_y))))))) (forall ((V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal)) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Transcendental_Oln (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal) V_x))) V_x)))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_d $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) V_d) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_z (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_x V_z)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_d))))) (forall ((V_X_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (= (tptp.c_SEQ_OBseq T_a V_X_2) (exists ((B_K $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) B_K) (forall ((B_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP V_X_2 B_n))) B_K)))))))) (forall ((V_X_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_SEQ_Oincseq T_a V_X_2) (forall ((B_m $$unsorted) (B_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat B_m) B_n)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP V_X_2 B_m)) (tptp.hAPP V_X_2 B_n)))))))) (forall ((V_a $$unsorted) (V_r $$unsorted)) (= (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_Complex_Ocomplex (tptp.c_Complex_Orcis V_r V_a)) (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal V_r))) (forall ((V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_x))) (= (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal _let_1) _let_1))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_RealVector_Onorm__class_Onorm T_a V_a))) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (= (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal _let_1) _let_1)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_c) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal V_x)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) V_c) (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal V_y)))) (= V_x _let_1))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_RealVector_Oreal__normed__algebra__1 T_a)) (= (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_w)) (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal V_w))))) (forall ((V_x $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal V_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)) V_x))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a V_a) (tptp.c_RealVector_Onorm__class_Onorm T_a V_b)))) (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.c_Groups_Ominus__class_Ominus T_a V_a V_b)))))) (forall ((V_x $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal) (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal V_x)))) (forall ((V_X_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (=> (tptp.c_SEQ_OBseq T_a V_X_2) (exists ((B_K $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) B_K) (forall ((B_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP V_X_2 B_n))) B_K)))))))) (forall ((V_X_2 $$unsorted) (V_K_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_RealVector_Oreal__normed__vector T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) V_K_2) (=> (forall ((B_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_RealVector_Onorm__class_Onorm T_a (tptp.hAPP V_X_2 B_n))) V_K_2))) (tptp.c_SEQ_OBseq T_a V_X_2))))) (forall ((V_r $$unsorted)) (= (tptp.c_RealVector_Onorm__class_Onorm tptp.tc_RealDef_Oreal V_r) (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal V_r))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_a V_x) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal V_x V_b) (exists ((B_d $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal) B_d) (forall ((B_y $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal (tptp.c_Groups_Ominus__class_Ominus tptp.tc_RealDef_Oreal V_x B_y)) B_d) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal V_a) B_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal B_y) V_b)))))))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (V_nb_2 $$unsorted)) (=> (forall ((B_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_i V_nb_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint (tptp.hAPP V_f_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat B_i (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (tptp.hAPP V_f_2 B_i)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.hAPP V_f_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_k_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k_2) (tptp.hAPP V_f_2 V_nb_2))) (exists ((B_i $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat B_i) V_nb_2)) (= (tptp.hAPP V_f_2 B_i) V_k_2))))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (V_nb_2 $$unsorted)) (=> (forall ((B_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_i V_nb_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint (tptp.hAPP V_f_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat B_i (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (tptp.hAPP V_f_2 B_i)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.hAPP V_f_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_k_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k_2) (tptp.hAPP V_f_2 V_nb_2))) (exists ((B_i $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat B_i) V_nb_2)) (= (tptp.hAPP V_f_2 B_i) V_k_2))))))) (forall ((V_k $$unsorted) (V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_H)) V_k)))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_v