TSTP Solution File: SCT159+1 by cvc5---1.0.5
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%------------------------------------------------------------------------------
% File : cvc5---1.0.5
% Problem : SCT159+1 : TPTP v8.1.2. Released v5.2.0.
% Transfm : none
% Format : tptp
% Command : do_cvc5 %s %d
% Computer : n021.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 : Thu Aug 31 14:12:13 EDT 2023
% Result : Theorem 98.28s 98.49s
% Output : Proof 98.33s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.18/0.17 % Problem : SCT159+1 : TPTP v8.1.2. Released v5.2.0.
% 0.18/0.18 % Command : do_cvc5 %s %d
% 0.19/0.39 % Computer : n021.cluster.edu
% 0.19/0.39 % Model : x86_64 x86_64
% 0.19/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.19/0.39 % Memory : 8042.1875MB
% 0.19/0.39 % OS : Linux 3.10.0-693.el7.x86_64
% 0.19/0.39 % CPULimit : 300
% 0.19/0.39 % WCLimit : 300
% 0.19/0.39 % DateTime : Thu Aug 24 16:23:25 EDT 2023
% 0.19/0.40 % CPUTime :
% 0.26/0.58 %----Proving TF0_NAR, FOF, or CNF
% 98.28/98.49 ------- convert to smt2 : /export/starexec/sandbox/tmp/tmp.gyk0JRNesH/cvc5---1.0.5_9272.p...
% 98.28/98.49 ------- get file name : TPTP file name is SCT159+1
% 98.28/98.49 ------- cvc5-fof : /export/starexec/sandbox/solver/bin/cvc5---1.0.5_9272.smt2...
% 98.28/98.49 --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10...
% 98.28/98.49 --- Run --no-e-matching --full-saturate-quant at 5...
% 98.28/98.49 --- Run --no-e-matching --enum-inst-sum --full-saturate-quant at 5...
% 98.28/98.49 --- Run --finite-model-find --uf-ss=no-minimal at 5...
% 98.28/98.49 --- Run --multi-trigger-when-single --full-saturate-quant at 5...
% 98.28/98.49 --- Run --trigger-sel=max --full-saturate-quant at 5...
% 98.28/98.49 --- Run --multi-trigger-when-single --multi-trigger-priority --full-saturate-quant at 5...
% 98.28/98.49 --- Run --multi-trigger-cache --full-saturate-quant at 5...
% 98.28/98.49 --- Run --prenex-quant=none --full-saturate-quant at 5...
% 98.28/98.49 --- Run --enum-inst-interleave --decision=internal --full-saturate-quant at 5...
% 98.28/98.49 --- Run --relevant-triggers --full-saturate-quant at 5...
% 98.28/98.49 --- Run --finite-model-find --e-matching --sort-inference --uf-ss-fair at 5...
% 98.28/98.49 --- Run --pre-skolem-quant=on --full-saturate-quant at 10...
% 98.28/98.49 --- Run --cbqi-vo-exp --full-saturate-quant at 10...
% 98.28/98.49 --- Run --no-cbqi --full-saturate-quant at 10...
% 98.28/98.49 --- Run --macros-quant --macros-quant-mode=all --full-saturate-quant...
% 98.28/98.49 % SZS status Theorem for SCT159+1
% 98.28/98.49 % SZS output start Proof for SCT159+1
% 98.28/98.49 (
% 98.28/98.49 (let ((_let_1 (tptp.class_Rings_Ozero__neq__one tptp.tc_Nat_Onat))) (let ((_let_2 (forall ((V_k_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)) (exists ((B_b $$unsorted) (B_B $$unsorted)) (and (= V_A_2 (tptp.c_Set_Oinsert T_a B_b B_B)) (not (tptp.c_member T_a B_b B_B)) (= (tptp.c_Finite__Set_Ocard T_a B_B) V_k_2) (=> (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= B_B (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))))) (let ((_let_3 (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_b_2 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= V_b_2 V_a_2))))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.hAPP tptp.c_Nat_OSuc _let_4))) (let ((_let_6 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_7 (= _let_6 _let_5))) (let ((_let_8 (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Ozero__neq__one T_a) (not (= (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))))) (let ((_let_9 (tptp.tc_fun tptp.tc_Product__Type_Ounit tptp.tc_HOL_Obool))) (let ((_let_10 (tptp.c_Orderings_Otop__class_Otop _let_9))) (let ((_let_11 (tptp.c_Finite__Set_Ocard tptp.tc_Product__Type_Ounit _let_10))) (let ((_let_12 (= _let_11 _let_6))) (let ((_let_13 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (let ((_let_14 (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_x_2 V_A_2) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)))))) (let ((_let_15 (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (forall ((B_x $$unsorted)) (=> (tptp.c_member T_a B_x V_A_2) (forall ((B_xa $$unsorted)) (=> (tptp.c_member T_a B_xa V_A_2) (=> (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_f_2 B_xa)) (= B_x B_xa)))))))))) (let ((_let_16 (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_x_2))))) (let ((_let_17 (tptp.c_Orderings_Obot__class_Obot _let_9))) (let ((_let_18 (tptp.c_Set_Oinsert tptp.tc_Product__Type_Ounit SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_5 _let_17))) (let ((_let_19 (= _let_10 _let_18))) (let ((_let_20 (tptp.hBOOL (tptp.hAPP _let_10 _let_4)))) (let ((_let_21 (tptp.hBOOL (tptp.hAPP _let_18 _let_4)))) (let ((_let_22 (not _let_19))) (let ((_let_23 (or _let_22 (tptp.hBOOL (tptp.hAPP _let_17 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_5))))) (let ((_let_24 (forall ((B_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Product__Type_Ounit tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (or (not (= (tptp.c_Orderings_Otop__class_Otop _let_1) (tptp.c_Set_Oinsert tptp.tc_Product__Type_Ounit B_b _let_2))) (tptp.hBOOL (tptp.hAPP _let_2 B_b)))))))) (let ((_let_25 (not _let_23))) (let ((_let_26 (or (not (= _let_4 (tptp.c_Finite__Set_Ocard tptp.tc_Product__Type_Ounit _let_17))) _let_24))) (let ((_let_27 (not _let_24))) (let ((_let_28 (not _let_26))) (let ((_let_29 (= _let_5 _let_11))) (let ((_let_30 (= _let_29 _let_28))) (let ((_let_31 (forall ((V_k_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)) (not (forall ((B_b $$unsorted) (B_B $$unsorted)) (or (not (= V_A_2 (tptp.c_Set_Oinsert T_a B_b B_B))) (tptp.hBOOL (tptp.hAPP B_B B_b)) (not (= V_k_2 (tptp.c_Finite__Set_Ocard T_a B_B))) (and (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (not (= B_B (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))))))) (let ((_let_32 (AND_INTRO (TRUST_SUBS_EQ (ASSUME :args (_let_14)) :args ((= tptp.c_member (lambda ((BOUND_VARIABLE_10722 $$unsorted) (BOUND_VARIABLE_10723 $$unsorted) (BOUND_VARIABLE_10724 $$unsorted)) (tptp.hBOOL (tptp.hAPP BOUND_VARIABLE_10724 BOUND_VARIABLE_10723)))))) (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_15)) (MACRO_SR_EQ_INTRO :args (_let_15 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Fun_Oinj__on (lambda ((BOUND_VARIABLE_10047 $$unsorted) (BOUND_VARIABLE_10048 $$unsorted) (BOUND_VARIABLE_10049 $$unsorted) (BOUND_VARIABLE_10050 $$unsorted)) (forall ((B_x $$unsorted) (BOUND_VARIABLE_6751 $$unsorted)) (or (not (tptp.c_member BOUND_VARIABLE_10047 B_x BOUND_VARIABLE_10050)) (not (tptp.c_member BOUND_VARIABLE_10047 BOUND_VARIABLE_6751 BOUND_VARIABLE_10050)) (not (= (tptp.hAPP BOUND_VARIABLE_10049 B_x) (tptp.hAPP BOUND_VARIABLE_10049 BOUND_VARIABLE_6751))) (= B_x BOUND_VARIABLE_6751)))))))))) (let ((_let_33 (EQ_RESOLVE (ASSUME :args (_let_2)) (TRANS (MACRO_SR_EQ_INTRO :args (_let_2 SB_DEFAULT SBA_FIXPOINT)) (MACRO_SR_EQ_INTRO _let_32 :args ((forall ((V_k_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)) (not (forall ((B_b $$unsorted) (B_B $$unsorted)) (or (not (= V_A_2 (tptp.c_Set_Oinsert T_a B_b B_B))) (tptp.c_member T_a B_b B_B) (not (= V_k_2 (tptp.c_Finite__Set_Ocard T_a B_B))) (and (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_k_2) (not (= B_B (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))))) SB_DEFAULT SBA_FIXPOINT)))))) (let ((_let_34 (= _let_5 _let_6))) (let ((_let_35 (= _let_6 _let_11))) (let ((_let_36 (SYMM (ASSUME :args (_let_7))))) (let ((_let_37 (SYMM (ASSUME :args (_let_12))))) (let ((_let_38 (and _let_35 _let_34))) (let ((_let_39 (_let_35 _let_34))) (let ((_let_40 (or))) (let ((_let_41 (_let_27))) (let ((_let_42 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_23 0)) (CONG (REFL :args (_let_23)) (MACRO_SR_PRED_INTRO :args ((= (not _let_22) _let_19))) :args _let_40)) :args ((or _let_19 _let_23))) (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (IMPLIES_ELIM (SCOPE (SKOLEMIZE (ASSUME :args _let_41)) :args _let_41)) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_27) _let_24))) (REFL :args (_let_25)) :args _let_40)) (MACRO_RESOLUTION_TRUST (CNF_OR_NEG :args (_let_26 1)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_EQUIV_POS1 :args (_let_30)) :args ((or _let_28 (not _let_29) (not _let_30)))) (MACRO_RESOLUTION_TRUST (RESOLUTION (CNF_AND_NEG :args (_let_38)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_37 _let_36) (SCOPE (TRANS (SYMM (SYMM _let_36)) (SYMM (SYMM _let_37))) :args _let_39)) :args _let_39)) :args (true _let_38)) _let_37 _let_36 :args (_let_29 false _let_35 false _let_34)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_33 :args (_let_4 _let_10 tptp.tc_Product__Type_Ounit QUANTIFIERS_INST_E_MATCHING ((tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)))) :args (_let_31)))) _let_33 :args (_let_30 false _let_31)) :args (_let_28 false _let_29 false _let_30)) :args (_let_27 true _let_26)) :args (_let_25 true _let_24)) :args (_let_19 true _let_23)))) (let ((_let_43 (_let_16))) (let ((_let_44 (ASSUME :args _let_43))) (let ((_let_45 (= _let_4 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_5))) (let ((_let_46 (not _let_21))) (let ((_let_47 (or _let_46 _let_45))) (let ((_let_48 (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_b_2))) (= V_a_2 V_b_2))))) (let ((_let_49 (EQ_RESOLVE (ASSUME :args (_let_3)) (TRANS (MACRO_SR_EQ_INTRO :args (_let_3 SB_DEFAULT SBA_FIXPOINT)) (MACRO_SR_EQ_INTRO _let_32 :args ((forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (or (not (tptp.c_member T_a V_b_2 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (= V_a_2 V_b_2))) SB_DEFAULT SBA_FIXPOINT)))))) (let ((_let_50 (_let_48))) (let ((_let_51 (= _let_6 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_5))) (let ((_let_52 (= _let_4 _let_6))) (let ((_let_53 (not _let_45))) (let ((_let_54 (tptp.hBOOL (tptp.hAPP _let_18 _let_6)))) (let ((_let_55 (not _let_54))) (let ((_let_56 (or _let_55 _let_51))) (let ((_let_57 (tptp.hBOOL (tptp.hAPP _let_10 _let_6)))) (let ((_let_58 (and _let_19 _let_57))) (let ((_let_59 (ASSUME :args (_let_57)))) (let ((_let_60 (APPLY_UF tptp.hBOOL))) (let ((_let_61 (APPLY_UF tptp.hAPP))) (let ((_let_62 (ASSUME :args (_let_19)))) (let ((_let_63 (SYMM (SYMM _let_62)))) (let ((_let_64 (not _let_52))) (let ((_let_65 (not _let_1))) (let ((_let_66 (or _let_65 _let_64))) (let ((_let_67 (forall ((T_a $$unsorted)) (or (not (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))))))) (let ((_let_68 (EQ_RESOLVE (ASSUME :args (_let_8)) (MACRO_SR_EQ_INTRO :args (_let_8 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_69 (not _let_51))) (let ((_let_70 (and _let_64 _let_51))) (let ((_let_71 (_let_64 _let_51))) (let ((_let_72 (ASSUME :args (_let_64)))) (let ((_let_73 (ASSUME :args (_let_51)))) (let ((_let_74 (REFL :args (_let_4)))) (let ((_let_75 (ASSUME :args (_let_46)))) (let ((_let_76 (ASSUME :args (_let_20)))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (NOT_AND (MACRO_SR_PRED_TRANSFORM (SCOPE (AND_INTRO _let_75 _let_62 _let_76) :args (_let_19 _let_20 _let_46)) (SCOPE (MACRO_SR_PRED_ELIM (TRANS (SYMM (TRUE_INTRO _let_76)) (CONG (CONG _let_63 _let_74 :args _let_61) :args _let_60) (FALSE_INTRO _let_75))) :args (_let_46 _let_19 _let_20)) :args ((not (and _let_19 _let_20 _let_46)) SB_LITERAL))) (CONG (REFL :args (_let_22)) (REFL :args ((not _let_20))) (MACRO_SR_PRED_INTRO :args ((= (not _let_46) _let_21))) :args _let_40)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_47)) :args ((or _let_45 _let_46 (not _let_47)))) (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (RESOLUTION (CNF_AND_NEG :args (_let_70)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_72 _let_73) (SCOPE (FALSE_ELIM (TRANS (CONG _let_74 (SYMM _let_73) :args (=)) (FALSE_INTRO _let_72))) :args _let_71)) :args _let_71)) :args (true _let_70)) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_64) _let_52))) (REFL :args (_let_69)) (REFL :args (_let_53)) :args _let_40)) :args ((or _let_52 _let_53 _let_69))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_66)) :args ((or _let_64 _let_65 (not _let_66)))) (ASSUME :args (_let_1)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_68 :args (tptp.tc_Nat_Onat QUANTIFIERS_INST_CBQI_PROP)) :args (_let_67)))) _let_68 :args (_let_66 false _let_67)) :args (_let_64 false _let_1 false _let_66)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_56)) :args ((or _let_55 _let_51 (not _let_56)))) (MACRO_RESOLUTION_TRUST (RESOLUTION (CNF_AND_NEG :args (_let_58)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_59 _let_62) (SCOPE (TRUE_ELIM (TRANS (CONG (CONG (SYMM _let_63) (REFL :args (_let_6)) :args _let_61) :args _let_60) (TRUE_INTRO _let_59))) :args (_let_57 _let_19))) :args (_let_19 _let_57))) :args (true _let_58)) _let_42 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_44 :args (_let_6 tptp.tc_Product__Type_Ounit QUANTIFIERS_INST_CBQI_CONFLICT)) :args _let_43)) _let_44 :args (_let_57 false _let_16)) :args (_let_54 false _let_19 false _let_57)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_49 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_5 _let_6 tptp.tc_Product__Type_Ounit QUANTIFIERS_INST_CBQI_CONFLICT)) :args _let_50))) _let_49 :args (_let_56 false _let_48)) :args (_let_51 false _let_54 false _let_56)) :args (_let_53 true _let_52 false _let_51)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_49 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_5 _let_4 tptp.tc_Product__Type_Ounit QUANTIFIERS_INST_CBQI_CONFLICT)) :args _let_50))) _let_49 :args (_let_47 false _let_48)) :args (_let_46 true _let_45 false _let_47)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_44 :args (_let_4 tptp.tc_Product__Type_Ounit QUANTIFIERS_INST_CBQI_CONFLICT)) :args _let_43)) _let_44 :args (_let_20 false _let_16)) _let_42 :args (false true _let_21 false _let_20 false _let_19)) :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))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (= (tptp.c_Finite__Set_Ocard T_b (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) (tptp.c_Finite__Set_Ocard T_a V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2) (tptp.c_Set_Oimage T_a T_b V_f_2 V_B_2)) (= V_A_2 V_B_2)))) _let_16 (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Otop T_a) (= (tptp.hAPP (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b T_a)) V_x_2) (tptp.c_Orderings_Otop__class_Otop T_a)))) _let_15 (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))))) (tptp.c_Arrow__Order__Mirabelle_OIIA tptp.v_F) (tptp.c_Arrow__Order__Mirabelle_Ounanimity tptp.v_F) (tptp.c_Finite__Set_Ofinite tptp.tc_Arrow__Order__Mirabelle_Oindi (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Arrow__Order__Mirabelle_Oindi tptp.tc_HOL_Obool))) (forall ((V_h_2 $$unsorted) (T_b $$unsorted) (V_Fa_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_Fa_2) (tptp.c_Finite__Set_Ofinite T_b (tptp.c_Set_Oimage T_a T_b V_h_2 V_Fa_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_a) (tptp.c_Finite__Set_Ofinite T_a V_A_2))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (= (tptp.c_Finite__Set_Ofinite _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Finite__Set_Ofinite T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_sum T_a T_b))) (= (tptp.c_Finite__Set_Ofinite _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (and (tptp.c_Finite__Set_Ofinite T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Finite__Set_Ofinite T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_a) (tptp.c_Finite__Set_Ofinite T_a V_A_2))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (=> (tptp.c_Finite__Set_Ofinite _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Finite__Set_Ofinite T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.c_Finite__Set_Ofinite T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofinite T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Finite__Set_Ofinite _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_a) (tptp.c_Finite__Set_Ofinite T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.c_Finite__Set_Ofinite T_a V_A_2)) (= (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a (tptp.c_Set_Oimage T_b T_a V_f_2 V_A_2)) (=> (tptp.c_Fun_Oinj__on T_b T_a V_f_2 V_A_2) (tptp.c_Finite__Set_Ofinite T_b V_A_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Finite__Set_Ofinite T_a _let_1) (=> (= (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Finite__Set_Ocard T_a _let_1)) (= V_A_2 _let_1))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Finite__Set_Ofinite T_a _let_1) (=> (= (tptp.c_Set_Oimage T_a T_a V_f_2 _let_1) _let_1) (tptp.c_Fun_Oinj__on T_a T_a V_f_2 _let_1))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Finite__Set_Ofinite T_a _let_1) (=> (tptp.c_Fun_Oinj__on T_a T_a V_f_2 _let_1) (= (tptp.c_Set_Oimage T_a T_a V_f_2 _let_1) _let_1))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (= (tptp.c_Finite__Set_Ocard T_b (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) (tptp.c_Finite__Set_Ocard T_a V_A_2)) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (= (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (= (tptp.c_Finite__Set_Ocard T_b (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) (tptp.c_Finite__Set_Ocard T_a V_A_2))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2) V_B_2) (=> (tptp.c_Fun_Oinj__on T_b T_a V_g_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Set_Oimage T_b T_a V_g_2 V_B_2) V_A_2) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (tptp.c_Finite__Set_Ofinite T_b V_B_2) (= (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Finite__Set_Ocard T_b V_B_2))))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Finite__Set_Ofinite T_a _let_1) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_Finite__Set_Ocard T_a _let_1))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (= (tptp.c_Set_Oimage T_b T_a (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) V_A_2))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (tptp.c_Fun_Oinj__on T_b T_a (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_a T_b _let_1 V_f_2) (tptp.hAPP V_f_2 V_x_2)) V_x_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (or (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.c_Finite__Set_Ofinite T_a V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_a (tptp.c_Set_Oimage T_b T_a V_f_2 V_A_2)) (tptp.c_Finite__Set_Ocard T_b V_A_2)) (not (tptp.c_Fun_Oinj__on T_b T_a V_f_2 V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_1)) (= (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2) (tptp.c_Set_Oimage T_a T_b V_f_2 V_B_2)) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_member T_b (tptp.hAPP V_f_2 V_a_2) (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) (tptp.c_member T_a V_a_2 V_A_2)))) (forall ((V_b_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 _let_1) (= (tptp.c_member T_b V_b_2 (tptp.c_Set_Oimage T_a T_b V_f_2 _let_1)) (exists ((B_x $$unsorted)) (and (= V_b_2 (tptp.hAPP V_f_2 B_x)) (forall ((B_y $$unsorted)) (=> (= V_b_2 (tptp.hAPP V_f_2 B_y)) (= B_y B_x))))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_b (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) (tptp.c_Finite__Set_Ocard T_a V_A_2)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_x))) (forall ((T_a $$unsorted)) (tptp.c_Finite__Set_Ofinite T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_x)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Obot T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Orderings_Obot__class_Obot T_a) V_x))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Obot T_a) (= (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b T_a)) V_x_2) (tptp.c_Orderings_Obot__class_Obot T_a)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2 V_x_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.