%------------------------------------------------------------------------------
% File     : cvc5---1.0.5
% Problem  : SEU388+2 : TPTP v8.1.2. Released v3.3.0.
% Transfm  : none
% Format   : tptp
% Command  : do_cvc5 %s %d

% Computer : n007.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 16:31:19 EDT 2023

% Result   : Theorem 1.85s 2.16s
% Output   : Proof 1.85s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.22/0.23  % Problem    : SEU388+2 : TPTP v8.1.2. Released v3.3.0.
% 0.22/0.24  % Command    : do_cvc5 %s %d
% 0.24/0.46  % Computer : n007.cluster.edu
% 0.24/0.46  % Model    : x86_64 x86_64
% 0.24/0.46  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.24/0.46  % Memory   : 8042.1875MB
% 0.24/0.46  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.24/0.46  % CPULimit   : 300
% 0.24/0.46  % WCLimit    : 300
% 0.24/0.46  % DateTime   : Wed Aug 23 18:40:25 EDT 2023
% 0.24/0.46  % CPUTime    : 
% 0.31/0.70  %----Proving TF0_NAR, FOF, or CNF
% 1.85/2.16  ------- convert to smt2 : /export/starexec/sandbox2/tmp/tmp.5e9HbbsAiO/cvc5---1.0.5_13089.p...
% 1.85/2.16  ------- get file name : TPTP file name is SEU388+2
% 1.85/2.16  ------- cvc5-fof : /export/starexec/sandbox2/solver/bin/cvc5---1.0.5_13089.smt2...
% 1.85/2.16  --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10...
% 1.85/2.16  % SZS status Theorem for SEU388+2
% 1.85/2.16  % SZS output start Proof for SEU388+2
% 1.85/2.16  (
% 1.85/2.16  (let ((_let_1 (not (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (= (tptp.in C (tptp.neighborhood_system A B)) (tptp.point_neighbourhood C A B)))))))))) (let ((_let_2 (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.topological_space B) (tptp.top_str B) (tptp.element C (tptp.the_carrier B))) (= (tptp.in A (tptp.a_2_0_yellow19 B C)) (exists ((D $$unsorted)) (and (tptp.point_neighbourhood D B C) (= A D)))))))) (let ((_let_3 (tptp.empty tptp.empty_set))) (let ((_let_4 (tptp.relation tptp.empty_set))) (let ((_let_5 (tptp.relation_empty_yielding tptp.empty_set))) (let ((_let_6 (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (= (tptp.neighborhood_system A B) (tptp.a_2_0_yellow19 A B)))))))) (let ((_let_7 (tptp.a_2_0_yellow19 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_49 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_51))) (let ((_let_8 (tptp.neighborhood_system SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_49 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_51))) (let ((_let_9 (= _let_8 _let_7))) (let ((_let_10 (tptp.in SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_50 _let_8))) (let ((_let_11 (tptp.in SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_50 _let_7))) (let ((_let_12 (tptp.element SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_51 (tptp.the_carrier SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_49)))) (let ((_let_13 (not _let_12))) (let ((_let_14 (tptp.top_str SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_49))) (let ((_let_15 (not _let_14))) (let ((_let_16 (tptp.topological_space SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_49))) (let ((_let_17 (not _let_16))) (let ((_let_18 (tptp.empty_carrier SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_49))) (let ((_let_19 (or _let_18 _let_17 _let_15 _let_13 _let_9))) (let ((_let_20 (forall ((A $$unsorted) (BOUND_VARIABLE_18427 $$unsorted)) (or (tptp.empty_carrier A) (not (tptp.topological_space A)) (not (tptp.top_str A)) (not (tptp.element BOUND_VARIABLE_18427 (tptp.the_carrier A))) (= (tptp.neighborhood_system A BOUND_VARIABLE_18427) (tptp.a_2_0_yellow19 A BOUND_VARIABLE_18427)))))) (let ((_let_21 (EQ_RESOLVE (ASSUME :args (_let_6)) (MACRO_SR_EQ_INTRO :args (_let_6 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_22 (tptp.point_neighbourhood SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_50 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_49 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_51))) (let ((_let_23 (= _let_10 _let_22))) (let ((_let_24 (or _let_18 _let_17 _let_15 _let_13 _let_23))) (let ((_let_25 (forall ((A $$unsorted) (BOUND_VARIABLE_31752 $$unsorted) (BOUND_VARIABLE_31750 $$unsorted)) (or (tptp.empty_carrier A) (not (tptp.topological_space A)) (not (tptp.top_str A)) (not (tptp.element BOUND_VARIABLE_31750 (tptp.the_carrier A))) (= (tptp.in BOUND_VARIABLE_31752 (tptp.neighborhood_system A BOUND_VARIABLE_31750)) (tptp.point_neighbourhood BOUND_VARIABLE_31752 A BOUND_VARIABLE_31750)))))) (let ((_let_26 (not _let_24))) (let ((_let_27 (EQ_RESOLVE (ASSUME :args (_let_1)) (MACRO_SR_EQ_INTRO :args (_let_1 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_28 (or))) (let ((_let_29 (not _let_25))) (let ((_let_30 (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (IMPLIES_ELIM (SCOPE (SKOLEMIZE _let_27) :args (_let_29))) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_29) _let_25))) (REFL :args (_let_26)) :args _let_28)) _let_27 :args (_let_26 true _let_25)))) (let ((_let_31 (REFL :args (_let_24)))) (let ((_let_32 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_24 3)) (CONG _let_31 (MACRO_SR_PRED_INTRO :args ((= (not _let_13) _let_12))) :args _let_28)) :args ((or _let_12 _let_24))) _let_30 :args (_let_12 true _let_24)))) (let ((_let_33 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_24 2)) (CONG _let_31 (MACRO_SR_PRED_INTRO :args ((= (not _let_15) _let_14))) :args _let_28)) :args ((or _let_14 _let_24))) _let_30 :args (_let_14 true _let_24)))) (let ((_let_34 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_24 1)) (CONG _let_31 (MACRO_SR_PRED_INTRO :args ((= (not _let_17) _let_16))) :args _let_28)) :args ((or _let_16 _let_24))) _let_30 :args (_let_16 true _let_24)))) (let ((_let_35 (MACRO_RESOLUTION_TRUST (CNF_OR_NEG :args (_let_24 0)) _let_30 :args ((not _let_18) true _let_24)))) (let ((_let_36 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_19)) :args ((or _let_18 _let_17 _let_15 _let_13 _let_9 (not _let_19)))) _let_35 _let_34 _let_33 _let_32 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_21 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_49 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_51 QUANTIFIERS_INST_E_MATCHING ((not (= (tptp.element BOUND_VARIABLE_18427 (tptp.the_carrier A)) false))))) :args (_let_20))) _let_21 :args (_let_19 false _let_20)) :args (_let_9 true _let_18 false _let_16 false _let_14 false _let_12 false _let_19)))) (let ((_let_37 (= _let_22 _let_11))) (let ((_let_38 (not _let_10))) (let ((_let_39 (MACRO_RESOLUTION_TRUST (CNF_OR_NEG :args (_let_24 4)) _let_30 :args ((not _let_23) true _let_24)))) (let ((_let_40 (_let_23))) (let ((_let_41 (or _let_18 _let_17 _let_15 _let_13 _let_37))) (let ((_let_42 (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (or (tptp.empty_carrier B) (not (tptp.topological_space B)) (not (tptp.top_str B)) (not (tptp.element C (tptp.the_carrier B))) (= (tptp.in A (tptp.a_2_0_yellow19 B C)) (tptp.point_neighbourhood A B C)))))) (let ((_let_43 (EQ_RESOLVE (ASSUME :args (_let_2)) (MACRO_SR_EQ_INTRO :args (_let_2 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_44 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_41)) :args ((or _let_18 _let_17 _let_15 _let_13 _let_37 (not _let_41)))) _let_35 _let_34 _let_33 _let_32 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_43 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_50 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_49 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_51 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((tptp.point_neighbourhood A B C)))) :args (_let_42)))) _let_43 :args (_let_41 false _let_42)) :args (_let_37 true _let_18 false _let_16 false _let_14 false _let_12 false _let_41)))) (let ((_let_45 (not _let_37))) (let ((_let_46 (not _let_11))) (let ((_let_47 (_let_37))) (let ((_let_48 (not _let_9))) (let ((_let_49 (and _let_10 _let_9))) (let ((_let_50 (_let_10 _let_9))) (let ((_let_51 (ASSUME :args (_let_10)))) (let ((_let_52 (ASSUME :args (_let_9)))) (let ((_let_53 (CONG (REFL :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_50)) (SYMM _let_52) :args (APPLY_UF tptp.in)))) (let ((_let_54 (MACRO_RESOLUTION_TRUST (REORDERING (RESOLUTION (CNF_AND_NEG :args (_let_49)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_51 _let_52) (SCOPE (TRUE_ELIM (TRANS _let_53 (TRUE_INTRO _let_51))) :args _let_50)) :args _let_50)) :args (true _let_49)) :args ((or _let_38 _let_11 _let_48))) _let_36 (REORDERING (CNF_EQUIV_POS2 :args _let_47) :args ((or _let_22 _let_46 _let_45))) _let_44 (CNF_EQUIV_NEG2 :args _let_40) _let_39 :args (_let_38 false _let_9 true _let_11 false _let_37 true _let_22 true _let_23)))) (let ((_let_55 (and _let_38 _let_9))) (let ((_let_56 (_let_38 _let_9))) (let ((_let_57 (ASSUME :args (_let_38)))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (RESOLUTION (CNF_AND_NEG :args (_let_55)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_57 _let_52) (SCOPE (FALSE_ELIM (TRANS _let_53 (FALSE_INTRO _let_57))) :args _let_56)) :args _let_56)) :args (true _let_55)) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_38) _let_10))) (REFL :args (_let_48)) (REFL :args (_let_46)) :args _let_28)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_EQUIV_POS1 :args _let_47) :args ((or (not _let_22) _let_11 _let_45))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_EQUIV_NEG1 :args _let_40) :args ((or _let_10 _let_22 _let_23))) _let_54 _let_39 :args (_let_22 true _let_10 true _let_23)) _let_44 :args (_let_11 false _let_22 false _let_37)) _let_54 _let_36 :args (false false _let_11 true _let_10 false _let_9)) :args ((forall ((A $$unsorted)) (=> (tptp.rel_str A) (=> (tptp.strict_rel_str A) (= A (tptp.rel_str_of (tptp.the_carrier A) (tptp.the_InternalRel A)))))) (forall ((A $$unsorted)) (=> (tptp.latt_str A) (=> (tptp.strict_latt_str A) (= A (tptp.latt_str_of (tptp.the_carrier A) (tptp.the_L_join A) (tptp.the_L_meet A)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.net_str B A)) (=> (tptp.strict_net_str B A) (= B (tptp.net_str_of A (tptp.the_carrier B) (tptp.the_InternalRel B) (tptp.the_mapping A B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (not (tptp.in B A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.proper_subset A B) (not (tptp.proper_subset B A)))) (forall ((A $$unsorted)) (=> (tptp.v1_membered A) (forall ((B $$unsorted)) (=> (tptp.element B A) (tptp.v1_xcmplx_0 B))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.reflexive_relstr A))) (let ((_let_2 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_2 _let_1 (tptp.complete_relstr A)) (and _let_2 _let_1 (tptp.up_complete_relstr A) (tptp.join_complete_relstr A))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_1 (tptp.boolean_relstr A)) (and _let_1 (tptp.reflexive_relstr A) (tptp.transitive_relstr A) (tptp.antisymmetric_relstr A) (tptp.with_suprema_relstr A) (tptp.with_infima_relstr A) (tptp.upper_bounded_relstr A) (tptp.distributive_relstr A) (tptp.heyting_relstr A)))))) (forall ((A $$unsorted)) (=> (tptp.v2_membered A) (forall ((B $$unsorted)) (=> (tptp.element B A) (and (tptp.v1_xcmplx_0 B) (tptp.v1_xreal_0 B)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.reflexive_relstr A))) (let ((_let_2 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_2 _let_1 (tptp.join_complete_relstr A)) (and _let_2 _let_1 (tptp.lower_bounded_relstr A))))))) (forall ((A $$unsorted)) (=> (tptp.v3_membered A) (forall ((B $$unsorted)) (=> (tptp.element B A) (and (tptp.v1_xcmplx_0 B) (tptp.v1_xreal_0 B) (tptp.v1_rat_1 B)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.lower_bounded_relstr A))) (let ((_let_2 (tptp.with_suprema_relstr A))) (let ((_let_3 (tptp.antisymmetric_relstr A))) (let ((_let_4 (tptp.transitive_relstr A))) (let ((_let_5 (tptp.reflexive_relstr A))) (let ((_let_6 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_6 _let_5 _let_4 _let_3 _let_2 _let_1 (tptp.up_complete_relstr A)) (and _let_6 _let_5 _let_4 _let_3 _let_2 (tptp.with_infima_relstr A) (tptp.complete_relstr A) _let_1 (tptp.upper_bounded_relstr A) (tptp.bounded_relstr A))))))))))) (forall ((A $$unsorted)) (=> (tptp.v4_membered A) (forall ((B $$unsorted)) (=> (tptp.element B A) (and (tptp.v1_xcmplx_0 B) (tptp.v1_xreal_0 B) (tptp.v1_int_1 B) (tptp.v1_rat_1 B)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.antisymmetric_relstr A))) (let ((_let_2 (tptp.reflexive_relstr A))) (let ((_let_3 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_3 _let_2 _let_1 (tptp.join_complete_relstr A)) (and _let_3 _let_2 _let_1 (tptp.with_infima_relstr A)))))))) (forall ((A $$unsorted)) (=> (tptp.v5_membered A) (forall ((B $$unsorted)) (=> (tptp.element B A) (and (tptp.v1_xcmplx_0 B) (tptp.natural B) (tptp.v1_xreal_0 B) (tptp.v1_int_1 B) (tptp.v1_rat_1 B)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.upper_bounded_relstr A))) (let ((_let_2 (tptp.antisymmetric_relstr A))) (let ((_let_3 (tptp.reflexive_relstr A))) (let ((_let_4 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_4 _let_3 _let_2 _let_1 (tptp.join_complete_relstr A)) (and _let_4 _let_3 _let_2 (tptp.with_suprema_relstr A) _let_1)))))))) (forall ((A $$unsorted)) (=> (tptp.empty A) (and (tptp.v1_membered A) (tptp.v2_membered A) (tptp.v3_membered A) (tptp.v4_membered A) (tptp.v5_membered A)))) (forall ((A $$unsorted)) (=> (tptp.v1_membered A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (tptp.v1_membered B))))) (forall ((A $$unsorted)) (=> (tptp.v2_membered A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (and (tptp.v1_membered B) (tptp.v2_membered B)))))) (forall ((A $$unsorted)) (=> (tptp.v3_membered A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (and (tptp.v1_membered B) (tptp.v2_membered B) (tptp.v3_membered B)))))) (forall ((A $$unsorted)) (=> (tptp.v4_membered A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (and (tptp.v1_membered B) (tptp.v2_membered B) (tptp.v3_membered B) (tptp.v4_membered B)))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (forall ((B $$unsorted)) (=> (tptp.element B A) (and (tptp.epsilon_transitive B) (tptp.epsilon_connected B) (tptp.ordinal B)))))) (forall ((A $$unsorted)) (=> (tptp.empty A) (tptp.finite A))) (forall ((A $$unsorted)) (=> (tptp.preboolean A) (and (tptp.cup_closed A) (tptp.diff_closed A)))) (forall ((A $$unsorted)) (=> (tptp.empty A) (tptp.function A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.function C))) (=> (tptp.relation_of2 C A B) (=> (and _let_1 (tptp.v1_partfun1 C A B)) (and _let_1 (tptp.quasi_total C A B)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.lattice A))) (let ((_let_2 (not (tptp.empty_carrier A)))) (=> (tptp.latt_str A) (=> (and _let_2 _let_1 (tptp.complete_latt_str A)) (and _let_2 (tptp.join_commutative A) (tptp.join_associative A) (tptp.meet_commutative A) (tptp.meet_associative A) (tptp.meet_absorbing A) (tptp.join_absorbing A) _let_1 (tptp.lower_bounded_semilattstr A) (tptp.upper_bounded_semilattstr A) (tptp.bounded_lattstr A))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (=> (tptp.with_suprema_relstr A) (not (tptp.empty_carrier A))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.latt_str A) (=> (and _let_1 (tptp.lattice A)) (and _let_1 (tptp.join_commutative A) (tptp.join_associative A) (tptp.meet_commutative A) (tptp.meet_associative A) (tptp.meet_absorbing A) (tptp.join_absorbing A)))))) (forall ((A $$unsorted)) (=> (tptp.v5_membered A) (tptp.v4_membered A))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (and (tptp.epsilon_transitive A) (tptp.epsilon_connected A)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation A))) (=> (and _let_1 (tptp.symmetric A) (tptp.transitive A)) (and _let_1 (tptp.reflexive A))))) (forall ((A $$unsorted)) (=> (tptp.empty A) (tptp.relation A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.cartesian_product2 A B))) (tptp.relation C))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (=> (tptp.empty B) (and (tptp.open_subset B A) (tptp.closed_subset B A))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.connected_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (and (tptp.directed_subset B A) (tptp.filtered_subset B A)))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_1 (tptp.complete_relstr A)) (and _let_1 (tptp.with_suprema_relstr A) (tptp.with_infima_relstr A)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.join_complete_relstr A))) (let ((_let_2 (tptp.up_complete_relstr A))) (let ((_let_3 (tptp.upper_bounded_relstr A))) (let ((_let_4 (tptp.antisymmetric_relstr A))) (let ((_let_5 (tptp.transitive_relstr A))) (let ((_let_6 (tptp.reflexive_relstr A))) (let ((_let_7 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_7 _let_6 _let_5 _let_4 _let_3 _let_2 _let_1) (and _let_7 _let_6 _let_5 _let_4 (tptp.lower_bounded_relstr A) _let_3 (tptp.bounded_relstr A) _let_2 _let_1 (tptp.with_suprema_relstr A) (tptp.with_infima_relstr A) (tptp.complete_relstr A)))))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (=> (tptp.empty_carrier A) (tptp.v1_yellow_3 A)))) (forall ((A $$unsorted)) (=> (tptp.v5_membered A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (and (tptp.v1_membered B) (tptp.v2_membered B) (tptp.v3_membered B) (tptp.v4_membered B) (tptp.v5_membered B)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.ordinal A))) (=> (and (tptp.empty A) _let_1) (and (tptp.epsilon_transitive A) (tptp.epsilon_connected A) _let_1 (tptp.natural A))))) (forall ((A $$unsorted)) (=> (tptp.finite A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (tptp.finite B))))) (forall ((A $$unsorted)) (=> (and (tptp.cup_closed A) (tptp.diff_closed A)) (tptp.preboolean A))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.function A))) (let ((_let_2 (tptp.relation A))) (=> (and _let_2 (tptp.empty A) _let_1) (and _let_2 _let_1 (tptp.one_to_one A)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.quasi_total C A B))) (let ((_let_2 (tptp.function C))) (=> (tptp.relation_of2 C A B) (=> (and _let_2 _let_1 (tptp.bijective C A B)) (and _let_2 (tptp.one_to_one C) _let_1 (tptp.onto C A B))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (=> (tptp.with_infima_relstr A) (not (tptp.empty_carrier A))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.latt_str A) (=> (and _let_1 (tptp.join_commutative A) (tptp.join_associative A) (tptp.meet_commutative A) (tptp.meet_associative A) (tptp.meet_absorbing A) (tptp.join_absorbing A)) (and _let_1 (tptp.lattice A)))))) (forall ((A $$unsorted)) (=> (tptp.v4_membered A) (tptp.v3_membered A))) (forall ((A $$unsorted)) (=> (and (tptp.epsilon_transitive A) (tptp.epsilon_connected A)) (tptp.ordinal A))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (=> (tptp.empty B) (tptp.boundary_set B A)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.up_complete_relstr A))) (let ((_let_2 (tptp.connected_relstr A))) (let ((_let_3 (tptp.lower_bounded_relstr A))) (let ((_let_4 (tptp.antisymmetric_relstr A))) (let ((_let_5 (tptp.transitive_relstr A))) (let ((_let_6 (tptp.reflexive_relstr A))) (let ((_let_7 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_7 _let_6 _let_5 _let_4 _let_3 _let_2 _let_1) (and _let_7 _let_6 _let_5 _let_4 (tptp.with_suprema_relstr A) (tptp.with_infima_relstr A) (tptp.complete_relstr A) _let_3 (tptp.upper_bounded_relstr A) (tptp.bounded_relstr A) _let_2 _let_1 (tptp.join_complete_relstr A)))))))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.reflexive_relstr A))) (let ((_let_2 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_2 _let_1 (tptp.trivial_carrier A)) (and _let_2 _let_1 (tptp.transitive_relstr A) (tptp.antisymmetric_relstr A) (tptp.complete_relstr A))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (=> (not (tptp.v1_yellow_3 A)) (not (tptp.empty_carrier A))))) (forall ((A $$unsorted)) (=> (tptp.element A tptp.omega) (and (tptp.epsilon_transitive A) (tptp.epsilon_connected A) (tptp.ordinal A) (tptp.natural A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.quasi_total C A B))) (let ((_let_2 (tptp.function C))) (=> (tptp.relation_of2 C A B) (=> (and _let_2 (tptp.one_to_one C) _let_1 (tptp.onto C A B)) (and _let_2 _let_1 (tptp.bijective C A B))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.latt_str A) (=> (and _let_1 (tptp.lower_bounded_semilattstr A) (tptp.upper_bounded_semilattstr A)) (and _let_1 (tptp.bounded_lattstr A)))))) (forall ((A $$unsorted)) (=> (tptp.v3_membered A) (tptp.v2_membered A))) (forall ((A $$unsorted)) (=> (tptp.empty A) (and (tptp.epsilon_transitive A) (tptp.epsilon_connected A) (tptp.ordinal A)))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (=> (tptp.empty B) (tptp.nowhere_dense B A)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.reflexive_relstr A))) (let ((_let_2 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_2 (tptp.trivial_carrier A) _let_1) (and _let_2 _let_1 (tptp.v2_waybel_3 A))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_1 (tptp.complete_relstr A)) (and _let_1 (tptp.bounded_relstr A)))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_1 (tptp.reflexive_relstr A)) (and _let_1 (not (tptp.v1_yellow_3 A))))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.quasi_total B A A))) (let ((_let_2 (tptp.function B))) (=> (tptp.relation_of2 B A A) (=> (and _let_2 (tptp.v1_partfun1 B A A) (tptp.reflexive B) _let_1) (and _let_2 (tptp.one_to_one B) _let_1 (tptp.onto B A A) (tptp.bijective B A A))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.latt_str A) (=> (and _let_1 (tptp.bounded_lattstr A)) (and _let_1 (tptp.lower_bounded_semilattstr A) (tptp.upper_bounded_semilattstr A)))))) (forall ((A $$unsorted)) (=> (tptp.v2_membered A) (tptp.v1_membered A))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (=> (tptp.nowhere_dense B A) (tptp.boundary_set B A)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.reflexive_relstr A))) (let ((_let_2 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_2 _let_1 (tptp.v3_waybel_3 A)) (and _let_2 _let_1 (tptp.up_complete_relstr A) (tptp.v2_waybel_3 A))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.with_infima_relstr A))) (let ((_let_2 (tptp.with_suprema_relstr A))) (let ((_let_3 (tptp.lower_bounded_relstr A))) (let ((_let_4 (tptp.antisymmetric_relstr A))) (let ((_let_5 (tptp.transitive_relstr A))) (let ((_let_6 (tptp.reflexive_relstr A))) (=> (tptp.rel_str A) (=> (and _let_6 _let_5 _let_4 _let_3 (tptp.v3_waybel_3 A) _let_2 _let_1) (and (not (tptp.empty_carrier A)) _let_6 _let_5 _let_4 _let_3 (tptp.upper_bounded_relstr A) (tptp.bounded_relstr A) (tptp.up_complete_relstr A) (tptp.join_complete_relstr A) (not (tptp.v1_yellow_3 A)) (tptp.v1_waybel_2 A) (tptp.v2_waybel_2 A) _let_2 _let_1 (tptp.complete_relstr A))))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (=> (tptp.bounded_relstr A) (and (tptp.lower_bounded_relstr A) (tptp.upper_bounded_relstr A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.empty B)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.quasi_total C A B))) (let ((_let_2 (tptp.function C))) (=> (tptp.relation_of2 C A B) (=> (and _let_2 _let_1) (and _let_2 (tptp.v1_partfun1 C A B) _let_1)))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.latt_str A) (=> (and _let_1 (tptp.boolean_lattstr A)) (and _let_1 (tptp.distributive_lattstr A) (tptp.lower_bounded_semilattstr A) (tptp.upper_bounded_semilattstr A) (tptp.bounded_lattstr A) (tptp.complemented_lattstr A)))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.boundary_set B A))) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (=> (and (tptp.closed_subset B A) _let_1) (and _let_1 (tptp.nowhere_dense B A)))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.reflexive_relstr A))) (let ((_let_2 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_2 _let_1 (tptp.trivial_carrier A)) (and _let_2 _let_1 (tptp.connected_relstr A))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_1 (tptp.heyting_relstr A)) (and _let_1 (tptp.reflexive_relstr A) (tptp.transitive_relstr A) (tptp.antisymmetric_relstr A) (tptp.with_suprema_relstr A) (tptp.with_infima_relstr A)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.lower_bounded_relstr A))) (let ((_let_2 (tptp.with_suprema_relstr A))) (let ((_let_3 (tptp.antisymmetric_relstr A))) (let ((_let_4 (tptp.transitive_relstr A))) (let ((_let_5 (tptp.reflexive_relstr A))) (=> (tptp.rel_str A) (=> (and _let_5 _let_4 _let_3 _let_2 _let_1 (tptp.up_complete_relstr A) (tptp.v2_waybel_3 A)) (and (not (tptp.empty_carrier A)) _let_5 _let_4 _let_3 _let_2 _let_1 (tptp.v3_waybel_3 A)))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (=> (and (tptp.lower_bounded_relstr A) (tptp.upper_bounded_relstr A)) (tptp.bounded_relstr A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (not (tptp.empty B))) (forall ((C $$unsorted)) (let ((_let_1 (tptp.quasi_total C A B))) (let ((_let_2 (tptp.function C))) (=> (tptp.relation_of2 C A B) (=> (and _let_2 _let_1) (and _let_2 (not (tptp.empty C)) (tptp.v1_partfun1 C A B) _let_1)))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.latt_str A) (=> (and _let_1 (tptp.distributive_lattstr A) (tptp.bounded_lattstr A) (tptp.complemented_lattstr A)) (and _let_1 (tptp.boolean_lattstr A)))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.nowhere_dense B A))) (let ((_let_2 (tptp.open_subset B A))) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (=> (and _let_2 _let_1) (and (tptp.empty B) _let_2 (tptp.closed_subset B A) (tptp.v1_membered B) (tptp.v2_membered B) (tptp.v3_membered B) (tptp.v4_membered B) (tptp.v5_membered B) (tptp.boundary_set B A) _let_1)))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_1 (tptp.heyting_relstr A)) (and _let_1 (tptp.distributive_relstr A)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.connected_relstr A))) (let ((_let_2 (tptp.antisymmetric_relstr A))) (let ((_let_3 (tptp.transitive_relstr A))) (let ((_let_4 (tptp.reflexive_relstr A))) (let ((_let_5 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_5 _let_4 _let_3 _let_2 (tptp.complete_relstr A) _let_1) (and _let_5 _let_4 _let_3 _let_2 _let_1 (tptp.v2_waybel_3 A)))))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.lattice A))) (let ((_let_2 (not (tptp.empty_carrier A)))) (=> (tptp.latt_str A) (=> (and _let_2 _let_1 (tptp.distributive_lattstr A)) (and _let_2 (tptp.join_commutative A) (tptp.join_associative A) (tptp.meet_commutative A) (tptp.meet_associative A) (tptp.meet_absorbing A) (tptp.join_absorbing A) _let_1 (tptp.modular_lattstr A))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_1 (tptp.heyting_relstr A)) (and _let_1 (tptp.upper_bounded_relstr A)))))) (forall ((A $$unsorted)) (=> (and (tptp.transitive_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.full_subrelstr B A))) (=> (tptp.subrelstr B A) (=> _let_1 (and (tptp.transitive_relstr B) _let_1))))))) (forall ((A $$unsorted)) (let ((_let_1 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_1 (tptp.boolean_relstr A)) (and _let_1 (tptp.reflexive_relstr A) (tptp.transitive_relstr A) (tptp.antisymmetric_relstr A) (tptp.with_suprema_relstr A) (tptp.with_infima_relstr A) (tptp.lower_bounded_relstr A) (tptp.upper_bounded_relstr A) (tptp.bounded_relstr A) (tptp.distributive_relstr A) (tptp.complemented_relstr A)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.with_suprema_relstr A))) (let ((_let_2 (tptp.reflexive_relstr A))) (=> (tptp.rel_str A) (=> (and _let_2 _let_1 (tptp.up_complete_relstr A)) (and (not (tptp.empty_carrier A)) _let_2 _let_1 (tptp.upper_bounded_relstr A))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.complemented_relstr A))) (let ((_let_2 (tptp.distributive_relstr A))) (let ((_let_3 (tptp.bounded_relstr A))) (let ((_let_4 (tptp.with_infima_relstr A))) (let ((_let_5 (tptp.with_suprema_relstr A))) (let ((_let_6 (tptp.antisymmetric_relstr A))) (let ((_let_7 (tptp.transitive_relstr A))) (let ((_let_8 (tptp.reflexive_relstr A))) (let ((_let_9 (not (tptp.empty_carrier A)))) (=> (tptp.rel_str A) (=> (and _let_9 _let_8 _let_7 _let_6 _let_5 _let_4 _let_3 _let_2 _let_1) (and _let_9 _let_8 _let_7 _let_6 _let_5 _let_4 (tptp.lower_bounded_relstr A) (tptp.upper_bounded_relstr A) _let_3 _let_2 _let_1 (tptp.boolean_relstr A)))))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.unordered_pair_as_carrier_subset A B C) (tptp.unordered_pair_as_carrier_subset A C B))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.unordered_pair A B) (tptp.unordered_pair B A))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_union2 A B) (tptp.set_union2 B A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.join_commutative A) (tptp.join_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.join_commut A B C) (tptp.join_commut A C B))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_intersection2 A B) (tptp.set_intersection2 B A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.meet_commut A B C) (tptp.meet_commut A C B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_union2 A B C) (tptp.subset_union2 A C B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_intersection2 A B C) (tptp.subset_intersection2 A C B))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.ordinal A) (tptp.ordinal B)) (or (tptp.ordinal_subset A B) (tptp.ordinal_subset B A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (= B (tptp.identity_relation A)) (forall ((C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair C D) B) (and (tptp.in C A) (= C D))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= A B) (and (tptp.subset A B) (tptp.subset B A)))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted) (C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (=> (tptp.ex_inf_of_relstr_set A B) (= (= C (tptp.meet_on_relstr A B)) (and (tptp.relstr_element_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.relstr_element_smaller A B D) (tptp.related A D C))))))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (= (tptp.identity_on_carrier A) (tptp.identity_as_relation_of (tptp.the_carrier A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (= C (tptp.relation_dom_restriction A B)) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.ordered_pair D E))) (= (tptp.in _let_1 C) (and (tptp.in D B) (tptp.in _let_1 A)))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.net_str B A)) (forall ((C $$unsorted)) (= (tptp.is_eventually_in A B C) (exists ((D $$unsorted)) (and (tptp.element D (tptp.the_carrier B)) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.the_carrier B)) (=> (tptp.related B D E) (tptp.in (tptp.apply_netmap A B E) C)))))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (= (tptp.bottom_of_relstr A) (tptp.join_on_relstr A tptp.empty_set)))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (forall ((B $$unsorted) (C $$unsorted)) (= (= C (tptp.relation_image A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.relation_dom A)) (tptp.in E B) (= D (tptp.apply A E)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (tptp.relation C) (= (= C (tptp.relation_rng_restriction A B)) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.ordered_pair D E))) (= (tptp.in _let_1 C) (and (tptp.in E A) (tptp.in _let_1 B)))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.antisymmetric A) (tptp.is_antisymmetric_in A (tptp.relation_field A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.net_str B A)) (forall ((C $$unsorted)) (= (tptp.is_often_in A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier B)) (exists ((E $$unsorted)) (and (tptp.element E (tptp.the_carrier B)) (tptp.related B D E) (tptp.in (tptp.apply_netmap A B E) C))))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (and (not (tptp.empty_carrier C)) (tptp.transitive_relstr C) (tptp.directed_relstr C) (tptp.net_str C A)) (= (tptp.subnet C A B) (exists ((D $$unsorted)) (let ((_let_1 (tptp.the_carrier B))) (let ((_let_2 (tptp.the_carrier C))) (and (tptp.function D) (tptp.quasi_total D _let_2 _let_1) (tptp.relation_of2_as_subset D _let_2 _let_1) (= (tptp.the_mapping A C) (tptp.function_of_composition _let_2 _let_1 (tptp.the_carrier A) D (tptp.the_mapping A B))) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.the_carrier B)) (exists ((F $$unsorted)) (and (tptp.element F (tptp.the_carrier C)) (forall ((G $$unsorted)) (let ((_let_1 (tptp.the_carrier C))) (=> (tptp.element G _let_1) (=> (tptp.related C F G) (tptp.related B E (tptp.apply_on_set_and_struct _let_1 B D G))))))))))))))))))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (forall ((B $$unsorted) (C $$unsorted)) (= (= C (tptp.relation_inverse_image A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.relation_dom A)) (tptp.in (tptp.apply A D) B)))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_semilatt_str A)) (= (tptp.lower_bounded_semilattstr A) (exists ((B $$unsorted)) (and (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (and (= (tptp.meet A B C) B) (= (tptp.meet A C B) B))))))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.the_carrier A))) (= (= C (tptp.topstr_closure A B)) (forall ((D $$unsorted)) (=> (tptp.in D (tptp.the_carrier A)) (= (tptp.in D C) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.powerset (tptp.the_carrier A))) (not (and (tptp.open_subset E A) (tptp.in D E) (tptp.disjoint B E))))))))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted) (C $$unsorted)) (= (= C (tptp.relation_image A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in (tptp.ordered_pair E D) A) (tptp.in E B))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (=> (tptp.rel_str B) (= (tptp.subrelstr B A) (and (tptp.subset (tptp.the_carrier B) (tptp.the_carrier A)) (tptp.subset (tptp.the_InternalRel B) (tptp.the_InternalRel A)))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (=> (tptp.net_str B A) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.strict_net_str D A) (tptp.subnetstr D A B)) (= (= D (tptp.preimage_subnetstr A B C)) (and (tptp.full_subrelstr D B) (tptp.subrelstr D B) (= (tptp.the_carrier D) (tptp.function_invverse_img_as_carrier_subset B A (tptp.the_mapping A B) C)))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted) (C $$unsorted)) (= (= C (tptp.relation_inverse_image A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in (tptp.ordered_pair D E) A) (tptp.in E B))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.connected A) (tptp.is_connected_in A (tptp.relation_field A))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (=> (tptp.subrelstr B A) (= (tptp.full_subrelstr B A) (= (tptp.the_InternalRel B) (tptp.relation_restriction_as_relation_of (tptp.the_InternalRel A) (tptp.the_carrier B)))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.latt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (= (tptp.latt_set_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.in D C) (tptp.below A B D)))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_semilatt_str A)) (=> (tptp.lower_bounded_semilattstr A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (= (= B (tptp.bottom_of_semilattstr A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (and (= (tptp.meet A B C) B) (= (tptp.meet A C B) B)))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.transitive A) (tptp.is_transitive_in A (tptp.relation_field A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.latt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (= (tptp.latt_element_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.in D C) (tptp.below A D B)))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.the_carrier A))) (= (= C (tptp.lim_points_of_net A