tptp.c_Int_OPls))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_v_H))) V_k)))))))) (not (tptp.c_Nat__Numeral_Oneg _let_26)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint tptp.c_Int_OPls tptp.c_Int_OPls)) (forall ((V_k $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_k tptp.c_Int_OPls) V_k)) (forall ((V_k $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint tptp.c_Int_OPls V_k) V_k)) (= tptp.c_Int_OPls _let_12) (forall ((V_w $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) tptp.c_Int_OPls) V_w) tptp.c_Int_OPls)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint tptp.c_Int_OPls) tptp.c_Int_OPls)) (forall ((V_k $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_k tptp.c_Int_OPls) V_k)) (= _let_12 _let_26) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint tptp.c_Int_OPls _let_12)) (= _let_22 _let_27) (= _let_27 _let_22) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Int_Onumber__class_Onumber__of T_a tptp.c_Int_OPls) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Int_Onumber__class_Onumber__of T_a tptp.c_Int_OPls)) V_a))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Int_Onumber__class_Onumber__of T_a tptp.c_Int_OPls)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Int_Onumber__class_Onumber__of T_a tptp.c_Int_OPls) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_v_2 $$unsorted)) (= (= (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_v_2) tptp.c_Int_OPls)))) (forall ((V_v_2 $$unsorted)) (= (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_v_2) tptp.c_Int_OPls)))) (forall ((V_v_H_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_v_2 V_v_H_2))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_2) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_H_2)) (and (=> _let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint tptp.c_Int_OPls V_v_H_2)) _let_1)))) (forall ((V_v_H_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_v_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_2)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_H_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_v_H_2))) (tptp.hBOOL (tptp.hAPP _let_1 tptp.c_Int_OPls)))))) (tptp.hBOOL (tptp.hAPP _let_13 _let_26)) (forall ((V_y_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Int_Onumber__class_Onumber__of T_a V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint tptp.c_Int_OPls) V_y_2))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_x_2)) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_x_2) tptp.c_Int_OPls))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Int_Onumber__class_Onumber__of T_a V_x_2) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_x_2 tptp.c_Int_OPls)))) (forall ((V_y_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Int_Onumber__class_Onumber__of T_a V_y_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint tptp.c_Int_OPls V_y_2)))) (forall ((V_v_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint tptp.c_Int_OPls V_v_2))) (forall ((V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_H))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat _let_2 _let_1))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_v_H tptp.c_Int_OPls))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_v tptp.c_Int_OPls))) (and (=> _let_5 (= _let_3 _let_1)) (=> (not _let_5) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_v V_v_H))))))))))))) (forall ((V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v_H)))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_v tptp.c_Int_OPls))) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_2) (= _let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_v_H)))))))) _let_25 (forall ((V_v $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat V_v) _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_v tptp.c_Int_OPls))) (and (=> _let_3 (= _let_2 _let_1)) (=> (not _let_3) (= _let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_Osucc V_v))))))))) (forall ((V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))) (forall ((V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k_2) tptp.c_Int_OPls) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k_2 tptp.c_Int_OPls))) (forall ((V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k_2) tptp.c_Int_OPls) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k_2 tptp.c_Int_OPls))) (forall ((V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint tptp.c_Int_OPls (tptp.c_Int_OBit0 V_k_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint tptp.c_Int_OPls V_k_2))) (= (tptp.c_Int_OBit0 tptp.c_Int_OPls) tptp.c_Int_OPls) (= (tptp.c_Int_Osucc tptp.c_Int_OPls) _let_11) (forall ((V_l $$unsorted)) (not (= tptp.c_Int_OPls (tptp.c_Int_OBit1 V_l)))) (forall ((V_l_2 $$unsorted)) (= (= tptp.c_Int_OPls (tptp.c_Int_OBit0 V_l_2)) (= tptp.c_Int_OPls V_l_2))) (forall ((V_k $$unsorted)) (not (= (tptp.c_Int_OBit1 V_k) tptp.c_Int_OPls))) (forall ((V_k_2 $$unsorted)) (= (= (tptp.c_Int_OBit0 V_k_2) tptp.c_Int_OPls) (= V_k_2 tptp.c_Int_OPls))) (forall ((V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k_2)) tptp.c_Int_OPls)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k_2) tptp.c_Int_OPls)))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint tptp.c_Int_OPls))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint tptp.c_Int_OPls))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2))))) (forall ((V_l $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint tptp.c_Int_OPls (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit0 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint tptp.c_Int_OPls V_l)))) (tptp.hBOOL (tptp.hAPP _let_13 _let_18)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_22 _let_19) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (not (= V_a_2 _let_1)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2))) (or (not (= V_x_2 _let_1)) (not (= V_y_2 _let_1))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1)) (tptp.c_Groups_Ozero__class_Ozero T_a))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2))) _let_1)) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))))) _let_21 (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_RealDef_Oreal))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal))) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2)) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1))))))) (forall ((V_n $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_n))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_2) _let_2)))))) (forall ((V_w $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b V_w))) (=> (and (tptp.class_Groups_Omonoid__mult T_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) _let_1) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) _let_1) _let_1))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2)) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_3 V_x) _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_y) _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= V_x V_y))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x) V_y)))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))))) (forall ((V_nb_2 $$unsorted)) (= (tptp.c_Parity_Oeven__odd__class_Oeven tptp.tc_Nat_Onat V_nb_2) (exists ((B_m $$unsorted)) (= V_nb_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) B_m))))) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat _let_15 _let_15) _let_19) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_z) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_z V_z))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_z) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_z V_z))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_1) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_n)) _let_1))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1)))) (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_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1) (= V_a_2 _let_1))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a _let_1 _let_1) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))) (forall ((V_z $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.c_Groups_Oplus__class_Oplus T_a V_z V_z)))) (forall ((V_z $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_z) (tptp.c_Groups_Oplus__class_Oplus T_a V_z V_z)))) (forall ((V_x $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat _let_1))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (let ((_let_4 (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal))) (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal (tptp.c_Int_OBit0 _let_1))) (tptp.hAPP (tptp.hAPP _let_4 V_x) _let_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal _let_1)) V_x)) _let_2))))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_RealDef_Oreal (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)))) (= _let_18 (tptp.hAPP _let_20 _let_19)) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)))) (= _let_19 (tptp.hAPP tptp.c_Int_Onat _let_18)) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oabs__class_Oabs T_a V_a)) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Groups_Oabs__class_Oabs T_a _let_1) _let_1)))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_OBit1 V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_k) V_k))) (forall ((V_w_2 $$unsorted)) (= (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_w_2))) (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_w_2)))) (forall ((V_w_2 $$unsorted)) (= (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_w_2))) (tptp.c_Nat__Numeral_Oneg (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_w_2)))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint V_n))) (and (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_n)))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_n)))))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit0 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_k V_l)))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_k V_l)))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k) (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit0 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint V_k V_l)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k_2)) (tptp.c_Int_OBit0 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k_2) V_l_2)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k_2)) (tptp.c_Int_OBit1 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k_2) V_l_2)))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k1_2)) (tptp.c_Int_OBit0 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k1_2) V_k2_2)))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k1_2)) (tptp.c_Int_OBit1 