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_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (= V_x_2 V_y_2)) (or (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (tptp.c_Orderings_Oord__class_Oless T_a V_y_2 V_x_2))))) (forall ((V_y_2 $$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_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b) V_f_2 V_g_2) (forall ((B_x $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq T_b (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.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 $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_x)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (and (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 $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (and (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2 V_x_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= V_x_2 V_y_2) (and (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (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 $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (or (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (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 $$unsorted)) (= (tptp.c_member T_a V_x_2 (tptp.c_fequal V_y_2)) (= V_x_2 V_y_2))) (forall ((V_y $$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.c_Orderings_Oord__class_Oless__eq T_a V_y V_x)))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_x)) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)))) (forall ((V_x_2 $$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_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (= 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(tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) (tptp.hAPP V_f_2 V_x_2)) V_x_2)))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.c_member T_b V_x_2 (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (tptp.c_member T_a (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) V_x_2) V_B_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_a V_B_2) (tptp.c_Finite__Set_Ocard T_a V_A_2)) (= V_A_2 V_B_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_B_2) (=> 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tptp.tc_HOL_Obool)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (not (= V_x_2 V_y_2)) (=> (tptp.c_member T_a V_x_2 V_A_2) (=> (tptp.c_member T_a V_y_2 V_A_2) (not (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.c_member T_a V_x_2 V_A_2) (=> (tptp.c_member T_a V_y_2 V_A_2) (= (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (= V_x_2 V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (=> (tptp.c_member T_a V_x_2 V_A_2) (=> (tptp.c_member T_a V_y_2 V_A_2) (= V_x_2 V_y_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2)))) (forall ((V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.c_member T_b V_y_2 (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) (= (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) V_y_2)) V_y_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_Finite__Set_Ocard T_a V_A_2)) (and (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.c_Finite__Set_Ofinite T_a V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2) V_B_2) (=> (tptp.c_Finite__Set_Ofinite T_b V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Finite__Set_Ocard T_b V_B_2)))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) V_B_2 (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) (tptp.c_Finite__Set_Ofinite T_b V_B_2)))) (forall ((T_a $$unsorted)) (= (tptp.c_Finite__Set_Ocard T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_Finite__Set_Ocard T_a V_A_2)) (tptp.c_Finite__Set_Ofinite T_a V_A_2))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 (tptp.c_Set_Oimage T_a T_a V_f_2 V_A_2)) (tptp.c_Fun_Oinj__on T_a T_a V_f_2 V_A_2)))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_a V_f_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) _let_1 V_A_2) (=> (tptp.c_Fun_Oinj__on T_a T_a V_f_2 V_A_2) (= _let_1 V_A_2)))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (tptp.c_Finite__Set_Ofinite T_b V_B_2) (= (exists ((B_f $$unsorted)) (and (tptp.c_Fun_Oinj__on T_a T_b B_f V_A_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Set_Oimage T_a T_b B_f V_A_2) V_B_2))) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Finite__Set_Ocard T_b V_B_2)))))) (forall ((V_n $$unsorted)) (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 $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Orderings_Obot__class_Obot _let_1) V_A_2))) (forall ((T_a $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_b_2 $$unsorted)) (=> (= V_b_2 (tptp.hAPP V_f_2 V_x_2)) (=> (tptp.c_member T_b V_x_2 V_A_2) (tptp.c_member T_a V_b_2 (tptp.c_Set_Oimage T_b T_a V_f_2 V_A_2))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (tptp.c_member T_a V_x_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_c_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (=> (tptp.c_member T_a V_c_2 V_A_2) (tptp.c_member T_a V_c_2 V_B_2)))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (not (tptp.c_member T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_A_H_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (exists ((B_f $$unsorted)) (and (tptp.c_Fun_Oinj__on T_a T_b B_f V_A_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Set_Oimage T_a T_b B_f V_A_2) V_A_H_2))) (exists ((B_g $$unsorted)) (= (tptp.c_Set_Oimage T_b T_a B_g V_A_H_2) V_A_2))))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted)) (= (exists ((B_m $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_m V_n_2) (tptp.hBOOL (tptp.hAPP V_P_2 B_m)))) (exists ((B_x $$unsorted)) (and (tptp.c_member tptp.tc_Nat_Onat B_x (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_c_2 $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_member T_a V_c_2 V_B_2))) (let ((_let_2 (tptp.c_member T_a V_c_2 V_A_2))) (=> (= V_A_2 V_B_2) (=> (=> _let_2 (not _let_1)) (not (=> (not _let_2) _let_1))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2) (= V_A_2 V_B_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_l_2 V_u_2))) (forall ((V_c_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (=> (tptp.c_member T_a V_c_2 V_A_2) (tptp.c_member T_a V_c_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_B_2) (and (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (not (= V_A_2 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (or (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_B_2) (= V_A_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_C_2) (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_C_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless _let_1 V_B_2 V_C_2) (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_C_2))))) (= (tptp.c_Orderings_Obot__class_Obot tptp.tc_Nat_Onat) _let_4) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.c_Orderings_Oord__class_Oless _let_1 V_f_2 V_g_2) (and (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_f_2 V_g_2) (not (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_g_2 V_f_2))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2) (tptp.c_Set_Oimage T_a T_b V_f_2 V_B_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Finite__Set_Ocard T_a V_B_2))))) _let_14 (forall ((V_x_2 $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (= (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_member T_a V_x_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (=> (= V_x_2 V_y_2) (= (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_member T_a V_y_2 V_A_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (=> (= V_a_2 V_b_2) (=> (tptp.c_member T_a V_b_2 V_A_2) (tptp.c_member T_a V_a_2 V_A_2)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_n))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (not (= V_m_2 V_n_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (not (= V_x V_y)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x V_y)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_y V_x)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_n))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_m) (not (= V_m V_n)))) (forall ((V_t $$unsorted) (V_s $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_s V_t) (not (= V_s V_t)))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_n_2) V_m_2)))) (=> (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) _let_1) (=> (=> (= V_m_2 V_n_2) _let_1) (=> (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2) _let_1) _let_1))))) (forall ((V_N_2 $$unsorted)) (= (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat V_N_2) (exists ((B_m $$unsorted)) (forall ((B_x $$unsorted)) (=> (tptp.c_member tptp.tc_Nat_Onat B_x V_N_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_x B_m)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_A_2)) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= V_A_2 V_B_2) (and (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_B_2 V_A_2))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_C_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= V_A_2 V_B_2) (not (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (not (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2))))))) (forall ((V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n V_n)) (forall ((V_n $$unsorted) (V_m $$unsorted)) (or (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (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.