B)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (= (tptp.in D C) (forall ((E $$unsorted)) (=> (tptp.point_neighbourhood E A D) (tptp.is_eventually_in A B E))))))))))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (forall ((B $$unsorted) (C $$unsorted)) (= (tptp.apply_binary A B C) (tptp.apply A (tptp.ordered_pair B C)))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.the_carrier A))) (= (tptp.point_neighbourhood C A B) (tptp.in B (tptp.interior A C))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (= (= D (tptp.unordered_triple A B C)) (forall ((E $$unsorted)) (= (tptp.in E D) (not (and (not (= E A)) (not (= E B)) (not (= E C)))))))) (forall ((A $$unsorted)) (= (tptp.finite A) (exists ((B $$unsorted)) (and (tptp.relation B) (tptp.function B) (= (tptp.relation_rng B) A) (tptp.in (tptp.relation_dom B) tptp.omega))))) (forall ((A $$unsorted)) (= (tptp.function A) (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.in (tptp.ordered_pair B C) A) (tptp.in (tptp.ordered_pair B D) A)) (= C D))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.quasi_total C A B))) (let ((_let_2 (= A tptp.empty_set))) (let ((_let_3 (= B tptp.empty_set))) (=> (tptp.relation_of2_as_subset C A B) (and (=> (=> _let_3 _let_2) (= _let_1 (= A (tptp.relation_dom_as_subset A B C)))) (=> _let_3 (or _let_2 (= _let_1 (= C tptp.empty_set)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.strict_latt_str B) (tptp.latt_str B)) (= (= B (tptp.boole_lattice A)) (and (= (tptp.the_carrier B) (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset A)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset A)) (and (= (tptp.apply_binary (tptp.the_L_join B) C D) (tptp.subset_union2 A C D)) (= (tptp.apply_binary (tptp.the_L_meet B) C D) (tptp.subset_intersection2 A C D))))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.join_semilatt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element C _let_1) (= (tptp.join A B C) (tptp.apply_binary_as_element _let_1 _let_1 _let_1 (tptp.the_L_join A) B C))))))))) (forall ((A $$unsorted)) (=> (exists ((B $$unsorted) (C $$unsorted)) (= A (tptp.ordered_pair B C))) (forall ((B $$unsorted)) (= (= B (tptp.pair_first A)) (forall ((C $$unsorted) (D $$unsorted)) (=> (= A (tptp.ordered_pair C D)) (= B C))))))) (forall ((A $$unsorted)) (= (tptp.succ A) (tptp.set_union2 A (tptp.singleton A)))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (= (tptp.topological_space A) (and (tptp.in (tptp.the_carrier A) (tptp.the_topology A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_topology A))) (let ((_let_2 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset (tptp.powerset _let_2))) (=> (tptp.subset B _let_1) (tptp.in (tptp.union_of_subsets _let_2 B) _let_1)))))) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (forall ((C $$unsorted)) (let ((_let_1 (tptp.the_topology A))) (let ((_let_2 (tptp.the_carrier A))) (=> (tptp.element C (tptp.powerset _let_2)) (=> (and (tptp.in B _let_1) (tptp.in C _let_1)) (tptp.in (tptp.subset_intersection2 _let_2 B C) _let_1)))))))))))) (forall ((A $$unsorted)) (= (tptp.relation A) (forall ((B $$unsorted)) (not (and (tptp.in B A) (forall ((C $$unsorted) (D $$unsorted)) (not (= B (tptp.ordered_pair C D))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_reflexive_in A B) (forall ((C $$unsorted)) (=> (tptp.in C B) (tptp.in (tptp.ordered_pair C C) A))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (tptp.relation_of2 C A B) (tptp.subset C (tptp.cartesian_product2 A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (= B (tptp.set_meet A)))) (let ((_let_2 (= A tptp.empty_set))) (and (=> (not _let_2) (= _let_1 (forall ((C $$unsorted)) (= (tptp.in C B) (forall ((D $$unsorted)) (=> (tptp.in D A) (tptp.in C D))))))) (=> _let_2 (= _let_1 (= B tptp.empty_set))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (= (tptp.empty_carrier A) (tptp.empty (tptp.the_carrier A))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= B (tptp.singleton A)) (forall ((C $$unsorted)) (= (tptp.in C B) (= C A))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.interior A B) (tptp.subset_complement _let_1 (tptp.topstr_closure A (tptp.subset_complement _let_1 B))))))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (= (tptp.open_subsets B A) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.the_carrier A))) (=> (tptp.in C B) (tptp.open_subset C A))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (= (tptp.directed_subset B A) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (not (and (tptp.in C B) (tptp.in D B) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.the_carrier A)) (not (and (tptp.in E B) (tptp.related A C E) (tptp.related A D E)))))))))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted) (C $$unsorted)) (= (= C (tptp.fiber A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (not (= D B)) (tptp.in (tptp.ordered_pair D B) A)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (= B (tptp.inclusion_relation A)) (and (= (tptp.relation_field B) A) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C A) (tptp.in D A)) (= (tptp.in (tptp.ordered_pair C D) B) (tptp.subset C D)))))))) (forall ((A $$unsorted)) (= (= A tptp.empty_set) (forall ((B $$unsorted)) (not (tptp.in B A))))) _let_6 (forall ((A $$unsorted)) (= (tptp.incl_POSet A) (tptp.rel_str_of A (tptp.inclusion_order A)))) (forall ((A $$unsorted) (B $$unsorted)) (= (= B (tptp.powerset A)) (forall ((C $$unsorted)) (= (tptp.in C B) (tptp.subset C A))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (= (tptp.upper_relstr_subset B A) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (and (tptp.in C B) (tptp.related A C D)) (tptp.in D B))))))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.latt_str A))) (let ((_let_2 (not (tptp.empty_carrier A)))) (=> (and _let_2 _let_1) (=> (and _let_2 (tptp.lattice A) (tptp.complete_latt_str A) _let_1) (forall ((B $$unsorted) (C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (= C (tptp.join_of_latt_set A B)) (and (tptp.latt_element_smaller A C B) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.latt_element_smaller A D B) (tptp.below A C D))))))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.latt_str A)) (forall ((B $$unsorted)) (= (tptp.meet_of_latt_set A B) (tptp.join_of_latt_set A (tptp.a_2_2_lattice3 A B)))))) (forall ((A $$unsorted)) (= (tptp.centered A) (and (not (= A tptp.empty_set)) (forall ((B $$unsorted)) (not (and (not (= B tptp.empty_set)) (tptp.subset B A) (tptp.finite B) (= (tptp.set_meet B) tptp.empty_set))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (= (tptp.poset_of_lattice A) (tptp.rel_str_of (tptp.the_carrier A) (tptp.k2_lattice3 A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_semilatt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element C _let_1) (= (tptp.meet A B C) (tptp.apply_binary_as_element _let_1 _let_1 _let_1 (tptp.the_L_meet A) B C))))))))) (forall ((A $$unsorted)) (=> (exists ((B $$unsorted) (C $$unsorted)) (= A (tptp.ordered_pair B C))) (forall ((B $$unsorted)) (= (= B (tptp.pair_second A)) (forall ((C $$unsorted) (D $$unsorted)) (=> (= A (tptp.ordered_pair C D)) (= B D))))))) (forall ((A $$unsorted)) (= (tptp.epsilon_transitive A) (forall ((B $$unsorted)) (=> (tptp.in B A) (tptp.subset B A))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (= (tptp.empty_carrier_subset A) tptp.empty_set))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (= A B) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.ordered_pair C D))) (= (tptp.in _let_1 A) (tptp.in _let_1 B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.element B A))) (let ((_let_2 (tptp.empty A))) (and (=> (not _let_2) (= _let_1 (tptp.in B A))) (=> _let_2 (= _let_1 (tptp.empty B))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.unordered_pair A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (or (= D A) (= D B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B A) (= (tptp.proper_element B A) (not (= B (tptp.union A)))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (= (tptp.closed_subsets B A) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.the_carrier A))) (=> (tptp.in C B) (tptp.closed_subset C A))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.well_founded_relation A) (forall ((B $$unsorted)) (not (and (tptp.subset B (tptp.relation_field A)) (not (= B tptp.empty_set)) (forall ((C $$unsorted)) (not (and (tptp.in C B) (tptp.disjoint (tptp.fiber A C) B)))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.set_union2 A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (or (tptp.in D A) (tptp.in D B)))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (= (tptp.transitive_relstr A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (and (tptp.related A B C) (tptp.related A C D)) (tptp.related A B D))))))))))) (forall ((A $$unsorted)) (= (tptp.boole_POSet A) (tptp.poset_of_lattice (tptp.boole_lattice A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.cartesian_product2 A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted) (F $$unsorted)) (and (tptp.in E A) (tptp.in F B) (= D (tptp.ordered_pair E F)))))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (= (tptp.compact_top_space A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (not (and (tptp.is_a_cover_of_carrier A B) (tptp.open_subsets B A) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (not (and (tptp.subset C B) (tptp.is_a_cover_of_carrier A C) (tptp.finite C)))))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (= (tptp.cast_to_el_of_LattPOSet A B) B))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.join_semilatt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.below A B C) (= (tptp.join A B C) C)))))))) (forall ((A $$unsorted)) (= (tptp.epsilon_connected A) (forall ((B $$unsorted) (C $$unsorted)) (not (and (tptp.in B A) (tptp.in C A) (not (tptp.in B C)) (not (= B C)) (not (tptp.in C B))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (= (tptp.cast_as_carrier_subset A) (tptp.the_carrier A)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (tptp.subset A B) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.ordered_pair C D))) (=> (tptp.in _let_1 A) (tptp.in _let_1 B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.subset A B) (forall ((C $$unsorted)) (=> (tptp.in C A) (tptp.in C B))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_well_founded_in A B) (forall ((C $$unsorted)) (not (and (tptp.subset C B) (not (= C tptp.empty_set)) (forall ((D $$unsorted)) (not (and (tptp.in D C) (tptp.disjoint (tptp.fiber A D) C))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.set_intersection2 A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D A) (tptp.in D B)))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (forall ((B $$unsorted) (C $$unsorted)) (let ((_let_1 (= C (tptp.apply A B)))) (let ((_let_2 (tptp.in B (tptp.relation_dom A)))) (and (=> _let_2 (= _let_1 (tptp.in (tptp.ordered_pair B C) A))) (=> (not _let_2) (= _let_1 (= C tptp.empty_set))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier (tptp.poset_of_lattice A))) (= (tptp.cast_to_el_of_lattice A B) B))))) (forall ((A $$unsorted)) (= (tptp.ordinal A) (and (tptp.epsilon_transitive A) (tptp.epsilon_connected A)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (= B (tptp.relation_dom A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (tptp.in (tptp.ordered_pair C D) A)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_antisymmetric_in A B) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C B) (tptp.in D B) (tptp.in (tptp.ordered_pair C D) A) (tptp.in (tptp.ordered_pair D C) A)) (= C D))))))) (forall ((A $$unsorted)) (= (tptp.cast_to_subset A) A)) (forall ((A $$unsorted) (B $$unsorted)) (= (= B (tptp.union A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (and (tptp.in C D) (tptp.in D A))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.well_ordering A) (and (tptp.reflexive A) (tptp.transitive A) (tptp.antisymmetric A) (tptp.connected A) (tptp.well_founded_relation A))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.equipotent A B) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (tptp.one_to_one C) (= (tptp.relation_dom C) A) (= (tptp.relation_rng C) B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.set_difference A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D A) (not (tptp.in D B))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (= (tptp.lower_bounded_relstr A) (exists ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (and (tptp.element B _let_1) (tptp.relstr_element_smaller A _let_1 B))))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (forall ((B $$unsorted)) (= (= B (tptp.relation_rng A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (and (tptp.in D (tptp.relation_dom A)) (= C (tptp.apply A D)))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (= (tptp.transitive_relstr A) (tptp.is_transitive_in (tptp.the_InternalRel A) (tptp.the_carrier A))))) (forall ((A $$unsorted)) (= (= A tptp.omega) (and (tptp.in tptp.empty_set A) (tptp.being_limit_ordinal A) (tptp.ordinal A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (=> (and (tptp.in tptp.empty_set B) (tptp.being_limit_ordinal B)) (tptp.subset A B))))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (= (tptp.open_subset B A) (tptp.in B (tptp.the_topology A))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (= B (tptp.relation_rng A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (tptp.in (tptp.ordered_pair D C) A)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (= (tptp.subset_complement A B) (tptp.set_difference A B)))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.ordered_pair A B) (tptp.unordered_pair (tptp.unordered_pair A B) (tptp.singleton A)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.well_orders A B) (and (tptp.is_reflexive_in A B) (tptp.is_transitive_in A B) (tptp.is_antisymmetric_in A B) (tptp.is_connected_in A B) (tptp.is_well_founded_in A B)))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.rel_str A)) (= (tptp.directed_relstr A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (exists ((D $$unsorted)) (and (tptp.element D (tptp.the_carrier A)) (tptp.related A B D) (tptp.related A C D)))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (= (tptp.antisymmetric_relstr A) (tptp.is_antisymmetric_in (tptp.the_InternalRel A) (tptp.the_carrier A))))) (forall ((A $$unsorted)) (= (tptp.being_limit_ordinal A) (= A (tptp.union A)))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.closed_subset B A) (tptp.open_subset (tptp.subset_difference _let_1 (tptp.cast_as_carrier_subset A) B) A))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_field A) (tptp.set_union2 (tptp.relation_dom A) (tptp.relation_rng A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_connected_in A B) (forall ((C $$unsorted) (D $$unsorted)) (not (and (tptp.in C B) (tptp.in D B) (not (= C D)) (not (tptp.in (tptp.ordered_pair C D) A)) (not (tptp.in (tptp.ordered_pair D C) A))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.relation_restriction A B) (tptp.set_intersection2 A (tptp.cartesian_product2 B B)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (= B (tptp.relation_inverse A)) (forall ((C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair C D) B) (tptp.in (tptp.ordered_pair D C) A)))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier B)) (forall ((D $$unsorted)) (let ((_let_1 (tptp.the_carrier D))) (=> (and (tptp.strict_net_str D A) (tptp.net_str D A)) (= (= D (tptp.netstr_restr_to_element A B C)) (and (forall ((E $$unsorted)) (= (tptp.in E (tptp.the_carrier D)) (exists ((F $$unsorted)) (and (tptp.element F (tptp.the_carrier B)) (= F E) (tptp.related B C F))))) (= (tptp.the_InternalRel D) (tptp.relation_restriction_as_relation_of (tptp.the_InternalRel B) _let_1)) (= (tptp.the_mapping A D) (tptp.partfun_dom_restriction (tptp.the_carrier B) (tptp.the_carrier A) (tptp.the_mapping A B) _let_1))))))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.relation_isomorphism A B C) (and (= (tptp.relation_dom C) (tptp.relation_field A)) (= (tptp.relation_rng C) (tptp.relation_field B)) (tptp.one_to_one C) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.relation_field A))) (= (tptp.in (tptp.ordered_pair D E) A) (and (tptp.in D _let_1) (tptp.in E _let_1) (tptp.in (tptp.ordered_pair (tptp.apply C D) (tptp.apply C E)) B))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.disjoint A B) (= (tptp.set_intersection2 A B) tptp.empty_set))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (= (tptp.ex_sup_of_relstr_set A B) (exists ((C $$unsorted)) (and (tptp.element C (tptp.the_carrier A)) (tptp.relstr_set_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.relstr_set_smaller A B D) (tptp.related A C D)))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (and (tptp.relstr_set_smaller A B D) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.the_carrier A)) (=> (tptp.relstr_set_smaller A B E) (tptp.related A D E))))) (= D C)))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (= (tptp.relation_of_lattice A) (tptp.a_1_0_filter_1 A)))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (= (tptp.one_to_one A) (forall ((B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_dom A))) (=> (and (tptp.in B _let_1) (tptp.in C _let_1) (= (tptp.apply A B) (tptp.apply A C))) (= B C))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted) (C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.relstr_element_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.in D B) (tptp.related A C D))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.latt_str A)) (= (tptp.meet_absorbing A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.join A (tptp.meet A B