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k1_2) V_k2_2)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k_2)) (tptp.c_Int_OBit1 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k_2) V_l_2)))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k1_2)) (tptp.c_Int_OBit1 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k1_2) V_k2_2)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (= (tptp.c_Int_OBit1 V_k_2) (tptp.c_Int_OBit1 V_l_2)) (= V_k_2 V_l_2))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (= (tptp.c_Int_OBit0 V_k_2) (tptp.c_Int_OBit0 V_l_2)) (= V_k_2 V_l_2))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (not (= (tptp.c_Int_OBit1 V_k) (tptp.c_Int_OBit0 V_l)))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (not (= (tptp.c_Int_OBit0 V_k) (tptp.c_Int_OBit1 V_l)))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Osucc (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit0 (tptp.c_Int_Osucc V_k)))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Osucc (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit1 V_k))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k)) V_l) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_OBit0 V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_k V_k))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit0 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_k V_l)))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k) (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_k V_l)))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_k V_l)))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k) (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit0 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_k (tptp.c_Int_Osucc V_l))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) V_l) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l)) V_l)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k_2) (tptp.c_Int_OBit0 V_l_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k_2 V_l_2))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k1_2) (tptp.c_Int_OBit0 V_k2_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k1_2 V_k2_2))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k_2) (tptp.c_Int_OBit0 V_l_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k_2 V_l_2))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k_2) (tptp.c_Int_OBit1 V_l_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k_2 V_l_2))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k1_2) (tptp.c_Int_OBit0 V_k2_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k1_2 V_k2_2))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k1_2) (tptp.c_Int_OBit1 V_k2_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k1_2 V_k2_2))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k_2) (tptp.c_Int_OBit1 V_l_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k_2) V_l_2)))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_k1_2) (tptp.c_Int_OBit1 V_k2_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint V_k1_2) V_k2_2)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k_2)) (tptp.c_Int_OBit0 V_l_2))) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k_2 V_l_2))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k1_2)) (tptp.c_Int_OBit0 V_k2_2))) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k1_2 V_k2_2))) (forall ((V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_w_2) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 _let_1)))) (forall ((V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint (tptp.c_Int_OBit0 V_w_2) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_w_2 _let_1)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat _let_1))) (let ((_let_4 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_Groups_Oplus__class_Oplus T_a V_x V_y)) _let_3) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_4 V_x) _let_3) (tptp.hAPP (tptp.hAPP _let_4 V_y) _let_3)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Int_Onumber__class_Onumber__of T_a _let_1)) V_x)) V_y))))))))) (forall ((V_k $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_k))) _let_1)) (= V_a _let_1))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)))))) (forall ((V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint _let_1) V_x)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_x (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (or (= V_x _let_1) (= V_x (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_RealDef_Oreal tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal) (tptp.hAPP _let_2 V_n)) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal _let_1)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat _let_1)) V_n)))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_RealDef_Oreal))) (let ((_let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat _let_1))) (let ((_let_4 (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal))) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal V_x V_y)) _let_3) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.c_Groups_Oplus__class_Oplus tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.hAPP _let_4 V_x) _let_3) (tptp.hAPP (tptp.hAPP _let_4 V_y) _let_3)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_RealDef_Oreal _let_1)) V_x)) V_y)))))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a V_w))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit0 