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_j V_k) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i V_k)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n V_m) (= V_m V_n)))) (forall ((V_N_2 $$unsorted)) (= (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat V_N_2) (exists ((B_m $$unsorted)) (forall ((B_x $$unsorted)) (=> (tptp.c_member tptp.tc_Nat_Onat B_x V_N_2) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat B_x B_m)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofinite T_a V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Finite__Set_Ocard T_a V_B_2)) (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_B_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (not (tptp.c_member T_a B_x V_A_2))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (exists ((B_x $$unsorted)) (tptp.c_member T_a B_x V_A_2)) (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_c_2 $$unsorted) (T_a $$unsorted)) (not (tptp.c_member T_a V_c_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.c_member T_a V_a_2 V_A_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (=> (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_member T_a V_x_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_x_2 V_A_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (tptp.c_member T_a V_x_2 V_B_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (=> (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_member T_a V_x_2 V_B_2)))) (forall ((T_b $$unsorted) (V_f_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_x_2 V_A_2) (=> (= V_b_2 (tptp.hAPP V_f_2 V_x_2)) (tptp.c_member T_b V_b_2 (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_member T_b (tptp.hAPP V_f_2 V_x_2) (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_z_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_z_2 (tptp.c_Set_Oimage T_b T_a V_f_2 V_A_2)) (exists ((B_x $$unsorted)) (and (tptp.c_member T_b B_x V_A_2) (= V_z_2 (tptp.hAPP V_f_2 B_x)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 _let_2) (= V_A_2 _let_2))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (= (tptp.c_Orderings_Otop__class_Otop _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Set_Oimage T_b T_a V_f_2 V_A_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Set_Oimage T_b T_a V_f_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Set_Oimage T_b T_a V_f_2 V_A_2)) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not (= V_n _let_1)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (not (= V_n_2 _let_1)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n_2)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2) (tptp.c_Set_Oimage T_a T_b V_f_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_B_2 (tptp.c_Set_Oimage T_b T_a V_f_2 V_A_2)) (exists ((B_AA $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) B_AA V_A_2) (= V_B_2 (tptp.c_Set_Oimage T_b T_a V_f_2 B_AA)))))) (forall ((V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n_2 _let_1) (= V_n_2 _let_1)))) (forall ((V_n $$unsorted)) (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_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (and (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2) (not (= V_m_2 V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (= V_m_2 V_n_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (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)) (=> (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (= V_m V_n)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (= (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2 V_b_2) (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2 V_d_2)) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_b_2) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 V_d_2) (= V_b_2 V_d_2)))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (= (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2 V_b_2) (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2 V_d_2)) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_b_2) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 V_d_2) (= V_a_2 V_c_2)))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_b_2) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 V_d_2) (= (= (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2 V_b_2) (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2 V_d_2)) (and (= V_a_2 V_c_2) (= V_b_2 V_d_2))))))) (forall ((V_m_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_b_2 $$unsorted)) (=> (= V_b_2 (tptp.hAPP V_f_2 V_x_2)) (tptp.c_member T_a V_b_2 (tptp.c_Set_Oimage T_b T_a V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((T_b $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (tptp.c_member T_a (tptp.hAPP V_f_2 V_x_2) (tptp.c_Set_Oimage T_b T_a V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b_2 V_a_2) (= (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2 V_b_2)) (not (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_b_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_b_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2 V_b_2) (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2 V_d_2)) (or (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b_2 V_a_2) (and (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c_2 V_a_2) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b_2 V_d_2)))))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted)) (= (forall ((B_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_m V_n_2) (tptp.hBOOL (tptp.hAPP V_P_2 B_m)))) (forall ((B_x $$unsorted)) (=> (tptp.c_member tptp.tc_Nat_Onat B_x (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_k_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_i_2 (tptp.c_SetInterval_Oord_OatMost T_a V_less__eq_2 V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_i_2) V_k_2)))) (forall ((V_k_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_i_2 (tptp.c_SetInterval_Oord_OatLeast T_a V_less__eq_2 V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_k_2) V_i_2)))) (forall ((V_k_2 $$unsorted) (V_less_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_i_2 (tptp.c_SetInterval_Oord_OlessThan T_a V_less_2 V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_i_2) V_k_2)))) (forall ((V_k_2 $$unsorted) (V_less_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_i_2 (tptp.c_SetInterval_Oord_OgreaterThan T_a V_less_2 V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_k_2) V_i_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_i_2 (tptp.c_SetInterval_Oord_OatLeastAtMost T_a V_less__eq_2 V_l_2 V_u_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_l_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_i_2) V_u_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_i_2 (tptp.c_SetInterval_Oord_OgreaterThanLessThan T_a V_less_2 V_l_2 V_u_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_l_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_i_2) V_u_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless _let_1 V_B_2 V_C_2) (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_C_2))))) (not (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat _let_13)) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_i_2 (tptp.c_SetInterval_Oord_OgreaterThanAtMost T_a V_less__eq_2 V_less_2 V_l_2 V_u_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_l_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_i_2) V_u_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_i_2 (tptp.c_SetInterval_Oord_OatLeastLessThan T_a V_less__eq_2 V_less_2 V_l_2 V_u_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_l_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_i_2) V_u_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x V_y) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x V_y))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (tptp.c_member T_a V_x_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (=> (tptp.c_Finite__Set_Ofinite _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (=> (not (= (tptp.c_Finite__Set_Ocard T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (tptp.c_Finite__Set_Ofinite T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_Fa_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_Fa_2 V_A_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_Fa_2) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_2 V_B_2 V_A_2) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_Fa_2 V_B_2)) _let_1) _let_1)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_Fa_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_Fa_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_a