C) C) C)))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset (tptp.powerset _let_1))) (= (tptp.is_a_cover_of_carrier A B) (= (tptp.cast_as_carrier_subset A) (tptp.union_of_subsets _let_1 B)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (tptp.relation C) (= (= C (tptp.relation_composition A B)) (forall ((D $$unsorted) (E $$unsorted)) (= (tptp.in (tptp.ordered_pair D E) C) (exists ((F $$unsorted)) (and (tptp.in (tptp.ordered_pair D F) A) (tptp.in (tptp.ordered_pair F E) B)))))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_transitive_in A B) (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (tptp.in C B) (tptp.in D B) (tptp.in E B) (tptp.in (tptp.ordered_pair C D) A) (tptp.in (tptp.ordered_pair D E) A)) (tptp.in (tptp.ordered_pair C E) A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.powerset A))) (= (= C (tptp.complements_of_subsets A B)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset A)) (= (tptp.in D C) (tptp.in (tptp.subset_complement A D) B))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier B)) (= (tptp.apply_netmap A B C) (tptp.apply_on_structs B A (tptp.the_mapping A B) C)))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.proper_subset A B) (and (tptp.subset A B) (not (= A B))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (= (tptp.ex_inf_of_relstr_set A B) (exists ((C $$unsorted)) (and (tptp.element C (tptp.the_carrier A)) (tptp.relstr_element_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.relstr_element_smaller A B D) (tptp.related A D C)))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (and (tptp.relstr_element_smaller A B D) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.the_carrier A)) (=> (tptp.relstr_element_smaller A B E) (tptp.related A E D))))) (= D C)))))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (=> (tptp.net_str B A) (forall ((C $$unsorted)) (=> (tptp.net_str C A) (= (tptp.subnetstr C A B) (and (tptp.subrelstr C B) (= (tptp.the_mapping A C) (tptp.relation_dom_restr_as_relation_of (tptp.the_carrier B) (tptp.the_carrier A) (tptp.the_mapping A B) (tptp.the_carrier C))))))))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (= (tptp.function_inverse A) (tptp.relation_inverse A))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted) (C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.relstr_set_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.in D B) (tptp.related A D C))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.related A B C) (tptp.in (tptp.ordered_pair B C) (tptp.the_InternalRel A))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.reflexive A) (tptp.is_reflexive_in A (tptp.relation_field A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.is_a_cluster_point_of_netstr A B C) (forall ((D $$unsorted)) (=> (tptp.point_neighbourhood D A C) (tptp.is_often_in A B D)))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted) (C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (=> (tptp.ex_sup_of_relstr_set A B) (= (= C (tptp.join_on_relstr A B)) (and (tptp.relstr_set_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.relstr_set_smaller A B D) (tptp.related A C D))))))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (=> (tptp.net_str B A) (forall ((C $$unsorted)) (=> (tptp.subnetstr C A B) (= (tptp.full_subnetstr C A B) (and (tptp.full_subrelstr C B) (tptp.subrelstr C B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.rel_str_of A B))) (=> (tptp.relation_of2 B A A) (and (tptp.strict_rel_str _let_1) (tptp.rel_str _let_1))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.net_str_of A B C D))) (let ((_let_2 (tptp.the_carrier A))) (=> (and (tptp.one_sorted_str A) (tptp.relation_of2 C B B) (tptp.function D) (tptp.quasi_total D B _let_2) (tptp.relation_of2 D B _let_2)) (and (tptp.strict_net_str _let_1 A) (tptp.net_str _let_1 A)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.latt_str_of A B C))) (let ((_let_2 (tptp.cartesian_product2 A A))) (=> (and (tptp.function B) (tptp.quasi_total B _let_2 A) (tptp.relation_of2 B _let_2 A) (tptp.function C) (tptp.quasi_total C _let_2 A) (tptp.relation_of2 C _let_2 A)) (and (tptp.strict_latt_str _let_1) (tptp.latt_str _let_1)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A) (not (tptp.empty_carrier B)) (tptp.lattice B) (tptp.latt_str B) (tptp.element C (tptp.the_carrier A)) (tptp.element D (tptp.the_carrier B))) (tptp.element (tptp.k10_filter_1 A B C D) (tptp.the_carrier (tptp.k8_filter_1 A B))))) true (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A) (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (tptp.element (tptp.lim_points_of_net A B) (tptp.powerset (tptp.the_carrier A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.latt_str A)) (tptp.element (tptp.join_of_latt_set A B) (tptp.the_carrier A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.latt_str A)) (tptp.element (tptp.meet_of_latt_set A B) (tptp.the_carrier A)))) true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (not (tptp.empty A)) (not (tptp.empty B)) (tptp.element C A) (tptp.element D B)) (tptp.element (tptp.ordered_pair_as_product_element A B C D) (tptp.cartesian_product2 A B)))) true true (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_lattice A))) (and (tptp.strict_latt_str _let_1) (tptp.latt_str _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.join_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.join A B C) _let_1)))) true true (forall ((A $$unsorted)) (tptp.element (tptp.k1_pcomps_1 A) (tptp.powerset (tptp.powerset A)))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (tptp.element (tptp.empty_carrier_subset A) (tptp.powerset (tptp.the_carrier A))))) true true true (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation A) (tptp.relation_of2_as_subset (tptp.relation_restriction_as_relation_of A B) B B))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset (tptp.the_carrier A)))) (=> (and (tptp.top_str A) (tptp.element B _let_1)) (tptp.element (tptp.interior A B) _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier B))) (let ((_let_2 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (not (tptp.empty_carrier B)) (tptp.one_sorted_str B) (tptp.function C) (tptp.quasi_total C _let_2 _let_1) (tptp.relation_of2 C _let_2 _let_1) (tptp.element D _let_2)) (tptp.element (tptp.apply_on_structs A B C D) _let_1))))) true (forall ((A $$unsorted)) (tptp.relation (tptp.inclusion_relation A))) true (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.the_carrier A))) (tptp.element (tptp.neighborhood_system A B) (tptp.powerset (tptp.the_carrier (tptp.boole_POSet (tptp.cast_as_carrier_subset A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.rel_str A) (tptp.element (tptp.join_on_relstr A B) (tptp.the_carrier A)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.inclusion_order A))) (and (tptp.reflexive _let_1) (tptp.antisymmetric _let_1) (tptp.transitive _let_1) (tptp.v1_partfun1 _let_1 A A) (tptp.relation_of2_as_subset _let_1 A A)))) true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted) (E $$unsorted) (F $$unsorted)) (let ((_let_1 (tptp.cartesian_product2 A B))) (=> (and (not (tptp.empty A)) (not (tptp.empty B)) (tptp.function D) (tptp.quasi_total D _let_1 C) (tptp.relation_of2 D _let_1 C) (tptp.element E A) (tptp.element F B)) (tptp.element (tptp.apply_binary_as_element A B C D E F) C)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.function_inverse A))) (=> (and (tptp.relation A) (tptp.function A)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.k2_lattice3 A))) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (and (tptp.reflexive _let_2) (tptp.antisymmetric _let_2) (tptp.transitive _let_2) (tptp.v1_partfun1 _let_2 _let_1 _let_1) (tptp.relation_of2_as_subset _let_2 _let_1 _let_1)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.meet A B C) _let_1)))) true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.partfun_dom_restriction A B C D))) (=> (and (tptp.function C) (tptp.relation_of2 C A B)) (and (tptp.function _let_1) (tptp.relation_of2_as_subset _let_1 A B))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (tptp.element (tptp.cast_as_carrier_subset A) (tptp.powerset (tptp.the_carrier A))))) true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.unordered_pair_as_carrier_subset A B C) (tptp.powerset _let_1))))) (forall ((A $$unsorted)) (tptp.element (tptp.cast_to_subset A) (tptp.powerset A))) true (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation A) (tptp.relation (tptp.relation_restriction A B)))) true (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.rel_str A) (tptp.element (tptp.meet_on_relstr A B) (tptp.the_carrier A)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.incl_POSet A))) (and (tptp.strict_rel_str _let_1) (tptp.rel_str _let_1)))) true (forall ((A $$unsorted)) (let ((_let_1 (tptp.poset_of_lattice A))) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (and (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.rel_str _let_1))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.join_commutative A) (tptp.join_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.join_commut A B C) _let_1)))) true (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B _let_1) (tptp.element (tptp.subset_complement A B) _let_1)))) true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (not (tptp.empty_carrier B)) (tptp.net_str B A) (tptp.element C (tptp.the_carrier B))) (tptp.element (tptp.apply_netmap A B C) (tptp.the_carrier A)))) true (forall ((A $$unsorted)) (=> (tptp.rel_str A) (tptp.element (tptp.bottom_of_relstr A) (tptp.the_carrier A)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (and (tptp.strict_rel_str _let_1) (tptp.rel_str _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier B))) (=> (and (not (tptp.empty A)) (not (tptp.empty_carrier B)) (tptp.rel_str B) (tptp.function C) (tptp.quasi_total C A _let_1) (tptp.relation_of2 C A _let_1) (tptp.element D A)) (tptp.element (tptp.apply_on_set_and_struct A B C D) _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A) (tptp.element B (tptp.the_carrier A))) (tptp.element (tptp.cast_to_el_of_LattPOSet A B) (tptp.the_carrier (tptp.poset_of_lattice A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.meet_commut A B C) _let_1)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (tptp.relation (tptp.relation_inverse A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2 C A B) (tptp.element (tptp.relation_dom_as_subset A B C) (tptp.powerset A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.subset_union2 A B C) _let_1)))) true true (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A) (tptp.element B (tptp.the_carrier (tptp.poset_of_lattice A)))) (tptp.element (tptp.cast_to_el_of_lattice A B) (tptp.the_carrier A)))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_semilatt_str A)) (tptp.element (tptp.bottom_of_semilattstr A) (tptp.the_carrier A)))) true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.the_carrier B))) (=> (and (tptp.one_sorted_str A) (tptp.one_sorted_str B) (tptp.function C) (tptp.quasi_total C _let_1 _let_2) (tptp.relation_of2 C _let_1 _let_2)) (tptp.element (tptp.function_invverse_img_as_carrier_subset A B C D) (tptp.powerset _let_1)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.relation_composition A B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2 C A B) (tptp.element (tptp.relation_rng_as_subset A B C) (tptp.powerset B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B (tptp.powerset _let_1)) (tptp.element (tptp.union_of_subsets A B) _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.subset_intersection2 A B C) _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.netstr_restr_to_element A B C))) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (not (tptp.empty_carrier B)) (tptp.net_str B A) (tptp.element C (tptp.the_carrier B))) (and (tptp.strict_net_str _let_1 A) (tptp.net_str _let_1 A))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.identity_as_relation_of A))) (and (tptp.v1_partfun1 _let_1 A A) (tptp.relation_of2_as_subset _let_1 A A)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset (tptp.the_carrier A)))) (=> (and (tptp.top_str A) (tptp.element B _let_1)) (tptp.element (tptp.topstr_closure A B) _let_1)))) (forall ((A $$unsorted)) (tptp.relation (tptp.identity_relation A))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B (tptp.powerset _let_1)) (tptp.element (tptp.meet_of_subsets A B) _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.subset_difference A B C) _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.preimage_subnetstr A B C))) (=> (and (tptp.one_sorted_str A) (tptp.net_str B A)) (and (tptp.strict_net_str _let_1 A) (tptp.subnetstr _let_1 A B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.function_of_composition A B C D E))) (=> (and (not (tptp.empty B)) (tptp.function D) (tptp.quasi_total D A B) (tptp.relation_of2 D A B) (tptp.function E) (tptp.quasi_total E B C) (tptp.relation_of2 E B C)) (and (tptp.function _let_1) (tptp.quasi_total _let_1 A C) (tptp.relation_of2_as_subset _let_1 A C))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.identity_on_carrier A))) (=> (tptp.one_sorted_str A) (and (tptp.function _let_2) (tptp.quasi_total _let_2 _let_1 _let_1) (tptp.relation_of2_as_subset _let_2 _let_1 _let_1)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation A) (tptp.relation (tptp.relation_dom_restriction A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset (tptp.powerset A)))) (=> (tptp.element B _let_1) (tptp.element (tptp.complements_of_subsets A B) _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier B))) (=> (and (not (tptp.empty A)) (not (tptp.empty_carrier B)) (tptp.one_sorted_str B) (tptp.function C) (tptp.quasi_total C A _let_1) (tptp.relation_of2 C A _let_1) (tptp.element D A)) (tptp.element (tptp.apply_on_set_and_struct2 A B C D) _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.k8_filter_1 A B))) (=> (and (not (tptp.empty_carrier A)) (tptp.latt_str A) (not (tptp.empty_carrier B)) (tptp.latt_str B)) (and (tptp.strict_latt_str _let_1) (tptp.latt_str _let_1))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.function C) (tptp.quasi_total C A B) (tptp.relation_of2 C A B) (tptp.element D A)) (tptp.element (tptp.apply_as_element A B C D) B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.relation (tptp.relation_rng_restriction A B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (tptp.relation_of2 C A B) (tptp.relation_of2_as_subset (tptp.relation_dom_restr_as_relation_of A B C D) A B))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (tptp.relation (tptp.relation_of_lattice A)))) true (forall ((A $$unsorted)) (=> (tptp.meet_semilatt_str A) (tptp.one_sorted_str A))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (tptp.one_sorted_str A))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (tptp.one_sorted_str A))) true (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (=> (tptp.net_str B A) (tptp.rel_str B))))) (forall ((A $$unsorted)) (=> (tptp.join_semilatt_str A) (tptp.one_sorted_str A))) (forall ((A $$unsorted)) (=> (tptp.latt_str A) (and (tptp.meet_semilatt_str A) (tptp.join_semilatt_str A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.the_carrier A))) (forall ((C $$unsorted)) (=> (tptp.point_neighbourhood C A B) (tptp.element C (tptp.powerset (tptp.the_carrier A))))))) true true (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (=> (tptp.subrelstr B A) (tptp.rel_str B))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.subnetstr C A B) (tptp.net_str C A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2_as_subset C A B) (tptp.element C (tptp.powerset (tptp.cartesian_product2 A B))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.subnet C A B) (and (not (tptp.empty_carrier C)) (tptp.transitive_relstr C) (tptp.directed_relstr C) (tptp.net_str C A)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.cartesian_product2 _let_1 _let_1))) (let ((_let_3 (tptp.the_L_meet A))) (=> (tptp.meet_semilatt_str A) (and (tptp.function _let_3) (tptp.quasi_total _let_3 _let_2 _let_1) (tptp.relation_of2_as_subset _let_3 _let_2 _let_1))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.rel_str A) (tptp.relation_of2_as_subset (tptp.the_InternalRel A) _let_1 _let_1)))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (tptp.element (tptp.the_topology A) (tptp.powerset (tptp.powerset (tptp.the_carrier A)))))) true (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.the_carrier B))) (let ((_let_3 (tptp.the_mapping A B))) (=> (and (tptp.one_sorted_str A) (tptp.net_str B A)) (and (tptp.function _let_3) (tptp.quasi_total _let_3 _let_2 _let_1) (tptp.relation_of2_as_subset _let_3 _let_2 _let_1))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.cartesian_product2 _let_1 _let_1))) (let ((_let_3 (tptp.the_L_join A))) (=> (tptp.join_semilatt_str A) (and (tptp.function _let_3) (tptp.quasi_total _let_3 _let_2 _let_1) (tptp.relation_of2_as_subset _let_3 _let_2 _let_1))))))) (exists ((A $$unsorted)) (tptp.meet_semilatt_str A)) (exists ((A $$unsorted)) (tptp.rel_str A)) (exists ((A $$unsorted)) (tptp.top_str A)) (exists ((A $$unsorted)) (tptp.one_sorted_str A)) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (exists ((B $$unsorted)) (tptp.net_str B A)))) (exists ((A $$unsorted)) (tptp.join_semilatt_str A)) (exists ((A $$unsorted)) (tptp.latt_str A)) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.the_carrier A))) (exists ((C $$unsorted)) (tptp.point_neighbourhood C A B)))) (forall ((A $$unsorted) (B $$unsorted)) (exists ((C $$unsorted)) (tptp.relation_of2 C A B))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (tptp.element B A))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (exists ((B $$unsorted)) (tptp.subrelstr B A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.net_str B A)) (exists ((C $$unsorted)) (tptp.subnetstr C A B)))) (forall ((A $$unsorted) (B $$unsorted)) (exists ((C $$unsorted)) (tptp.relation_of2_as_subset C A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (exists ((C $$unsorted)) (tptp.subnet C A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.finite B) (tptp.finite (tptp.set_intersection2 A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_composition B A))) (=> (and (tptp.empty A) (tptp.relation B)) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.finite A) (tptp.finite (tptp.set_intersection2 A B)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_inverse A))) (=> (tptp.empty A) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.finite A) (tptp.finite (tptp.set_difference A B)))) (and _let_3 _let_4 _let_5) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.function A) (tptp.finite B)) (tptp.finite (tptp.relation_image A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_dom_restriction A B))) (=> (and (tptp.relation A) (tptp.relation_empty_yielding A)) (and (tptp.relation _let_1) (tptp.relation_empty_yielding _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_InternalRel A))) (=> (and (not (tptp.v1_yellow_3 A)) (tptp.rel_str A)) (and (not (tptp.empty _let_1)) (tptp.relation _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.finite A) (tptp.finite B)) (tptp.finite (tptp.cartesian_product2 A B)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (=> (and (not (tptp.empty_carrier A)) (tptp.rel_str A)) (and (not (tptp.empty _let_1)) (tptp.lower_relstr_subset _let_1 A) (tptp.upper_relstr_subset _let_1 A))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.the_mapping A B))) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (not (tptp.empty_carrier B)) (tptp.net_str B A)) (and (not (tptp.empty _let_1)) (tptp.relation _let_1) (tptp.function _let_1) (tptp.quasi_total _let_1 (tptp.the_carrier B) (tptp.the_carrier A)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.preimage_subnetstr A B C))) (=> (and (tptp.one_sorted_str A) (tptp.transitive_relstr B) (tptp.net_str B A)) (and (tptp.transitive_relstr _let_1) (tptp.strict_net_str _let_1 A) (tptp.full_subnetstr _let_1 A B))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.singleton A))) (and (not (tptp.empty _let_1)) (tptp.finite _let_1)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.powerset A))) (and (not (tptp.empty _let_1)) (tptp.cup_closed _let_1) (tptp.diff_closed _let_1) (tptp.preboolean _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_composition A B))) (=> (and (tptp.relation A) (tptp.function A) (tptp.relation B) (tptp.function B)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_lattice A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_latt_str _let_1) (tptp.join_commutative _let_1) (tptp.join_associative _let_1) (tptp.meet_commutative _let_1) (tptp.meet_associative _let_1) (tptp.meet_absorbing _let_1) (tptp.join_absorbing _let_1) (tptp.lattice _let_1) (tptp.distributive_lattstr _let_1) (tptp.modular_lattstr _let_1) (tptp.lower_bounded_semilattstr _let_1) (tptp.upper_bounded_semilattstr _let_1) (tptp.bounded_lattstr _let_1) (tptp.complemented_lattstr _let_1) (tptp.boolean_lattstr _let_1) (tptp.complete_latt_str _let_1)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_lattice A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_latt_str _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.rel_str_of A B))) (=> (and (not (tptp.empty A)) (tptp.relation_of2 B A A)) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1))))) (forall ((A $$unsorted)) (not (tptp.empty (tptp.succ A)))) (and (tptp.epsilon_transitive tptp.omega) (tptp.epsilon_connected tptp.omega) (tptp.ordinal tptp.omega) (not (tptp.empty tptp.omega))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.empty_carrier_subset A))) (=> (tptp.one_sorted_str A) (and (tptp.empty _let_1) (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1) (tptp.v5_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.set_intersection2 A B)))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (not (tptp.empty (tptp.the_carrier A))))) (forall ((A $$unsorted)) (not (tptp.empty (tptp.powerset A)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1) (tptp.complete_relstr _let_1) (tptp.lower_bounded_relstr _let_1) (tptp.upper_bounded_relstr _let_1) (tptp.bounded_relstr _let_1) (tptp.up_complete_relstr _let_1) (tptp.join_complete_relstr _let_1) (tptp.distributive_relstr _let_1)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.lower_bounded_relstr _let_1) (tptp.upper_bounded_relstr _let_1) (tptp.bounded_relstr _let_1) (tptp.up_complete_relstr _let_1) (tptp.join_complete_relstr _let_1) (not (tptp.v1_yellow_3 _let_1)) (tptp.distributive_relstr _let_1) (tptp.heyting_relstr _let_1) (tptp.complemented_relstr _let_1) (tptp.boolean_relstr _let_1) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1) (tptp.complete_relstr _let_1)))) _let_3 (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.singleton A))) (let ((_let_2 (tptp.rel_str_of _let_1 B))) (=> (tptp.relation_of2 B _let_1 _let_1) (and (not (tptp.empty_carrier _let_2)) (tptp.strict_rel_str _let_2) (tptp.trivial_carrier _let_2)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.poset_of_lattice A))) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (not (tptp.empty (tptp.ordered_pair A B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.netstr_restr_to_element A B C))) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (not (tptp.empty_carrier B)) (tptp.directed_relstr B) (tptp.net_str B A) (tptp.element C (tptp.the_carrier B))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_net_str _let_1 A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.v1_membered A) (tptp.v1_membered (tptp.set_intersection2 A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.v1_membered A) (tptp.v1_membered (tptp.set_intersection2 B A)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 A B))) (=> (tptp.v2_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.succ A))) (=> (and (tptp.ordinal A) (tptp.natural A)) (and (not (tptp.empty _let_1)) (tptp.epsilon_transitive _let_1) (tptp.epsilon_connected _let_1) (tptp.ordinal _let_1) (tptp.natural _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.unordered_pair A B))) (and (not (tptp.empty _let_1)) (tptp.finite _let_1)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.identity_relation A))) (and (tptp.relation _let_1) (tptp.function _let_1)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.cartesian_product2 _let_1 _let_1))) (let ((_let_3 (tptp.the_L_join A))) (=> (and (not (tptp.empty_carrier A)) (tptp.join_commutative A) (tptp.join_semilatt_str A)) (and (tptp.relation _let_3) (tptp.function _let_3) (tptp.quasi_total _let_3 _let_2 _let_1) (tptp.v1_binop_1 _let_3 _let_1) (tptp.v1_partfun1 _let_3 _let_2 _let_1))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_lattice A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_latt_str _let_1) (tptp.join_commutative _let_1) (tptp.join_associative _let_1) (tptp.meet_commutative _let_1) (tptp.meet_associative _let_1) (tptp.meet_absorbing _let_1) (tptp.join_absorbing _let_1) (tptp.lattice _let_1)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.the_InternalRel A))) (=> (and (tptp.reflexive_relstr A) (tptp.transitive_relstr A) (tptp.antisymmetric_relstr A) (tptp.rel_str A)) (and (tptp.relation _let_2) (tptp.reflexive _let_2) (tptp.antisymmetric _let_2) (tptp.transitive _let_2) (tptp.v1_partfun1 _let_2 _let_1 _let_1)))))) (and _let_4 _let_5 (tptp.function tptp.empty_set) (tptp.one_to_one tptp.empty_set) _let_3 (tptp.epsilon_transitive tptp.empty_set) (tptp.epsilon_connected tptp.empty_set) (tptp.ordinal tptp.empty_set)) (forall ((A $$unsorted)) (let ((_let_1 (tptp.identity_relation A))) (and (tptp.relation _let_1) (tptp.function _let_1) (tptp.reflexive _let_1) (tptp.symmetric _let_1) (tptp.antisymmetric _let_1) (tptp.transitive _let_1)))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (not (tptp.empty (tptp.cast_as_carrier_subset A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.set_union2 A B)))) (forall ((A $$unsorted)) (not (tptp.empty (tptp.singleton A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (tptp.closed_subset (tptp.topstr_closure A B) A))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (=> (and (tptp.with_suprema_relstr A) (tptp.rel_str A)) (and (not (tptp.empty _let_1)) (tptp.directed_subset _let_1 A))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (=> (not (tptp.empty A)) (and (not (tptp.empty_carrier _let_1)) (not (tptp.trivial_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.lower_bounded_relstr _let_1) (tptp.upper_bounded_relstr _let_1) (tptp.bounded_relstr _let_1) (tptp.up_complete_relstr _let_1) (tptp.join_complete_relstr _let_1) (not (tptp.v1_yellow_3 _let_1)) (tptp.distributive_relstr _let_1) (tptp.heyting_relstr _let_1) (tptp.complemented_relstr _let_1) (tptp.boolean_relstr _let_1) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1) (tptp.complete_relstr _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.empty A)) (not (tptp.empty (tptp.set_union2 A B))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.rel_str A)) (not (tptp.empty (tptp.cast_as_carrier_subset A))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.poset_of_lattice A))) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.upper_bounded_semilattstr A) (tptp.latt_str A)) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.upper_bounded_relstr _let_1) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 B A))) (=> (tptp.v2_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 A B))) (=> (tptp.v3_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 B A))) (=> (tptp.v3_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 A B))) (=> (tptp.v4_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 B A))) (=> (tptp.v4_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 A B))) (=> (tptp.v5_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1) (tptp.v5_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 B A))) (=> (tptp.v5_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1) (tptp.v5_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.v1_membered A) (tptp.v1_membered (tptp.set_difference A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_difference A B))) (=> (tptp.v2_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_difference A B))) (=> (tptp.v3_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_inverse A))) (=> (and (tptp.relation A) (tptp.function A) (tptp.one_to_one A)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.cartesian_product2 _let_1 _let_1))) (let ((_let_3 (tptp.the_L_join A))) (=> (and (not (tptp.empty_carrier A)) (tptp.join_associative A) (tptp.join_semilatt_str A)) (and (tptp.relation _let_3) (tptp.function _let_3) (tptp.quasi_total _let_3 _let_2 _let_1) (tptp.v2_binop_1 _let_3 _let_1) (tptp.v1_partfun1 _let_3 _let_2 _let_1))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_lattice A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_latt_str _let_1) (tptp.join_commutative _let_1) (tptp.join_associative _let_1) (tptp.meet_commutative _let_1) (tptp.meet_associative _let_1) (tptp.meet_absorbing _let_1) (tptp.join_absorbing _let_1) (tptp.lattice _let_1) (tptp.distributive_lattstr _let_1) (tptp.modular_lattstr _let_1) (tptp.lower_bounded_semilattstr _let_1) (tptp.upper_bounded_semilattstr _let_1) (tptp.bounded_lattstr _let_1) (tptp.complemented_lattstr _let_1) (tptp.boolean_lattstr _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.latt_str_of A B C))) (let ((_let_2 (tptp.cartesian_product2 A A))) (=> (and (not (tptp.empty A)) (tptp.function B) (tptp.quasi_total B _let_2 A) (tptp.relation_of2 B _let_2 A) (tptp.function C) (tptp.quasi_total C _let_2 A) (tptp.relation_of2 C _let_2 A)) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_latt_str _let_1)))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.rel_str_of A B))) (=> (and (tptp.reflexive B) (tptp.antisymmetric B) (tptp.transitive B) (tptp.v1_partfun1 B A A) (tptp.relation_of2 B A A)) (and (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.succ A))) (=> (tptp.ordinal A) (and (not (tptp.empty _let_1)) (tptp.epsilon_transitive _let_1) (tptp.epsilon_connected _let_1) (tptp.ordinal _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.set_difference A B)))) (forall ((A $$unsorted) (B $$unsorted)) (not (tptp.empty (tptp.unordered_pair A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.closed_subset B A) (tptp.element B (tptp.powerset _let_1))) (tptp.open_subset (tptp.subset_complement _let_1 B) A)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (=> (and (not (tptp.empty_carrier A)) (tptp.upper_bounded_relstr A) (tptp.rel_str A)) (and (not (tptp.empty _let_1)) (tptp.directed_subset _let_1 A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.empty A)) (not (tptp.empty (tptp.set_union2 B A))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.poset_of_lattice A))) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.lower_bounded_semilattstr A) (tptp.latt_str A)) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.lower_bounded_relstr _let_1) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_difference A B))) (=> (tptp.v4_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_difference A B))) (=> (tptp.v5_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1) (tptp.v5_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_dom_restriction A B))) (=> (and (tptp.relation A) (tptp.function A)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.cartesian_product2 _let_1 _let_1))) (let ((_let_3 (tptp.the_L_meet A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_semilatt_str A)) (and (tptp.relation _let_3) (tptp.function _let_3) (tptp.quasi_total _let_3 _let_2 _let_1) (tptp.v1_binop_1 _let_3 _let_1) (tptp.v1_partfun1 _let_3 _let_2 _let_1))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.poset_of_lattice A))) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.union A))) (=> (tptp.ordinal A) (and (tptp.epsilon_transitive _let_1) (tptp.epsilon_connected _let_1) (tptp.ordinal _let_1))))) (and _let_3 _let_4) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (not (tptp.empty B))) (not (tptp.empty (tptp.cartesian_product2 A B))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.open_subset B A) (tptp.element B (tptp.powerset _let_1))) (tptp.closed_subset (tptp.subset_complement _let_1 B) A)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (=> (and (tptp.with_infima_relstr A) (tptp.rel_str A)) (and (not (tptp.empty _let_1)) (tptp.filtered_subset _let_1 A))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.poset_of_lattice A))) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.complete_latt_str A) (tptp.latt_str A)) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.lower_bounded_relstr _let_1) (tptp.upper_bounded_relstr _let_1) (tptp.bounded_relstr _let_1) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1) (tptp.complete_relstr _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_rng_restriction A B))) (=> (and (tptp.relation B) (tptp.function B)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.cartesian_product2 _let_1 _let_1))) (let ((_let_3 (tptp.the_L_meet A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_associative A) (tptp.meet_semilatt_str A)) (and (tptp.relation _let_3) (tptp.function _let_3) (tptp.quasi_total _let_3 _let_2 _let_1) (tptp.v2_binop_1 _let_3 _let_1) (tptp.v1_partfun1 _let_3 _let_2 _let_1))))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (tptp.closed_subset (tptp.cast_as_carrier_subset A) A))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation A)) (not (tptp.empty (tptp.relation_dom A))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (=> (and (not (tptp.empty_carrier A)) (tptp.lower_bounded_relstr A) (tptp.rel_str A)) (and (not (tptp.empty _let_1)) (tptp.filtered_subset _let_1 A))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.incl_POSet A))) (and (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1)))) (and _let_3 (tptp.v1_membered tptp.empty_set) (tptp.v2_membered tptp.empty_set) (tptp.v3_membered tptp.empty_set) (tptp.v4_membered tptp.empty_set) (tptp.v5_membered tptp.empty_set)) (forall ((A $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation A)) (not (tptp.empty (tptp.relation_rng A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (tptp.open_subset (tptp.interior A B) A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.net_str_of A B C D))) (let ((_let_2 (tptp.the_carrier A))) (=> (and (tptp.one_sorted_str A) (not (tptp.empty B)) (tptp.relation_of2 C B B) (tptp.function D) (tptp.quasi_total D B _let_2) (tptp.relation_of2 D B _let_2)) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_net_str _let_1 A)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.incl_POSet A))) (=> (not (tptp.empty A)) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_dom A))) (=> (tptp.empty A) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_rng A))) (=> (tptp.empty A) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (=> (and (tptp.topological_space A) (tptp.top_str A)) (and (tptp.open_subset _let_1 A) (tptp.closed_subset _let_1 A))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.lower_bounded_relstr _let_1) (tptp.upper_bounded_relstr _let_1) (tptp.bounded_relstr _let_1) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1) (tptp.complete_relstr _let_1)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.lower_bounded_relstr _let_1) (tptp.upper_bounded_relstr _let_1) (tptp.bounded_relstr _let_1) (tptp.directed_relstr _let_1) (tptp.up_complete_relstr _let_1) (tptp.join_complete_relstr _let_1) (not (tptp.v1_yellow_3 _let_1)) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1) (tptp.complete_relstr _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.finite A) (tptp.finite B)) (tptp.finite (tptp.set_union2 A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_composition A B))) (=> (and (tptp.empty A) (tptp.relation B)) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (tptp.dense (tptp.cast_as_carrier_subset A) A))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (and (not (tptp.empty_carrier _let_1)) (tptp.strict_rel_str _let_1) (tptp.reflexive_relstr _let_1) (tptp.transitive_relstr _let_1) (tptp.antisymmetric_relstr _let_1) (tptp.with_suprema_relstr _let_1) (tptp.with_infima_relstr _let_1) (tptp.complete_relstr _let_1) (tptp.lower_bounded_relstr _let_1) (tptp.upper_bounded_relstr _let_1) (tptp.bounded_relstr _let_1) (tptp.up_complete_relstr _let_1) (tptp.join_complete_relstr _let_1) (tptp.distributive_relstr _let_1) (tptp.complemented_relstr _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.lattice B) (tptp.latt_str B)) (= (tptp.in A (tptp.a_1_0_filter_1 B)) (exists ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier B))) (and (tptp.element C _let_1) (tptp.element D _let_1) (= A (tptp.ordered_pair_as_product_element _let_1 _let_1 C D)) (tptp.below_refl B C D))))))) _let_2 (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.latt_str B)) (= (tptp.in A (tptp.a_2_2_lattice3 B C)) (exists ((D $$unsorted)) (and (tptp.element D (tptp.the_carrier B)) (= A D) (tptp.latt_set_smaller B D C)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.lattice B) (tptp.complete_latt_str B) (tptp.latt_str B)) (= (tptp.in A (tptp.a_2_3_lattice3 B C)) (exists ((D $$unsorted)) (and (tptp.element D (tptp.the_carrier B)) (= A D) (tptp.latt_set_smaller B D C)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.one_sorted_str B) (not (tptp.empty_carrier C)) (tptp.net_str C B) (tptp.element D (tptp.the_carrier C))) (= (tptp.in A (tptp.a_3_0_waybel_9 B C D)) (exists ((E $$unsorted)) (and (tptp.element E (tptp.the_carrier C)) (= A E) (tptp.related C D E)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation_of2 B A A) (forall ((C $$unsorted) (D $$unsorted)) (=> (= (tptp.rel_str_of A B) (tptp.rel_str_of C D)) (and (= A C) (= B D)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (tptp.one_sorted_str A) (tptp.relation_of2 C B B) (tptp.function D) (tptp.quasi_total D B _let_1) (tptp.relation_of2 D B _let_1)) (forall ((E $$unsorted) (F $$unsorted) (G $$unsorted) (H $$unsorted)) (=> (= (tptp.net_str_of A B C D) (tptp.net_str_of E F G H)) (and (= A E) (= B F) (= C G) (= D H))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.cartesian_product2 A A))) (=> (and (tptp.function B) (tptp.quasi_total B _let_1 A) (tptp.relation_of2 B _let_1 A) (tptp.function C) (tptp.quasi_total C _let_1 A) (tptp.relation_of2 C _let_1 A)) (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (= (tptp.latt_str_of A B C) (tptp.latt_str_of D E F)) (and (= A D) (= B E) (= C F))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_union2 A A) A)) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_intersection2 A A) A)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_union2 A B B) B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_intersection2 A B B) B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (= (tptp.subset_complement A (tptp.subset_complement A B)) B))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_inverse (tptp.relation_inverse A)) A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (= (tptp.complements_of_subsets A (tptp.complements_of_subsets A B)) B))) (forall ((A $$unsorted) (B $$unsorted)) (not (tptp.proper_subset A A))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.reflexive A) (forall ((B $$unsorted)) (=> (tptp.in B (tptp.relation_field A)) (tptp.in (tptp.ordered_pair B B) A)))))) (forall ((A $$unsorted)) (not (= (tptp.singleton A) tptp.empty_set))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (= (tptp.set_union2 (tptp.singleton A) B) B))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.disjoint (tptp.singleton A) B) (tptp.in A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.in A B)) (tptp.disjoint (tptp.singleton A) B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_dom (tptp.relation_rng_restriction A B)) (tptp.relation_dom B)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.transitive A) (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.in (tptp.ordered_pair B C) A) (tptp.in (tptp.ordered_pair C D) A)) (tptp.in (tptp.ordered_pair B D) A)))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.subset (tptp.singleton A) B) (tptp.in A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (not (and (tptp.well_ordering B) (tptp.equipotent A (tptp.relation_field B)) (forall ((C $$unsorted)) (=> (tptp.relation C) (not (tptp.well_orders C A)))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= (tptp.set_difference A B) tptp.empty_set) (tptp.subset A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.in C B) (tptp.in C A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.antisymmetric A) (forall ((B $$unsorted) (C $$unsorted)) (=> (and (tptp.in (tptp.ordered_pair B C) A) (tptp.in (tptp.ordered_pair C B) A)) (= B C)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (or (tptp.in C A) (tptp.subset A (tptp.set_difference B (tptp.singleton C)))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (forall ((C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element C _let_1) (= (tptp.in C (tptp.subset_complement _let_1 B)) (not 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$$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (not (tptp.empty B)) (tptp.open_subset B A) (tptp.closed_subset B A))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (not (tptp.empty B)) (tptp.finite B))))) (forall ((A $$unsorted)) (=> (not (tptp.empty A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset A)) (not (tptp.empty B)) (tptp.finite B))))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.relation_empty_yielding A) (tptp.function A))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.empty B) (tptp.v1_membered B) (tptp.v2_membered B) (tptp.v3_membered B) (tptp.v4_membered B) (tptp.v5_membered B) (tptp.boundary_set B A))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (exists ((B $$unsorted)) (and (tptp.net_str B A) (tptp.strict_net_str B 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(tptp.transitive_relstr A) (tptp.directed_relstr A))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (not (tptp.empty B)))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.empty B) (tptp.open_subset B A) (tptp.closed_subset B A) (tptp.v1_membered B) (tptp.v2_membered B) (tptp.v3_membered B) (tptp.v4_membered B) (tptp.v5_membered B) (tptp.boundary_set B A) (tptp.nowhere_dense B A))))) (exists ((A $$unsorted)) (and (tptp.rel_str A) (not (tptp.empty_carrier A)) (tptp.strict_rel_str A) (tptp.reflexive_relstr A) (tptp.transitive_relstr A) (tptp.antisymmetric_relstr A) (tptp.with_suprema_relstr A) (tptp.with_infima_relstr A) (tptp.upper_bounded_relstr A) (tptp.distributive_relstr A) (tptp.heyting_relstr A))) (exists ((A $$unsorted)) (and (tptp.latt_str A) (not (tptp.empty_carrier A)) (tptp.strict_latt_str A))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.closed_subset B A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.net_str B A)) (exists ((C $$unsorted)) (and (tptp.subnetstr C A B) (tptp.strict_net_str C A) (tptp.full_subnetstr C A B))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (not (tptp.empty B)) (tptp.closed_subset B A))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.lower_relstr_subset B A) (tptp.upper_relstr_subset B A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (not (tptp.empty_carrier B)) (tptp.net_str B A)) (exists ((C $$unsorted)) (and (tptp.subnetstr C A B) (not (tptp.empty_carrier C)) (tptp.strict_net_str C A) (tptp.full_subnetstr C A B))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.rel_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (not (tptp.empty B)) (tptp.lower_relstr_subset B A) (tptp.upper_relstr_subset B A))))) (exists ((A $$unsorted)) (and (tptp.latt_str A) (not (tptp.empty_carrier A)) (tptp.strict_latt_str A) (tptp.join_commutative A) (tptp.join_associative A) (tptp.meet_commutative A) (tptp.meet_associative A) (tptp.meet_absorbing A) (tptp.join_absorbing A) (tptp.lattice A))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.reflexive_relstr A) (tptp.transitive_relstr A) (tptp.rel_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (not (tptp.empty B)) (tptp.directed_subset B A) (tptp.lower_relstr_subset B A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A) (not (tptp.empty_carrier B)) (tptp.lattice B) (tptp.latt_str B) (tptp.element C (tptp.the_carrier A)) (tptp.element D (tptp.the_carrier B))) (= (tptp.k10_filter_1 A B C D) (tptp.ordered_pair C D)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (not (tptp.empty A)) (not (tptp.empty B)) (tptp.element C A) (tptp.element D B)) (= (tptp.ordered_pair_as_product_element A B C D) (tptp.ordered_pair C D)))) (forall ((A $$unsorted)) (= (tptp.k1_pcomps_1 A) (tptp.powerset A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_restriction_as_relation_of A B) (tptp.relation_restriction A B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.the_carrier B))) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (not (tptp.empty_carrier B)) (tptp.one_sorted_str B) (tptp.function C) (tptp.quasi_total C _let_1 _let_2) (tptp.relation_of2 C _let_1 _let_2) (tptp.element D _let_1)) (= (tptp.apply_on_structs A B C D) (tptp.apply C D)))))) (forall ((A $$unsorted)) (= (tptp.inclusion_order A) (tptp.inclusion_relation A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted) (E $$unsorted) (F $$unsorted)) (let ((_let_1 (tptp.cartesian_product2 A B))) (=> (and (not (tptp.empty A)) (not (tptp.empty B)) (tptp.function D) (tptp.quasi_total D _let_1 C) (tptp.relation_of2 D _let_1 C) (tptp.element E A) (tptp.element F B)) (= (tptp.apply_binary_as_element A B C D E F) (tptp.apply_binary D E F))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (= (tptp.k2_lattice3 A) (tptp.relation_of_lattice A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.function C) (tptp.relation_of2 C A B)) (= (tptp.partfun_dom_restriction A B C D) (tptp.relation_dom_restriction C D)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.unordered_pair_as_carrier_subset A B C) (tptp.unordered_pair B C))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.join_commutative A) (tptp.join_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.join_commut A B C) (tptp.join A B C))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier B))) (=> (and (not (tptp.empty A)) (not (tptp.empty_carrier B)) (tptp.rel_str B) (tptp.function C) (tptp.quasi_total C A _let_1) (tptp.relation_of2 C A _let_1) (tptp.element D A)) (= (tptp.apply_on_set_and_struct A B C D) (tptp.apply C D))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.meet_commut A B C) (tptp.meet A B C))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2 C A B) (= (tptp.relation_dom_as_subset A B C) (tptp.relation_dom C)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_union2 A B C) (tptp.set_union2 B C))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier B))) (let ((_let_2 (tptp.the_carrier A))) (=> (and (tptp.one_sorted_str A) (tptp.one_sorted_str B) (tptp.function C) (tptp.quasi_total C _let_2 _let_1) (tptp.relation_of2 C _let_2 _let_1)) (= (tptp.function_invverse_img_as_carrier_subset A B C D) (tptp.relation_inverse_image C D)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2 C A B) (= (tptp.relation_rng_as_subset A B C) (tptp.relation_rng C)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (= (tptp.union_of_subsets A B) (tptp.union B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_intersection2 A B C) (tptp.set_intersection2 B C))))) (forall ((A $$unsorted)) (= (tptp.identity_as_relation_of A) (tptp.identity_relation A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (= (tptp.meet_of_subsets A B) (tptp.set_meet B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_difference A B C) (tptp.set_difference B C))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (not (tptp.empty B)) (tptp.function D) (tptp.quasi_total D A B) (tptp.relation_of2 D A B) (tptp.function E) (tptp.quasi_total E B C) (tptp.relation_of2 E B C)) (= (tptp.function_of_composition A B C D E) (tptp.relation_composition D E)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.the_carrier B))) (=> (and (not (tptp.empty A)) (not (tptp.empty_carrier B)) (tptp.one_sorted_str B) (tptp.function C) (tptp.quasi_total C A _let_1) (tptp.relation_of2 C A _let_1) (tptp.element D A)) (= (tptp.apply_on_set_and_struct2 A B C D) (tptp.apply C D))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.function C) (tptp.quasi_total C A B) (tptp.relation_of2 C A B) (tptp.element D A)) (= (tptp.apply_as_element A B C D) (tptp.apply C D)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (tptp.relation_of2 C A B) (= (tptp.relation_dom_restr_as_relation_of A B C D) (tptp.relation_dom_restriction C D)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (tptp.relation_of2_as_subset C A B) (tptp.relation_of2 C A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.ordinal A) (tptp.ordinal B)) (= (tptp.ordinal_subset A B) (tptp.subset A B)))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.equipotent A B) (tptp.are_equipotent A B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_absorbing A) (tptp.join_absorbing A) (tptp.latt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.below_refl A B C) (tptp.below A B C))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.reflexive_relstr A) (tptp.rel_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.related_reflexive A B C) (tptp.related A B C))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.ordinal A) (tptp.ordinal B)) (tptp.ordinal_subset A A))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset A A)) (forall ((A $$unsorted) (B $$unsorted)) (tptp.equipotent A A)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_absorbing A) (tptp.join_absorbing A) (tptp.latt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.below_refl A B B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.reflexive_relstr A) (tptp.rel_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.related_reflexive A B B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in C A))) (=> (and _let_1 (exists ((F $$unsorted)) (and (= C F) (tptp.in D F) (forall ((G $$unsorted)) (=> (tptp.in G F) (tptp.in (tptp.ordered_pair D G) B))))) _let_1 (exists ((H $$unsorted)) (and (= C H) (tptp.in E H) (forall ((I $$unsorted)) (=> (tptp.in I H) (tptp.in (tptp.ordered_pair E I) B)))))) (= D E)))) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in D A))) (= (tptp.in (tptp.ordered_pair D E) C) (and _let_1 _let_1 (exists ((J $$unsorted)) (and (= D J) (tptp.in E J) (forall ((K $$unsorted)) (=> (tptp.in K J) (tptp.in (tptp.ordered_pair E K) B)))))))))))))) (forall ((A $$unsorted)) (=> (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.singleton B))) (let ((_let_2 (tptp.in B A))) (=> (and _let_2 (= C _let_1) _let_2 (= D _let_1)) (= C D))))) (exists ((B $$unsorted)) (and (tptp.relation B) (tptp.function B) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.in C A))) (= (tptp.in (tptp.ordered_pair C D) B) (and _let_1 _let_1 (= D (tptp.singleton C)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in C (tptp.complements_of_subsets (tptp.the_carrier A) B)))) (=> (and _let_1 (forall ((F $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element F (tptp.powerset _let_1)) (=> (= F C) (= D (tptp.subset_complement _let_1 F)))))) _let_1 (forall ((G $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element G (tptp.powerset _let_1)) (=> (= G C) (= E (tptp.subset_complement _let_1 G))))))) (= D E)))) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in D (tptp.complements_of_subsets (tptp.the_carrier A) B)))) (= (tptp.in (tptp.ordered_pair D E) C) (and _let_1 _let_1 (forall ((H $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element H (tptp.powerset _let_1)) (=> (= H D) (= E (tptp.subset_complement _let_1 H))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in C B))) (=> (and _let_1 (forall ((F $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element F (tptp.powerset _let_1)) (=> (= F C) (= D (tptp.subset_complement _let_1 F)))))) _let_1 (forall ((G $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element G (tptp.powerset _let_1)) (=> (= G C) (= E (tptp.subset_complement _let_1 G))))))) (= D E)))) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in D B))) (= (tptp.in (tptp.ordered_pair D E) C) (and _let_1 _let_1 (forall ((H $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element H (tptp.powerset _let_1)) (=> (= H D) (= E (tptp.subset_complement _let_1 H))))))))))))))) (forall ((A $$unsorted)) (=> (exists ((B $$unsorted)) (and (tptp.ordinal B) (tptp.in B A))) (exists ((B $$unsorted)) (and (tptp.ordinal B) (tptp.in B A) (forall ((C $$unsorted)) (=> (tptp.ordinal C) (=> (tptp.in C A) (tptp.ordinal_subset B C)))))))) (=> (and (=> (tptp.in tptp.empty_set tptp.omega) (forall ((A $$unsorted)) (=> (tptp.element A (tptp.powerset (tptp.powerset tptp.empty_set))) (not (and (not (= A tptp.empty_set)) (forall ((B $$unsorted)) (not (and (tptp.in B A) (forall ((C $$unsorted)) (=> (and (tptp.in C A) (tptp.subset B C)) (= C