V_w)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Ozero__class_Ozero T_a) _let_1) _let_1))))) (forall ((V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_RealDef_Oreal))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.c_Groups_Oabs__class_Oabs tptp.tc_RealDef_Oreal V_x) _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_RealDef_Oreal (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_RealDef_Oreal) V_x) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1)))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a V_w))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit1 V_w)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oone__class_Oone T_a) _let_1) _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit1 tptp.c_Int_OPls))) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit1 tptp.c_Int_OPls))) V_a))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit1 tptp.c_Int_OPls)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint tptp.c_Int_OPls (tptp.c_Int_OBit1 V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint tptp.c_Int_OPls) V_k_2)))) (forall ((V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Int_OBit1 V_k_2)) tptp.c_Int_OPls)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint V_k_2 tptp.c_Int_OPls))) (= _let_15 _let_16) (= _let_16 _let_15) (= _let_14 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint _let_11)) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat _let_1))) (let ((_let_4 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_Groups_Ominus__class_Ominus T_a V_x V_y)) _let_3) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP (tptp.hAPP _let_4 V_x) _let_3) (tptp.hAPP (tptp.hAPP _let_4 V_y) _let_3)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Int_Onumber__class_Onumber__of T_a _let_1)) V_x)) V_y))))))))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Osucc V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint V_k (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Int_Onat V_x)))) (=> (tptp.class_Rings_Osemiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x)) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_x))) (tptp.c_Groups_Oplus__class_Oplus T_a _let_2 _let_2))))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oplus__class_Oplus T_a _let_1 _let_1)) (tptp.c_Int_Onumber__class_Onumber__of T_a V_w)) (tptp.c_Int_Onumber__class_Onumber__of T_a (tptp.c_Int_OBit0 V_w)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_a)) V_a))))) (tptp.hBOOL (tptp.hAPP _let_13 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint (tptp.c_Int_OBit1 _let_11)))) (forall ((T_2 $$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)))) (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__1__strict tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__semiring__strict tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__ab__semigroup__add tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__ab__group__add__abs 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_Oordered__cancel__semiring 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_Rings_Oordered__comm__semiring tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__semiring__1 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__semiring 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_Oordered__semiring tptp.tc_Int_Oint) (tptp.class_Rings_Oordered__ring__abs 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 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_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_Rings_Omult__zero tptp.tc_Int_Oint) (tptp.class_Orderings_Oorder tptp.tc_Int_Oint) (tptp.class_Int_Oring__char__0 tptp.tc_Int_Oint) (tptp.class_Int_Onumber__ring tptp.tc_Int_Oint) (tptp.class_Rings_Osemiring tptp.tc_Int_Oint) (tptp.class_Orderings_Oord tptp.tc_Int_Oint) (tptp.class_Rings_Oring__1 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_Int_Onumber tptp.tc_Int_Oint) (tptp.class_Groups_Oone 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__ab__semigroup__add 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_Rings_Oordered__cancel__semiring tptp.tc_Nat_Onat) (tptp.class_Rings_Oordered__comm__semiring tptp.tc_Nat_Onat) (tptp.class_Groups_Ocancel__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Rings_Olinordered__semiring 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_Oordered__semiring 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 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_Power_Opower tptp.tc_Nat_Onat) (tptp.class_Groups_Ozero tptp.tc_Nat_Onat) (tptp.class_Int_Onumber tptp.tc_Nat_Onat) (tptp.class_Groups_Oone tptp.tc_Nat_Onat) (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_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le tptp.tc_RealDef_Oreal) (tptp.class_Rings_Olinordered__comm__semiring__strict tptp.tc_RealDef_Oreal) (tptp.class_Rings_Olinordered__semiring__1__strict tptp.tc_RealDef_Oreal) (tptp.class_RealVector_Oreal__normed__div__algebra tptp.tc_RealDef_Oreal) (tptp.class_Rings_Olinordered__semiring__strict tptp.tc_RealDef_Oreal) (tptp.class_RealVector_Oreal__normed__algebra__1 tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oordered__ab__semigroup__add tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oordered__ab__group__add__abs tptp.tc_RealDef_Oreal) (tptp.class_RealVector_Oreal__normed__algebra tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oordered__comm__monoid__add tptp.tc_RealDef_Oreal) (tptp.class_Groups_Olinordered__ab__group__add tptp.tc_RealDef_Oreal) (tptp.class_Groups_Ocancel__ab__semigroup__add tptp.tc_RealDef_Oreal) (tptp.class_Rings_Oring__1__no__zero__divisors tptp.tc_RealDef_Oreal) (tptp.class_Rings_Oordered__cancel__semiring tptp.tc_RealDef_Oreal) (tptp.class_RealVector_Oreal__normed__vector tptp.tc_RealDef_Oreal) (tptp.class_Rings_Olinordered__ring__strict tptp.tc_RealDef_Oreal) (tptp.class_Rings_Oring__no__zero__divisors tptp.tc_RealDef_Oreal) (tptp.class_Rings_Oordered__comm__semiring tptp.tc_RealDef_Oreal) (tptp.class_Rings_Olinordered__semiring__1 tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oordered__ab__group__add tptp.tc_RealDef_Oreal) (tptp.class_Groups_Ocancel__semigroup__add tptp.tc_RealDef_Oreal) (tptp.class_Rings_Olinordered__semiring tptp.tc_RealDef_Oreal) (tptp.class_Rings_Olinordered__semidom tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oab__semigroup__mult tptp.tc_RealDef_Oreal) (tptp.class_Groups_Ocomm__monoid__mult tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oab__semigroup__add tptp.tc_RealDef_Oreal) (tptp.class_Rings_Oordered__semiring tptp.tc_RealDef_Oreal) (tptp.class_Rings_Oordered__ring__abs tptp.tc_RealDef_Oreal) (tptp.class_Rings_Ono__zero__divisors tptp.tc_RealDef_Oreal) (tptp.class_Groups_Ocomm__monoid__add tptp.tc_RealDef_Oreal) (tptp.class_Rings_Olinordered__ring tptp.tc_RealDef_Oreal) (tptp.class_Rings_Olinordered__idom tptp.tc_RealDef_Oreal) (tptp.class_Rings_Ocomm__semiring__1 tptp.tc_RealDef_Oreal) (tptp.class_Rings_Ocomm__semiring tptp.tc_RealDef_Oreal) (tptp.class_Nat_Osemiring__char__0 tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oab__group__add tptp.tc_RealDef_Oreal) (tptp.class_Rings_Ozero__neq__one tptp.tc_RealDef_Oreal) (tptp.class_Rings_Oordered__ring tptp.tc_RealDef_Oreal) (tptp.class_Orderings_Opreorder tptp.tc_RealDef_Oreal) (tptp.class_Orderings_Olinorder tptp.tc_RealDef_Oreal) (tptp.class_Groups_Omonoid__mult tptp.tc_RealDef_Oreal) (tptp.class_Groups_Omonoid__add tptp.tc_RealDef_Oreal) (tptp.class_Rings_Osemiring__1 tptp.tc_RealDef_Oreal) (tptp.class_Rings_Osemiring__0 tptp.tc_RealDef_Oreal) (tptp.class_Groups_Ogroup__add tptp.tc_RealDef_Oreal) (tptp.class_Rings_Omult__zero tptp.tc_RealDef_Oreal) (tptp.class_Orderings_Oorder tptp.tc_RealDef_Oreal) (tptp.class_Int_Oring__char__0 tptp.tc_RealDef_Oreal) (tptp.class_Int_Onumber__ring tptp.tc_RealDef_Oreal) (tptp.class_Rings_Osemiring tptp.tc_RealDef_Oreal) (tptp.class_Orderings_Oord tptp.tc_RealDef_Oreal) (tptp.class_Rings_Oring__1 tptp.tc_RealDef_Oreal) (tptp.class_Power_Opower tptp.tc_RealDef_Oreal) (tptp.class_Groups_Ozero tptp.tc_RealDef_Oreal) (tptp.class_Rings_Oring tptp.tc_RealDef_Oreal) (tptp.class_Int_Onumber tptp.tc_RealDef_Oreal) (tptp.class_Groups_Oone tptp.tc_RealDef_Oreal) (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct tptp.tc_Complex_Ocomplex) (tptp.class_RealVector_Oreal__normed__div__algebra tptp.tc_Complex_Ocomplex) (tptp.class_RealVector_Oreal__normed__algebra__1 tptp.tc_Complex_Ocomplex) (tptp.class_RealVector_Oreal__normed__algebra tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Ocancel__ab__semigroup__add tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Oring__1__no__zero__divisors tptp.tc_Complex_Ocomplex) (tptp.class_RealVector_Oreal__normed__vector tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Oring__no__zero__divisors tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Ocancel__semigroup__add tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Oab__semigroup__mult tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Ocomm__monoid__mult tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Oab__semigroup__add tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Ono__zero__divisors tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Ocomm__monoid__add tptp.tc_Complex_Ocomplex) _let_10 (tptp.class_Rings_Ocomm__semiring tptp.tc_Complex_Ocomplex) (tptp.class_Nat_Osemiring__char__0 tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Oab__group__add tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Ozero__neq__one tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Omonoid__mult tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Omonoid__add tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Osemiring__1 tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Osemiring__0 tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Ogroup__add tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Omult__zero tptp.tc_Complex_Ocomplex) (tptp.class_Int_Oring__char__0 tptp.tc_Complex_Ocomplex) (tptp.class_Int_Onumber__ring tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Osemiring tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Oring__1 tptp.tc_Complex_Ocomplex) (tptp.class_Power_Opower tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Ozero tptp.tc_Complex_Ocomplex) (tptp.class_Rings_Oring tptp.tc_Complex_Ocomplex) (tptp.class_Int_Onumber tptp.tc_Complex_Ocomplex) (tptp.class_Groups_Oone tptp.tc_Complex_Ocomplex) _let_9 true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 1.63/1.85  )
% 1.63/1.85  % SZS output end Proof for SWW197+1
% 1.63/1.85  % cvc5---1.0.5 exiting
% 1.63/1.85  % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------