T_b V_f_2 V_z_2 V_g_2 V_Fa_2) (=> (tptp.c_Finite__Set_Ofinite T_b V_A_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) V_B_2 V_A_2) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_Fa_2 V_B_2)) _let_1) _let_1)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Fa_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_Fa_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_a T_b V_f_2 V_z_2 V_g_2 V_Fa_2) (=> (tptp.c_Finite__Set_Ofinite T_b V_A_2) (=> (tptp.c_member T_b V_x_2 V_A_2) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) _let_1) _let_1)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.hAPP tptp.c_Nat_OSuc V_n))) (forall ((V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n))) (forall ((V_n $$unsorted)) (not (= V_n (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_n) V_n))) (forall ((V_nat_H_2 $$unsorted) (V_nat_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Nat_OSuc V_nat_2) (tptp.hAPP tptp.c_Nat_OSuc V_nat_H_2)) (= V_nat_2 V_nat_H_2))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (= (tptp.hAPP tptp.c_Nat_OSuc V_x) (tptp.hAPP tptp.c_Nat_OSuc V_y)) (= V_x V_y))) (forall ((V_N_2 $$unsorted)) (tptp.c_Fun_Oinj__on tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc V_N_2)) (forall ((V_x_2 $$unsorted) (V_Fa_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_Fa_2) (= (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_x_2) V_x_2))) (forall ((V_x_2 $$unsorted) (V_Fa_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_a T_b V_f_2 V_z_2 V_g_2 V_Fa_2) (= (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_x_2) V_x_2))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_i_2 V_j_2)) (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_i_2) (tptp.hAPP tptp.c_Nat_OSuc V_j_2)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_n) V_n))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (not (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_n_2) V_m_2))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (= (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 _let_1) (or (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2) (= V_m_2 _let_1))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_n_2) (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n_2 V_m_2))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n))) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m _let_1) (=> (not (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n)) (= V_m _let_1))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 (tptp.hAPP tptp.c_Nat_OSuc V_m_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (= V_m_2 V_n_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (= V_n_2 V_m_2)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_m)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n (tptp.hAPP tptp.c_Nat_OSuc V_m)) (= V_m V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (=> (not (= _let_1 V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_j V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_i) V_k)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc V_n)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)) (= V_m V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n))) (forall ((V_m $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)))) (forall ((V_nat_H_1 $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_nat_H_1) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_nat_H $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_nat_H)))) (forall ((V_m $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)))) (forall ((V_n_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2) (exists ((B_m $$unsorted)) (= V_n_2 (tptp.hAPP tptp.c_Nat_OSuc B_m))))) (forall ((V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1)) (= V_n_2 _let_1)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (or (= V_m_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (exists ((B_j $$unsorted)) (and (= V_m_2 (tptp.hAPP tptp.c_Nat_OSuc B_j)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_j V_n_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (= V_n_2 V_m_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2) V_n_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_n_2) V_m_2))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Fa_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_Fa_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_Fa_2) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (tptp.c_member T_a V_x_2 V_A_2) (= (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) _let_1) _let_1)))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (V_Fa_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_Fa_2) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP V_f_2 B_x) B_y)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y)))) (=> (tptp.c_Finite__Set_Ofinite T_a V_N_2) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.hAPP V_Fa_2 V_N_2)) (tptp.hAPP V_Fa_2 (tptp.c_Set_Oimage T_a T_a V_h_2 V_N_2)))))))) (= (tptp.c_SetInterval_Oord__class_OgreaterThan tptp.tc_Nat_Onat _let_4) (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc _let_13)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_a T_b) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2)) (tptp.c_Finite__Set_Ofinite T_b V_B_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (= (tptp.c_SetInterval_Oord__class_OgreaterThan T_a V_x_2) (tptp.c_SetInterval_Oord__class_OgreaterThan T_a V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (= (tptp.c_member T_a V_i_2 (tptp.c_SetInterval_Oord__class_OgreaterThan T_a V_k_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_k_2 V_i_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_SetInterval_Oord__class_OgreaterThan T_a V_x_2) (tptp.c_SetInterval_Oord__class_OgreaterThan T_a V_y_2)) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2 V_x_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_a T_b) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2)) (and (tptp.c_Finite__Set_Ofinite T_a V_A_2) (tptp.c_Finite__Set_Ofinite T_b V_B_2)))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (tptp.c_Finite__Set_Ofinite T_b V_B_2) (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_a T_b) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_a T_b) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2)) (tptp.c_Finite__Set_Ofinite T_a V_A_2))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Sum__Type_OPlus T_a T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun (tptp.tc_sum T_a T_b) tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_sum T_a T_b) tptp.tc_HOL_Obool))) (and (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_B_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard (tptp.tc_sum T_a T_b) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2)))) (let ((_let_2 (and (tptp.c_Finite__Set_Ofinite T_a V_A_2) (tptp.c_Finite__Set_Ofinite T_b V_B_2)))) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Finite__Set_Ocard T_b V_B_2)))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_z $$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 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x V_z) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y V_z)))) (forall ((V_y $$unsorted) (V_z $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x V_z) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y V_z)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x V_y) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_x (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_y V_z)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_m))) (forall ((V_z $$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_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n) V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_k)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_m_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_n_2)) (= V_m_2 V_n_2))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m_2 V_k_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n_2 V_k_2)) (= V_m_2 V_n_2))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__semigroup__add T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_c))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_b_2) (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_c_2)) (= V_b_2 V_c_2)))) (forall ((V_c_2 $$unsorted) (V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a V_b_2 V_a_2) (tptp.c_Groups_Oplus__class_Oplus T_a V_c_2 V_a_2)) (= V_b_2 V_c_2)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)) (= V_b V_c)))) (forall ((V_c $$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_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_a) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_a)) (= V_b V_c)))) (forall ((V_k_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_k_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat V_A_2))))) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) V_A_2 _let_1) (= V_A_2 _let_1)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n V_m))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2) (exists ((B_k $$unsorted)) (= V_n_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m_2 B_k))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_m_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_n_2)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i V_j) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_m)))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i V_j) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_j)))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i V_j) (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.