B))))))))))) (forall ((D $$unsorted)) (=> (tptp.ordinal D) (=> (=> (tptp.in D tptp.omega) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.powerset (tptp.powerset D))) (not (and (not (= E tptp.empty_set)) (forall ((F $$unsorted)) (not (and (tptp.in F E) (forall ((G $$unsorted)) (=> (and (tptp.in G E) (tptp.subset F G)) (= G F))))))))))) (=> (tptp.in (tptp.succ D) tptp.omega) (forall ((H $$unsorted)) (=> (tptp.element H (tptp.powerset (tptp.powerset (tptp.succ D)))) (not (and (not (= H tptp.empty_set)) (forall ((I $$unsorted)) (not (and (tptp.in I H) (forall ((J $$unsorted)) (=> (and (tptp.in J H) (tptp.subset I J)) (= J I)))))))))))))) (forall ((D $$unsorted)) (=> (tptp.ordinal D) (=> (and (tptp.being_limit_ordinal D) (forall ((K $$unsorted)) (=> (tptp.ordinal K) (=> (tptp.in K D) (=> (tptp.in K tptp.omega) (forall ((L $$unsorted)) (=> (tptp.element L (tptp.powerset (tptp.powerset K))) (not (and (not (= L tptp.empty_set)) (forall ((M $$unsorted)) (not (and (tptp.in M L) (forall ((N $$unsorted)) (=> (and (tptp.in N L) (tptp.subset M N)) (= N M))))))))))))))) (or (= D tptp.empty_set) (=> (tptp.in D tptp.omega) (forall ((O $$unsorted)) (=> (tptp.element O (tptp.powerset (tptp.powerset D))) (not (and (not (= O tptp.empty_set)) (forall ((P $$unsorted)) (not (and (tptp.in P O) (forall ((Q $$unsorted)) (=> (and (tptp.in Q O) (tptp.subset P Q)) (= Q P)))))))))))))))) (forall ((D $$unsorted)) (=> (tptp.ordinal D) (=> (tptp.in D tptp.omega) (forall ((R $$unsorted)) (=> (tptp.element R (tptp.powerset (tptp.powerset D))) (not (and (not (= R tptp.empty_set)) (forall ((S $$unsorted)) (not (and (tptp.in S R) (forall ((T $$unsorted)) (=> (and (tptp.in T R) (tptp.subset S T)) (= T S)))))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation B) (tptp.relation C) (tptp.function C)) (exists ((D $$unsorted)) (and (tptp.relation D) (forall ((E $$unsorted) (F $$unsorted)) (= (tptp.in (tptp.ordered_pair E F) D) (and (tptp.in E A) (tptp.in F A) (tptp.in (tptp.ordered_pair (tptp.apply C E) (tptp.apply C F)) B)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in C A))) (=> (and _let_1 (exists ((F $$unsorted)) (and (= C F) (tptp.in D F) (forall ((G $$unsorted)) (=> (tptp.in G F) (tptp.in (tptp.ordered_pair D G) B))))) _let_1 (exists ((H $$unsorted)) (and (= C H) (tptp.in E H) (forall ((I $$unsorted)) (=> (tptp.in I H) (tptp.in (tptp.ordered_pair E I) B)))))) (= D E)))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (let ((_let_1 (tptp.in E A))) (and _let_1 _let_1 (exists ((J $$unsorted)) (and (= E J) (tptp.in D J) (forall ((K $$unsorted)) (=> (tptp.in K J) (tptp.in (tptp.ordered_pair D K) B)))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (forall ((C $$unsorted)) (=> (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (and (= D E) (exists ((G $$unsorted) (H $$unsorted)) (and (= (tptp.ordered_pair G H) E) (tptp.in G A) (exists ((I $$unsorted)) (and (= G I) (tptp.in H I) (forall ((J $$unsorted)) (=> (tptp.in J I) (tptp.in (tptp.ordered_pair H J) B))))))) (= D F) (exists ((K $$unsorted) (L $$unsorted)) (and (= (tptp.ordered_pair K L) F) (tptp.in K A) (exists ((M $$unsorted)) (and (= K M) (tptp.in L M) (forall ((N $$unsorted)) (=> (tptp.in N M) (tptp.in (tptp.ordered_pair L N) B)))))))) (= E F))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (exists ((F $$unsorted)) (and (tptp.in F (tptp.cartesian_product2 A C)) (= F E) (exists ((O $$unsorted) (P $$unsorted)) (and (= (tptp.ordered_pair O P) E) (tptp.in O A) (exists ((Q $$unsorted)) (and (= O Q) (tptp.in P Q) (forall ((R $$unsorted)) (=> (tptp.in R Q) (tptp.in (tptp.ordered_pair P R) B)))))))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.transitive_relstr A) (tptp.rel_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.finite C) (tptp.element C (tptp.powerset B))) (=> (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (and (= D E) (exists ((G $$unsorted)) (and (= G E) (exists ((H $$unsorted)) (and (tptp.element H (tptp.the_carrier A)) (tptp.in H B) (tptp.relstr_set_smaller A G H))))) (= D F) (exists ((I $$unsorted)) (and (= I F) (exists ((J $$unsorted)) (and (tptp.element J (tptp.the_carrier A)) (tptp.in J B) (tptp.relstr_set_smaller A I J)))))) (= E F))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (exists ((F $$unsorted)) (and (tptp.in F (tptp.powerset C)) (= F E) (exists ((K $$unsorted)) (and (= K E) (exists ((L $$unsorted)) (and (tptp.element L (tptp.the_carrier A)) (tptp.in L B) (tptp.relstr_set_smaller A K L))))))))))))) (forall ((A $$unsorted)) (=> (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.singleton B))) (let ((_let_2 (tptp.in B A))) (=> (and _let_2 (= C _let_1) _let_2 (= D _let_1)) (= C D))))) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (let ((_let_1 (tptp.in D A))) (and _let_1 _let_1 (= C (tptp.singleton D)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (= C D) (exists ((F $$unsorted) (G $$unsorted)) (and (= (tptp.ordered_pair F G) D) (tptp.in F A) (= G (tptp.singleton F)))) (= C E) (exists ((H $$unsorted) (I $$unsorted)) (and (= (tptp.ordered_pair H I) E) (tptp.in H A) (= I (tptp.singleton H))))) (= D E))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.cartesian_product2 A B)) (= E D) (exists ((J $$unsorted) (K $$unsorted)) (and (= (tptp.ordered_pair J K) D) (tptp.in J A) (= K (tptp.singleton J))))))))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (=> (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (= B C) (exists ((E $$unsorted)) (and (tptp.ordinal E) (= C E) (=> (tptp.in E tptp.omega) (forall ((F $$unsorted)) (=> (tptp.element F (tptp.powerset (tptp.powerset E))) (not (and (not (= F tptp.empty_set)) (forall ((G $$unsorted)) (not (and (tptp.in G F) (forall ((H $$unsorted)) (=> (and (tptp.in H F) (tptp.subset G H)) (= H G))))))))))))) (= B D) (exists ((I $$unsorted)) (and (tptp.ordinal I) (= D I) (=> (tptp.in I tptp.omega) (forall ((J $$unsorted)) (=> (tptp.element J (tptp.powerset (tptp.powerset I))) (not (and (not (= J tptp.empty_set)) (forall ((K $$unsorted)) (not (and (tptp.in K J) (forall ((L $$unsorted)) (=> (and (tptp.in L J) (tptp.subset K L)) (= L K)))))))))))))) (= C D))) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (and (tptp.in D (tptp.succ A)) (= D C) (exists ((M $$unsorted)) (and (tptp.ordinal M) (= C M) (=> (tptp.in M tptp.omega) (forall ((N $$unsorted)) (=> (tptp.element N (tptp.powerset (tptp.powerset M))) (not (and (not (= N tptp.empty_set)) (forall ((O $$unsorted)) (not (and (tptp.in O N) (forall ((P $$unsorted)) (=> (and (tptp.in P N) (tptp.subset O P)) (= P O))))))))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (= C D) (exists ((F $$unsorted)) (and (tptp.element F (tptp.powerset (tptp.the_carrier A))) (= F D) (tptp.closed_subset F A) (tptp.subset B D))) (= C E) (exists ((G $$unsorted)) (and (tptp.element G (tptp.powerset (tptp.the_carrier A))) (= G E) (tptp.closed_subset G A) (tptp.subset B E)))) (= D E))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.powerset (tptp.the_carrier A))) (= E D) (exists ((H $$unsorted)) (and (tptp.element H (tptp.powerset (tptp.the_carrier A))) (= H D) (tptp.closed_subset H A) (tptp.subset B D))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (=> (and (= C D) (tptp.in (tptp.set_difference _let_1 D) B) (= C E) (tptp.in (tptp.set_difference _let_1 E) B)) (= D E)))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.powerset (tptp.the_carrier A))) (= E D) (tptp.in (tptp.set_difference (tptp.cast_as_carrier_subset A) D) B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.ordinal A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.succ A))))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (= C D) (exists ((F $$unsorted)) (and (tptp.in F B) (= D (tptp.set_difference F (tptp.singleton A))))) (= C E) (exists ((G $$unsorted)) (and (tptp.in G B) (= E (tptp.set_difference G (tptp.singleton A)))))) (= D E))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.powerset A)) (= E D) (exists ((H $$unsorted)) (and (tptp.in H B) (= D (tptp.set_difference H (tptp.singleton A))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in C (tptp.complements_of_subsets (tptp.the_carrier A) B)))) (=> (and _let_1 (forall ((F $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element F (tptp.powerset _let_1)) (=> (= F C) (= D (tptp.subset_complement _let_1 F)))))) _let_1 (forall ((G $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element G (tptp.powerset _let_1)) (=> (= G C) (= E (tptp.subset_complement _let_1 G))))))) (= D E)))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (let ((_let_1 (tptp.in E (tptp.complements_of_subsets (tptp.the_carrier A) B)))) (and _let_1 _let_1 (forall ((H $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element H (tptp.powerset _let_1)) (=> (= H E) (= D (tptp.subset_complement _let_1 H))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (forall ((C $$unsorted)) (=> (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (and (= D E) (exists ((G $$unsorted) (H $$unsorted)) (and (= (tptp.ordered_pair G H) E) (tptp.in G (tptp.complements_of_subsets (tptp.the_carrier A) B)) (forall ((I $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element I (tptp.powerset _let_1)) (=> (= I G) (= H (tptp.subset_complement _let_1 I)))))))) (= D F) (exists ((J $$unsorted) (K $$unsorted)) (and (= (tptp.ordered_pair J K) F) (tptp.in J (tptp.complements_of_subsets (tptp.the_carrier A) B)) (forall ((L $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element L (tptp.powerset _let_1)) (=> (= L J) (= K (tptp.subset_complement _let_1 L))))))))) (= E F))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (exists ((F $$unsorted)) (and (tptp.in F (tptp.cartesian_product2 (tptp.complements_of_subsets (tptp.the_carrier A) B) C)) (= F E) (exists ((M $$unsorted) (N $$unsorted)) (and (= (tptp.ordered_pair M N) E) (tptp.in M (tptp.complements_of_subsets (tptp.the_carrier A) B)) (forall ((O $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element O (tptp.powerset _let_1)) (=> (= O M) (= N (tptp.subset_complement _let_1 O))))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in C B))) (=> (and _let_1 (forall ((F $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element F (tptp.powerset _let_1)) (=> (= F C) (= D (tptp.subset_complement _let_1 F)))))) _let_1 (forall ((G $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element G (tptp.powerset _let_1)) (=> (= G C) (= E (tptp.subset_complement _let_1 G))))))) (= D E)))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (let ((_let_1 (tptp.in E B))) (and _let_1 _let_1 (forall ((H $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element H (tptp.powerset _let_1)) (=> (= H E) (= D (tptp.subset_complement _let_1 H))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (forall ((C $$unsorted)) (=> (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (and (= D E) (exists ((G $$unsorted) (H $$unsorted)) (and (= (tptp.ordered_pair G H) E) (tptp.in G B) (forall ((I $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element I (tptp.powerset _let_1)) (=> (= I G) (= H (tptp.subset_complement _let_1 I)))))))) (= D F) (exists ((J $$unsorted) (K $$unsorted)) (and (= (tptp.ordered_pair J K) F) (tptp.in J B) (forall ((L $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element L (tptp.powerset _let_1)) (=> (= L J) (= K (tptp.subset_complement _let_1 L))))))))) (= E F))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (exists ((F $$unsorted)) (and (tptp.in F (tptp.cartesian_product2 B C)) (= F E) (exists ((M $$unsorted) (N $$unsorted)) (and (= (tptp.ordered_pair M N) E) (tptp.in M B) (forall ((O $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element O (tptp.powerset _let_1)) (=> (= O M) (= N (tptp.subset_complement _let_1 O))))))))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation B) (tptp.relation C) (tptp.function C)) (=> (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (and (= D E) (exists ((G $$unsorted) (H $$unsorted)) (and (= E (tptp.ordered_pair G H)) (tptp.in (tptp.ordered_pair (tptp.apply C G) (tptp.apply C H)) B))) (= D F) (exists ((I $$unsorted) (J $$unsorted)) (and (= F (tptp.ordered_pair I J)) (tptp.in (tptp.ordered_pair (tptp.apply C I) (tptp.apply C J)) B)))) (= E F))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (exists ((F $$unsorted)) (and (tptp.in F (tptp.cartesian_product2 A A)) (= F E) (exists ((K $$unsorted) (L $$unsorted)) (and (= E (tptp.ordered_pair K L)) (tptp.in (tptp.ordered_pair (tptp.apply C K) (tptp.apply C L)) B))))))))))) (forall ((A $$unsorted)) (=> (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (= B C) (tptp.ordinal C) (= B D) (tptp.ordinal D)) (= C D))) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (and (tptp.in D A) (= D C) (tptp.ordinal C)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.element B (tptp.powerset (tptp.powerset A))) (tptp.relation C) (tptp.function C)) (=> (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (and (= D E) (tptp.in (tptp.relation_image C E) B) (= D F) (tptp.in (tptp.relation_image C F) B)) (= E F))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (exists ((F $$unsorted)) (and (tptp.in F (tptp.powerset (tptp.relation_dom C))) (= F E) (tptp.in (tptp.relation_image C E) B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.ordinal B) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (= C D) (exists ((F $$unsorted)) (and (tptp.ordinal F) (= D F) (tptp.in F A))) (= C E) (exists ((G $$unsorted)) (and (tptp.ordinal G) (= E G) (tptp.in G A)))) (= D E))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.succ B)) (= E D) (exists ((H $$unsorted)) (and (tptp.ordinal H) (= D H) (tptp.in H A))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (forall ((C $$unsorted)) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (and (tptp.in E (tptp.cartesian_product2 A C)) (exists ((F $$unsorted) (G $$unsorted)) (and (= (tptp.ordered_pair F G) E) (tptp.in F A) (exists ((H $$unsorted)) (and (= F H) (tptp.in G H) (forall ((I $$unsorted)) (=> (tptp.in I H) (tptp.in (tptp.ordered_pair G I) B)))))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.transitive_relstr A) (tptp.rel_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.finite C) (tptp.element C (tptp.powerset B))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (and (tptp.in E (tptp.powerset C)) (exists ((F $$unsorted)) (and (= F E) (exists ((G $$unsorted)) (and (tptp.element G (tptp.the_carrier A)) (tptp.in G B) (tptp.relstr_set_smaller A F G))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.cartesian_product2 A B)) (exists ((E $$unsorted) (F $$unsorted)) (and (= (tptp.ordered_pair E F) D) (tptp.in E A) (= F (tptp.singleton E))))))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (and (tptp.in C (tptp.succ A)) (exists ((D $$unsorted)) (and (tptp.ordinal D) (= C D) (=> (tptp.in D tptp.omega) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.powerset (tptp.powerset D))) (not (and (not (= E tptp.empty_set)) (forall ((F $$unsorted)) (not (and (tptp.in F E) (forall ((G $$unsorted)) (=> (and (tptp.in G E) (tptp.subset F G)) (= G F))))))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.powerset (tptp.the_carrier A))) (exists ((E $$unsorted)) (and (tptp.element E (tptp.powerset (tptp.the_carrier A))) (= E D) (tptp.closed_subset E A) (tptp.subset B D))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.powerset (tptp.the_carrier A))) (tptp.in (tptp.set_difference (tptp.cast_as_carrier_subset A) D) B))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.ordinal A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.succ A))))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.powerset A)) (exists ((E $$unsorted)) (and (tptp.in E B) (= D (tptp.set_difference E (tptp.singleton A))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (forall ((C $$unsorted)) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (and (tptp.in E (tptp.cartesian_product2 (tptp.complements_of_subsets (tptp.the_carrier A) B) C)) (exists ((F $$unsorted) (G $$unsorted)) (and (= (tptp.ordered_pair F G) E) (tptp.in F (tptp.complements_of_subsets (tptp.the_carrier A) B)) (forall ((H $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element H (tptp.powerset _let_1)) (=> (= H F) (= G (tptp.subset_complement _let_1 H))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (forall ((C $$unsorted)) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (and (tptp.in E (tptp.cartesian_product2 B C)) (exists ((F $$unsorted) (G $$unsorted)) (and (= (tptp.ordered_pair F G) E) (tptp.in F B) (forall ((H $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element H (tptp.powerset _let_1)) (=> (= H F) (= G (tptp.subset_complement _let_1 H))))))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation B) (tptp.relation C) (tptp.function C)) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (and (tptp.in E (tptp.cartesian_product2 A A)) (exists ((F $$unsorted) (G $$unsorted)) (and (= E (tptp.ordered_pair F G)) (tptp.in (tptp.ordered_pair (tptp.apply C F) (tptp.apply C G)) B))))))))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (and (tptp.in C A) (tptp.ordinal C)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.element B (tptp.powerset (tptp.powerset A))) (tptp.relation C) (tptp.function C)) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (and (tptp.in E (tptp.powerset (tptp.relation_dom C))) (tptp.in (tptp.relation_image C E) B))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.ordinal B) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.succ B)) (exists ((E $$unsorted)) (and (tptp.ordinal E) (= D E) (tptp.in E A))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.finite C))) (=> (and (not (tptp.empty_carrier A)) (tptp.transitive_relstr