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k V_l) (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 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_k) V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k V_n))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (=> (tptp.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.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_k) V_n) (not (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (not (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k V_n)))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_j) V_i))) (forall ((V_i $$unsorted) (V_j $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_i) V_i))) (forall ((V_n_2 $$unsorted) (V_m_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_m_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k_2 V_n_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_m)))) (forall ((V_m $$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_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_k)))) (forall ((V_l $$unsorted) (V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_k V_l) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_k) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_j V_l))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_l $$unsorted) (V_k $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_k V_l) (=> (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_l) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_k V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_k $$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)) (=> (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n) V_m) (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m_2 V_n_2) _let_1) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_m $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (forall ((V_n $$unsorted)) (= (tptp.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_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 $$unsorted)) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_c))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_c_2 V_a_2) (tptp.c_Groups_Oplus__class_Oplus T_a V_c_2 V_b_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_b_2)))) (forall ((V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_c_2) (tptp.c_Groups_Oplus__class_Oplus T_a V_b_2 V_c_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_b_2)))) (forall ((V_a $$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 $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_a) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_b)) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_b)))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_c)) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_b)))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c V_d) (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_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_b) (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_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__semigroup__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_b) (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_b_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__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_b_2)) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2 V_b_2)))) (forall ((V_b_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__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_b_2 V_c_2)) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2 V_b_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a _let_1) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b _let_1) (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) _let_1)))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_c) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b V_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b V_c) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1 V_x_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1 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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2) _let_1) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2 _let_1))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2)) (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1 V_a_2))))) (forall ((V_d $$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.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 $$unsorted)) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c V_d) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_b V_d)))))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2) _let_1) (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 _let_1))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2)) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m_2 V_n_2) _let_2) (or (and (= V_m_2 _let_2) (= V_n_2 _let_1)) (and (= V_m_2 _let_1) (= V_n_2 _let_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= _let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m_2 V_n_2)) (or (and (= V_m_2 _let_2) (= V_n_2 _let_1)) (and (= V_m_2 _let_1) (= V_n_2 _let_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m_2 V_n_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_n_2))))) (forall ((V_m $$unsorted) (V_i $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_i V_m)))) (forall ((V_m $$unsorted) (V_i $$unsorted)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_i (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m V_i)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (exists ((B_k $$unsorted)) (= V_n_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_m_2 B_k)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a _let_1 V_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a _let_1 V_b) (tptp.c_Orderings_Oord__class_Oless T_a _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b))))))) (forall ((V_b $$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.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.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.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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a _let_1) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b _let_1) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_b) _let_1)))))) (forall ((V_b $$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.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_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (tptp.c_Finite__Set_Ofinite T_b V_B_2) (= (tptp.c_Finite__Set_Ocard (tptp.tc_sum T_a T_b) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Finite__Set_Ocard T_b V_B_2)))))) (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_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_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (not (= V_x V_y)) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)) (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Ozero__class_Ozero T_a) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_d $$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 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a V_c) (tptp.c_Groups_Oplus__class_Oplus T_a V_c V_a)))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (= (= V_b_2 (tptp.c_Groups_Oplus__class_Oplus T_a V_b_2 V_a_2)) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_nat $$unsorted)) (= (tptp.c_Nat_Onat_Onat__size (tptp.hAPP tptp.c_Nat_OSuc V_nat)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Nat_Onat_Onat__size V_nat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_f_2 $$unsorted) (V_m_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.hAPP V_f_2 V_n_2))))))))) (= (tptp.c_Nat_Onat_Onat__size _let_4) _let_4) (forall ((V_f_2 $$unsorted) (V_p_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n_2 V_p_2) (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_n_2 V_p_2))) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2 V_p_2))))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (= (tptp.hAPP V_f_2 _let_1) (tptp.c_Groups_Ozero__class_Ozero T_a)) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc _let_1) V_k_2)) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat _let_1 V_k_2))))))) (forall ((V_f_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP V_f_2 V_m_2) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2) V_n_2))))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (not (tptp.c_Finite__Set_Ofinite T_a V_A_2)) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_g_2 V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (not (tptp.c_Finite__Set_Ofinite T_a V_A_2)) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_f_2 V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_f_2 $$unsorted) (V_Fa_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_Fa_2) (= (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat V_f_2 V_Fa_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (forall ((B_x $$unsorted)) (=> (tptp.c_member T_a B_x V_Fa_2) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (= (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat V_f_2 V_A_2) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (exists ((B_x $$unsorted)) (and (tptp.c_member T_a B_x V_A_2) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((B_xa $$unsorted)) (=> (tptp.c_member T_a B_xa V_A_2) (=> (not (= B_x B_xa)) (= (tptp.hAPP V_f_2 B_xa) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a V_f_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_g_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a V_g_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_nat $$unsorted)) (= (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_nat)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat V_nat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_b)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted)) (=> (tptp.c_member T_a B_x V_A_2) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x)))) (tptp.c_Orderings_Oord__class_Oless T_b (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_f_2 V_A_2) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_g_2 V_A_2))))))) (forall ((V_n $$unsorted)) (= (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat V_n) V_n)) (= (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat _let_4) _let_4) (forall ((V_n_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat V_f_2 V_A_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (exists ((B_x $$unsorted)) (and (tptp.c_member T_a B_x V_A_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 B_x)))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (= V_B_2 (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2)) (=> (forall ((B_a $$unsorted)) (=> (tptp.c_member T_a B_a V_A_2) (= (tptp.hAPP V_g_2 B_a) (tptp.hAPP V_h_2 (tptp.hAPP V_f_2 B_a))))) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c V_h_2 V_B_2) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_c V_g_2 V_A_2))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Orderings_Oord T_a)) (=> (= V_a_2 V_c_2) (=> (= V_b_2 V_d_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c_2 B_x) (=> (tptp.c_Orderings_Oord__class_Oless T_a B_x V_d_2) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))))) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2 V_b_2)) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_g_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2 V_d_2)))))))) (forall ((V_f_2 $$unsorted) (V_m_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 _let_1)))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat _let_1 V_m_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_3) (= _let_2 (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.hAPP V_f_2 _let_1)))))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_l_2 V_u_2))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_c_2 V_d_2)) (or (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2 V_b_2)) (and (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c_2 V_a_2) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b_2 V_d_2)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b_2 V_a_2) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2 V_b_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2)) (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2 V_b_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_l_2 (tptp.hAPP tptp.c_Nat_OSuc V_u_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_l_2 V_u_2))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted)) (= (exists ((B_m $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat B_m V_n_2) (tptp.hBOOL (tptp.hAPP V_P_2 B_m)))) (exists ((B_x $$unsorted)) (and (tptp.c_member tptp.tc_Nat_Onat B_x (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted)) (= (forall ((B_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat B_m V_n_2) (tptp.hBOOL (tptp.hAPP V_P_2 B_m)))) (forall ((B_x $$unsorted)) (=> (tptp.c_member tptp.tc_Nat_Onat B_x (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_i_2 V_j_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_i_2) (tptp.hAPP tptp.c_Nat_OSuc V_j_2)))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (= (tptp.hAPP V_f_2 _let_1) _let_1) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc _let_1) V_k_2)) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat _let_1 V_k_2)))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_c_2 V_d_2)) (and (or (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2 V_b_2)) (and (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c_2 V_a_2) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b_2 V_d_2) (or (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 V_a_2) (tptp.c_Orderings_Oord__class_Oless T_a V_b_2 V_d_2)))) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c_2 V_d_2))))) (forall ((V_f_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.hAPP V_f_2 V_m_2) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2) V_n_2))))))) (forall ((V_p_2 $$unsorted) (V_f_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n_2 V_p_2))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 _let_2) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 _let_1)) (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a V_f_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat _let_2 _let_1))))))))) (= (tptp.c_HOL_Obool_Obool__size tptp.c_fTrue) _let_4) _let_12 (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.c_Orderings_Oord__class_Oless T_a V_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (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 ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) _let_8 (forall ((T_a $$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))))) _let_7 (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Nat_OSuc V_n) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) V_n))) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Nat_OSuc V_n) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat V_n (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (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 ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Groups_Oplus__class_Oplus T_a _let_1 _let_1))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (= (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat V_f_2 V_A_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (exists ((B_x $$unsorted)) (and (tptp.c_member T_a B_x V_A_2) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (forall ((B_xa $$unsorted)) (=> (tptp.c_member T_a B_xa V_A_2) (=> (not (= B_x B_xa)) (= (tptp.hAPP V_f_2 B_xa) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))) (= (tptp.c_HOL_Obool_Obool__size tptp.c_fFalse) _let_4) (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_member T_a V_a_2 (tptp.c_Int_Oring__1__class_OInts T_a)) (= (tptp.c_Orderings_Oord__class_Oless T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oone__class_Oone 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_n_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_n_2)) (exists ((B_k $$unsorted)) (and (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat B_k V_n_2) (forall ((B_i $$unsorted)) (=> (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 ((T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (tptp.c_member T_a (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Int_Oring__1__class_OInts T_a)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (=> (tptp.c_member T_a V_a_2 _let_1) (=> (tptp.c_member T_a V_b_2 _let_1) (tptp.c_member T_a (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_b_2) _let_1)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (tptp.c_member T_a (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Int_Oring__1__class_OInts T_a)))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Int_Oring__char__0 T_a) (=> (tptp.c_member T_a V_a_2 (tptp.c_Int_Oring__1__class_OInts T_a)) (= (= (tptp.c_Groups_Oplus__class_Oplus T_a V_a_2 V_a_2) _let_1) (= V_a_2 _let_1)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Oring__char__0 T_a) (=> (tptp.c_member T_a V_a_2 (tptp.c_Int_Oring__1__class_OInts T_a)) (not (= (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oplus__class_Oplus T_a (tptp.c_Groups_Oone__class_Oone T_a) V_a_2) V_a_2) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (=> (exists ((B_x $$unsorted)) (tptp.c_member T_a B_x V_A_2)) (= (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) _let_1)) (= (tptp.c_Finite__Set_Ocard T_a V_A_2) _let_1))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (not (tptp.c_member T_a V_x_2 V_A_2)) (= (tptp.c_Finite__Set_Ocard T_a (tptp.c_Set_Oinsert T_a V_x_2 V_A_2)) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) (tptp.c_Finite__Set_Ocard T_a V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 V_A_2)) (=> (not (= V_a_2 V_b_2)) (tptp.c_member T_a V_a_2 V_A_2)))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (=> (not (tptp.c_member T_a V_a_2 V_B_2)) (= V_a_2 V_b_2)) (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 V_B_2)))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (tptp.c_Finite__Set_Ofinite T_a (tptp.c_Set_Oinsert T_a V_a_2 V_A_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Fa_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_a T_b V_f_2 V_z_2 V_g_2 V_Fa_2) (=> (tptp.c_Finite__Set_Ofinite T_b V_A_2) (= (tptp.hAPP V_Fa_2 (tptp.c_Set_Oinsert T_b V_x_2 V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP V_Fa_2 V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_D_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_C_2 V_D_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Set_Oinsert T_a V_a_2 V_C_2) (tptp.c_Set_Oinsert T_a V_a_2 V_D_2))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_b_2 V_B_2))))) (forall ((V_a_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_B_2 (tptp.c_Set_Oinsert T_a V_a_2 V_B_2))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_b_2 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= V_b_2 V_a_2))) _let_3 (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2) V_B_2) (and (tptp.c_member T_a V_x_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (not (tptp.c_member T_a V_x_2 V_A_2)) (= (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_x_2 V_B_2)) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2 V_A_2))) (=> (tptp.c_member T_a V_x_2 V_A_2) (= (tptp.c_Set_Oinsert T_b (tptp.hAPP V_f_2 V_x_2) _let_1) _let_1)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_a V_x_2 _let_2))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 _let_3) (or (= V_A_2 _let_2) (= V_A_2 _let_3))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Set_Oimage T_b T_a V_f_2 (tptp.c_Set_Oinsert T_b V_a_2 V_B_2)) (tptp.c_Set_Oinsert T_a (tptp.hAPP V_f_2 V_a_2) (tptp.c_Set_Oimage T_b T_a V_f_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2))) (= (tptp.c_Set_Oinsert T_a V_x_2 _let_1) _let_1))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Set_Oinsert T_a V_y_2 V_A_2)) (tptp.c_Set_Oinsert T_a V_y_2 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Set_Oinsert T_a V_y_2 V_A_2) V_x_2)) (or (= V_y_2 V_x_2) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Finite__Set_Ofinite T_a (tptp.c_Set_Oinsert T_a V_a_2 V_A_2)) (tptp.c_Finite__Set_Ofinite T_a V_A_2))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_a_2 V_B_2))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 V_A_2)) (or (= V_a_2 V_b_2) (tptp.c_member T_a V_a_2 V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.c_member T_a V_x_2 V_A_2)) (=> (not (tptp.c_member T_a V_x_2 V_B_2)) (= (= (tptp.c_Set_Oinsert T_a V_x_2 V_A_2) (tptp.c_Set_Oinsert T_a V_x_2 V_B_2)) (= V_A_2 V_B_2))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_a_2 V_B_2) (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_a_2 V_A_2) (= (tptp.c_Set_Oinsert T_a V_a_2 V_A_2) V_A_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (= (tptp.c_Set_Oinsert T_a V_a_2 _let_1) (tptp.c_Set_Oinsert T_a V_b_2 _let_1)) (= V_a_2 V_b_2)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (= (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 _let_1)) (tptp.c_Set_Oinsert T_a V_c_2 (tptp.c_Set_Oinsert T_a V_d_2 _let_1))) (or (and (= V_a_2 V_c_2) (= V_b_2 V_d_2)) (and (= V_a_2 V_d_2) (= V_b_2 V_c_2)))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (not (= (tptp.c_Set_Oinsert T_a V_a_2 V_A_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (not (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Set_Oinsert T_a V_a_2 V_A_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (= V_a_2 V_b_2) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.c_Set_Oinsert T_a V_c_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (and (= V_a_2 V_b_2) (= V_b_2 V_c_2))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_a_2) (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Finite__Set_Ocard T_a (tptp.c_Set_Oinsert T_a V_x_2 V_A_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 _let_1) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 _let_1) (tptp.c_Set_Oinsert tptp.tc_Nat_Onat _let_1 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)))))) (forall ((V_m_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2 (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (tptp.c_Set_Oinsert tptp.tc_Nat_Onat V_m_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Fa_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_Fa_2) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_Fa_2 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_Fa_2 V_A_2))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (not (tptp.c_member T_a V_x_2 V_A_2)) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_g_2 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2)) (tptp.c_Groups_Oplus__class_Oplus T_b (tptp.hAPP V_g_2 V_x_2) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_g_2 V_A_2))))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_Fa_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.c_Finite__Set_Ofinite T_a V_Fa_2) (=> (not (tptp.c_member T_a V_a_2 V_Fa_2)) (= (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_f_2 (tptp.c_Set_Oinsert T_a V_a_2 V_Fa_2)) (tptp.c_Groups_Oplus__class_Oplus T_b (tptp.hAPP V_f_2 V_a_2) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b V_f_2 V_Fa_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (not (tptp.c_member T_a V_x_2 V_A_2)) (= (tptp.c_Finite__Set_Ocard T_a (tptp.c_Set_Oinsert T_a V_x_2 V_A_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Finite__Set_Ocard T_a V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a V_A_2))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_a (tptp.c_Set_Oinsert T_a V_x_2 V_A_2)))) (let ((_let_3 (tptp.c_member T_a V_x_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (and (=> _let_3 (= _let_2 _let_1)) (=> (not _let_3) (= _let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2))) (and (=> _let_2 (= _let_1 (tptp.c_Set_Oinsert tptp.tc_Nat_Onat V_n_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2 V_n_2)))) (=> (not _let_2) (= _let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Set_Othe__elem T_a (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2)) _let_2 (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Fa_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_Fa_2) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (not (tptp.c_member T_a V_x_2 V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_Fa_2 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_Fa_2 V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofinite T_a V_A_2) (=> (tptp.c_member T_a V_x_2 V_A_2) (= (tptp.c_Finite__Set_Ocard T_a V_A_2) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) (tptp.c_Finite__Set_Ocard T_a (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_1) (tptp.class_Orderings_Opreorder (tptp.tc_fun T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Finite__Set_Ofinite T_2) (tptp.class_Finite__Set_Ofinite T_1)) (tptp.class_Finite__Set_Ofinite (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$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 $$unsorted)) (=> (tptp.class_Orderings_Otop T_1) (tptp.class_Orderings_Otop (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Oord T_1) (tptp.class_Orderings_Oord (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Obot T_1) (tptp.class_Orderings_Obot (tptp.tc_fun T_2 T_1)))) (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__ab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__comm__monoid__add tptp.tc_Nat_Onat) (tptp.class_Groups_Ocancel__ab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Groups_Ocancel__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Rings_Olinordered__semidom tptp.tc_Nat_Onat) (tptp.class_Groups_Oab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Groups_Ocomm__monoid__add tptp.tc_Nat_Onat) (tptp.class_Rings_Ocomm__semiring__1 tptp.tc_Nat_Onat) _let_1 (tptp.class_Orderings_Opreorder tptp.tc_Nat_Onat) (tptp.class_Orderings_Olinorder tptp.tc_Nat_Onat) (tptp.class_Groups_Omonoid__add tptp.tc_Nat_Onat) (tptp.class_Orderings_Oorder tptp.tc_Nat_Onat) (tptp.class_Orderings_Oord tptp.tc_Nat_Onat) (tptp.class_Orderings_Obot tptp.tc_Nat_Onat) (tptp.class_Groups_Ozero tptp.tc_Nat_Onat) (tptp.class_Groups_Oone tptp.tc_Nat_Onat) (tptp.class_Orderings_Opreorder tptp.tc_HOL_Obool) (tptp.class_Finite__Set_Ofinite tptp.tc_HOL_Obool) (tptp.class_Orderings_Oorder tptp.tc_HOL_Obool) (tptp.class_Orderings_Otop tptp.tc_HOL_Obool) (tptp.class_Orderings_Oord tptp.tc_HOL_Obool) (tptp.class_Orderings_Obot tptp.tc_HOL_Obool) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Finite__Set_Ofinite T_2) (tptp.class_Finite__Set_Ofinite T_1)) (tptp.class_Finite__Set_Ofinite (tptp.tc_sum T_2 T_1)))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_1) (tptp.class_Finite__Set_Ofinite (tptp.tc_Option_Ooption T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Finite__Set_Ofinite T_2) (tptp.class_Finite__Set_Ofinite T_1)) (tptp.class_Finite__Set_Ofinite (tptp.tc_prod T_2 T_1)))) (tptp.class_Finite__Set_Ofinite tptp.tc_Product__Type_Ounit) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.c_fequal V_x_2) V_y_2))) (= V_x_2 V_y_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (or (not (= V_x_2 V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_fequal V_x_2) V_y_2)))) (not (tptp.hBOOL tptp.c_fFalse)) (tptp.hBOOL tptp.c_fTrue) (forall ((B_h $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Arrow__Order__Mirabelle_Oindi tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on tptp.tc_Arrow__Order__Mirabelle_Oindi tptp.tc_Nat_Onat B_h _let_1) (=> (= (tptp.c_Set_Oimage tptp.tc_Arrow__Order__Mirabelle_Oindi tptp.tc_Nat_Onat B_h _let_1) (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_Finite__Set_Ocard tptp.tc_Arrow__Order__Mirabelle_Oindi _let_1))) tptp.v_thesis____)))) (not tptp.v_thesis____) true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 98.33/98.50 )
% 98.33/98.50 % SZS output end Proof for SCT159+1
% 98.33/98.50 % cvc5---1.0.5 exiting
% 98.33/98.50 % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------