A) (tptp.rel_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A))) _let_1 (tptp.element C (tptp.powerset B))) (=> (and _let_1 (exists ((D $$unsorted)) (and (tptp.element D (tptp.the_carrier A)) (tptp.in D B) (tptp.relstr_set_smaller A tptp.empty_set D))) (forall ((E $$unsorted) (F $$unsorted)) (=> (and (tptp.in E C) (tptp.subset F C) (exists ((G $$unsorted)) (and (tptp.element G (tptp.the_carrier A)) (tptp.in G B) (tptp.relstr_set_smaller A F G)))) (exists ((H $$unsorted)) (and (tptp.element H (tptp.the_carrier A)) (tptp.in H B) (tptp.relstr_set_smaller A (tptp.set_union2 F (tptp.singleton E)) H)))))) (exists ((I $$unsorted)) (and (tptp.element I (tptp.the_carrier A)) (tptp.in I B) (tptp.relstr_set_smaller A C I))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (=> (and (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (tptp.in C A) (exists ((F $$unsorted)) (and (= C F) (tptp.in D F) (forall ((G $$unsorted)) (=> (tptp.in G F) (tptp.in (tptp.ordered_pair D G) B))))) (exists ((H $$unsorted)) (and (= C H) (tptp.in E H) (forall ((I $$unsorted)) (=> (tptp.in I H) (tptp.in (tptp.ordered_pair E I) B)))))) (= D E))) (forall ((C $$unsorted)) (not (and (tptp.in C A) (forall ((D $$unsorted)) (not (exists ((J $$unsorted)) (and (= C J) (tptp.in D J) (forall ((K $$unsorted)) (=> (tptp.in K J) (tptp.in (tptp.ordered_pair D K) B))))))))))) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (= (tptp.relation_dom C) A) (forall ((D $$unsorted)) (=> (tptp.in D A) (exists ((L $$unsorted)) (and (= D L) (tptp.in (tptp.apply C D) L) (forall ((M $$unsorted)) (=> (tptp.in M L) (tptp.in (tptp.ordered_pair (tptp.apply C D) M) B)))))))))))) (forall ((A $$unsorted)) (=> (and (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.singleton B))) (=> (and (tptp.in B A) (= C _let_1) (= D _let_1)) (= C D)))) (forall ((B $$unsorted)) (not (and (tptp.in B A) (forall ((C $$unsorted)) (not (= C (tptp.singleton B)))))))) (exists ((B $$unsorted)) (and (tptp.relation B) (tptp.function B) (= (tptp.relation_dom B) A) (forall ((C $$unsorted)) (=> (tptp.in C A) (= (tptp.apply B C) (tptp.singleton C)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (=> (and (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (tptp.in C (tptp.complements_of_subsets (tptp.the_carrier A) B)) (forall ((F $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element F (tptp.powerset _let_1)) (=> (= F C) (= D (tptp.subset_complement _let_1 F)))))) (forall ((G $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element G (tptp.powerset _let_1)) (=> (= G C) (= E (tptp.subset_complement _let_1 G))))))) (= D E))) (forall ((C $$unsorted)) (not (and (tptp.in C (tptp.complements_of_subsets (tptp.the_carrier A) B)) (forall ((D $$unsorted)) (not (forall ((H $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element H (tptp.powerset _let_1)) (=> (= H C) (= D (tptp.subset_complement _let_1 H)))))))))))) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (= (tptp.relation_dom C) (tptp.complements_of_subsets (tptp.the_carrier A) B)) (forall ((D $$unsorted)) (=> (tptp.in D (tptp.complements_of_subsets (tptp.the_carrier A) B)) (forall ((I $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element I (tptp.powerset _let_1)) (=> (= I D) (= (tptp.apply C D) (tptp.subset_complement _let_1 I))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.one_sorted_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (=> (and (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (tptp.in C B) (forall ((F $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element F (tptp.powerset _let_1)) (=> (= F C) (= D (tptp.subset_complement _let_1 F)))))) (forall ((G $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element G (tptp.powerset _let_1)) (=> (= G C) (= E (tptp.subset_complement _let_1 G))))))) (= D E))) (forall ((C $$unsorted)) (not (and (tptp.in C B) (forall ((D $$unsorted)) (not (forall ((H $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element H (tptp.powerset _let_1)) (=> (= H C) (= D (tptp.subset_complement _let_1 H)))))))))))) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (= (tptp.relation_dom C) B) (forall ((D $$unsorted)) (=> (tptp.in D B) (forall ((I $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element I (tptp.powerset _let_1)) (=> (= I D) (= (tptp.apply C D) (tptp.subset_complement _let_1 I))))))))))))) (=> (forall ((A $$unsorted)) (=> (tptp.ordinal A) (=> (forall ((B $$unsorted)) (=> (tptp.ordinal B) (=> (tptp.in B A) (=> (tptp.in B tptp.omega) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.powerset B))) (not (and (not (= C tptp.empty_set)) (forall ((D $$unsorted)) (not (and (tptp.in D C) (forall ((E $$unsorted)) (=> (and (tptp.in E C) (tptp.subset D E)) (= E D)))))))))))))) (=> (tptp.in A tptp.omega) (forall ((F $$unsorted)) (=> (tptp.element F (tptp.powerset (tptp.powerset A))) (not (and (not (= F tptp.empty_set)) (forall ((G $$unsorted)) (not (and (tptp.in G F) (forall ((H $$unsorted)) (=> (and (tptp.in H F) (tptp.subset G H)) (= H G)))))))))))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (=> (tptp.in A tptp.omega) (forall ((I $$unsorted)) (=> (tptp.element I (tptp.powerset (tptp.powerset A))) (not (and (not (= I tptp.empty_set)) (forall ((J $$unsorted)) (not (and (tptp.in J I) (forall ((K $$unsorted)) (=> (and (tptp.in K I) (tptp.subset J K)) (= K J)))))))))))))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.relation B) (tptp.function B) (= (tptp.relation_dom B) A) (forall ((C $$unsorted)) (=> (tptp.in C A) (= (tptp.apply B C) (tptp.singleton C))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (exists ((C $$unsorted)) (and (tptp.element C (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier A))) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.element E (tptp.powerset (tptp.the_carrier A))) (= E D) (tptp.closed_subset E A) (tptp.subset B D)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (exists ((C $$unsorted)) (and (tptp.element C (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier A))) (= (tptp.in D C) (tptp.in (tptp.set_difference (tptp.cast_as_carrier_subset A) D) B)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.disjoint A B) (tptp.disjoint B A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.equipotent A B) (tptp.equipotent B A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair A B) (tptp.cartesian_product2 C D)) (and (tptp.in A C) (tptp.in B D)))) (forall ((A $$unsorted)) (tptp.in A (tptp.succ A))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (= B tptp.empty_set))) (let ((_let_2 (= (tptp.complements_of_subsets A B) tptp.empty_set))) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (and (not (and (not _let_1) _let_2)) (not (and (not _let_2) _let_1))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (not (and (= (tptp.unordered_pair A B) (tptp.unordered_pair C D)) (not (= A C)) (not (= A D))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_rng (tptp.relation_rng_restriction B C))) (and (tptp.in A B) (tptp.in A (tptp.relation_rng C)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_rng_restriction A B)) A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng_restriction A B) B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_rng_restriction A B)) (tptp.relation_rng B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (and (tptp.subset (tptp.cartesian_product2 A C) (tptp.cartesian_product2 B C)) (tptp.subset (tptp.cartesian_product2 C A) (tptp.cartesian_product2 C B))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_rng (tptp.relation_rng_restriction A B)) (tptp.set_intersection2 (tptp.relation_rng B) A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset C D)) (tptp.subset (tptp.cartesian_product2 A C) (tptp.cartesian_product2 B D)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (=> (not (= B tptp.empty_set)) (= (tptp.meet_of_subsets A (tptp.complements_of_subsets A B)) (tptp.subset_complement A (tptp.union_of_subsets A B)))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (=> (tptp.element B (tptp.powerset (tptp.the_carrier _let_1))) (= (tptp.upper_relstr_subset B _let_1) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.subset C D) (tptp.subset D A) (tptp.in C B)) (tptp.in D B))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (= (tptp.cast_as_carrier_subset A) (tptp.the_carrier A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2_as_subset C A B) (and (tptp.subset (tptp.relation_dom C) A) (tptp.subset (tptp.relation_rng C) B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (=> (not (= B tptp.empty_set)) (= (tptp.union_of_subsets A (tptp.complements_of_subsets A B)) (tptp.subset_complement A (tptp.meet_of_subsets A B)))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier B)) (= (tptp.the_carrier (tptp.netstr_restr_to_element A B C)) (tptp.a_3_0_waybel_9 A B C)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset A B) (= (tptp.set_union2 A B) B))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.in A B) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C B) (tptp.subset D C)) (tptp.in D B))) (forall ((C $$unsorted)) (=> (tptp.in C B) (tptp.in (tptp.powerset C) B))) (forall ((C $$unsorted)) (not (and (tptp.subset C B) (not (tptp.are_equipotent C B)) (not (tptp.in C B)))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (= (tptp.compact_top_space A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset (tptp.powerset _let_1))) (not (and (tptp.centered B) (tptp.closed_subsets B A) (= (tptp.meet_of_subsets _let_1 B) tptp.empty_set))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.subset A B) (tptp.finite B)) (tptp.finite A))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset (tptp.powerset _let_1))) (= (tptp.finite (tptp.complements_of_subsets _let_1 B)) (tptp.finite B))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.relation_dom_restriction (tptp.relation_rng_restriction A C) B) (tptp.relation_rng_restriction A (tptp.relation_dom_restriction C B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_image C B)) (exists ((D $$unsorted)) (and (tptp.in D (tptp.relation_dom C)) (tptp.in (tptp.ordered_pair D A) C) (tptp.in D B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_image B A) (tptp.relation_rng B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (tptp.subset (tptp.relation_image B (tptp.relation_inverse_image B A)) A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_image B A) (tptp.relation_image B (tptp.set_intersection2 (tptp.relation_dom B) A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset A (tptp.relation_dom B)) (tptp.subset A (tptp.relation_inverse_image B (tptp.relation_image B A)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_image A (tptp.relation_dom A)) (tptp.relation_rng A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (=> (tptp.subset A (tptp.relation_rng B)) (= (tptp.relation_image B (tptp.relation_inverse_image B A)) A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (tptp.relation_of2_as_subset D C A) (=> (tptp.subset (tptp.relation_rng D) B) (tptp.relation_of2_as_subset D C B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.finite A) (tptp.finite (tptp.set_intersection2 A B)))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.subset_intersection2 _let_1 B (tptp.cast_as_carrier_subset A)) B)))))) (forall ((A $$unsorted)) (=> (and (tptp.antisymmetric_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (= (tptp.ex_sup_of_relstr_set A B) (exists ((C $$unsorted)) (and (tptp.element C (tptp.the_carrier A)) (tptp.relstr_set_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.relstr_set_smaller A B D) (tptp.related A C D)))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_rng (tptp.relation_composition A B)) (tptp.relation_image B (tptp.relation_rng A))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_inverse_image C B)) (exists ((D $$unsorted)) (and (tptp.in D (tptp.relation_rng C)) (tptp.in (tptp.ordered_pair A D) C) (tptp.in D B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_inverse_image B A) (tptp.relation_dom B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (tptp.relation_of2_as_subset D C A) (=> (tptp.subset A B) (tptp.relation_of2_as_subset D C B)))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset (tptp.powerset _let_1))) (= (tptp.closed_subsets B A) (tptp.open_subsets (tptp.complements_of_subsets _let_1 B) A))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier B)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier B)) (forall ((E $$unsorted)) (let ((_let_1 (tptp.netstr_restr_to_element A B C))) (=> (tptp.element E (tptp.the_carrier _let_1)) (=> (= D E) (= (tptp.apply_netmap A B D) (tptp.apply_netmap A _let_1 E)))))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_restriction C B)) (and (tptp.in A C) (tptp.in A (tptp.cartesian_product2 B B)))))) (forall ((A $$unsorted)) (=> (and (tptp.antisymmetric_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (= (tptp.ex_inf_of_relstr_set A B) (exists ((C $$unsorted)) (and (tptp.element C (tptp.the_carrier A)) (tptp.relstr_element_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.relstr_element_smaller A B D) (tptp.related A D C)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (not (and (not (= A tptp.empty_set)) (tptp.subset A (tptp.relation_rng B)) (= (tptp.relation_inverse_image B A) tptp.empty_set))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (=> (tptp.subset A B) (tptp.subset (tptp.relation_inverse_image C A) (tptp.relation_inverse_image C B))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (=> (tptp.finite A) (tptp.finite (tptp.relation_image B A))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.subset_complement _let_1 B) (tptp.subset_difference _let_1 (tptp.cast_as_carrier_subset A) B))))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset (tptp.powerset _let_1))) (= (tptp.open_subsets B A) (tptp.closed_subsets (tptp.complements_of_subsets _let_1 B) A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_restriction B A) (tptp.relation_dom_restriction (tptp.relation_rng_restriction A B) A)))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset (tptp.set_intersection2 A B) A)) (forall ((A $$unsorted)) (=> (tptp.finite A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (not (and (not (= B tptp.empty_set)) (forall ((C $$unsorted)) (not (and (tptp.in C B) (forall ((D $$unsorted)) (=> (and (tptp.in D B) (tptp.subset C D)) (= D C)))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_restriction B A) (tptp.relation_rng_restriction A (tptp.relation_dom_restriction B A))))) (forall ((A $$unsorted)) (= (tptp.bottom_of_relstr (tptp.boole_POSet A)) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (=> (tptp.in A (tptp.relation_field (tptp.relation_restriction C B))) (and (tptp.in A (tptp.relation_field C)) (tptp.in A B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset A C)) (tptp.subset A (tptp.set_intersection2 B C)))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (=> (tptp.net_str B A) (forall ((C $$unsorted)) (=> (tptp.subnetstr C A B) (tptp.subset (tptp.the_carrier C) (tptp.the_carrier B)))))))) (forall ((A $$unsorted)) (= (tptp.set_union2 A tptp.empty_set) A)) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.the_carrier (tptp.boole_lattice A))) (forall ((C $$unsorted)) (let ((_let_1 (tptp.boole_lattice A))) (=> (tptp.element C (tptp.the_carrier _let_1)) (and (= (tptp.join _let_1 B C) (tptp.set_union2 B C)) (= (tptp.meet _let_1 B C) (tptp.set_intersection2 B C)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (tptp.element A B))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.transitive_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (= (and (not (tptp.empty B)) (tptp.directed_subset B A)) (forall ((C $$unsorted)) (=> (and (tptp.finite C) (tptp.element C (tptp.powerset B))) (exists ((D $$unsorted)) (and (tptp.element D (tptp.the_carrier A)) (tptp.in D B) (tptp.relstr_set_smaller A C D)))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset B C)) (tptp.subset A C))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.incl_POSet A))) (and (= (tptp.the_carrier _let_1) A) (= (tptp.the_InternalRel _let_1) (tptp.inclusion_order A))))) (= (tptp.powerset tptp.empty_set) (tptp.singleton tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (=> (tptp.in (tptp.ordered_pair A B) C) (and (tptp.in A (tptp.relation_dom C)) (tptp.in B (tptp.relation_rng C)))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_field (tptp.relation_restriction B A)))) (=> (tptp.relation B) (and (tptp.subset _let_1 (tptp.relation_field B)) (tptp.subset _let_1 A))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (=> (tptp.net_str B A) (forall ((C $$unsorted)) (=> (tptp.subnetstr C A B) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier B)) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.the_carrier B)) (forall ((F $$unsorted)) (=> (tptp.element F (tptp.the_carrier C)) (forall ((G $$unsorted)) (=> (tptp.element G (tptp.the_carrier C)) (=> (and (= D F) (= E G) (tptp.related C F G)) (tptp.related B D E)))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.in A (tptp.relation_dom (tptp.relation_composition C B))) (and (tptp.in A (tptp.relation_dom C)) (tptp.in (tptp.apply C A) (tptp.relation_dom B)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.function D) (tptp.quasi_total D A B) (tptp.relation_of2_as_subset D A B)) (forall ((E $$unsorted)) (=> (and (tptp.relation E) (tptp.function E)) (=> (tptp.in C A) (or (= B tptp.empty_set) (= (tptp.apply (tptp.relation_composition D E) C) (tptp.apply E (tptp.apply D C))))))))) (forall ((A $$unsorted)) (=> (tptp.epsilon_transitive A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (=> (tptp.proper_subset A B) (tptp.in A B)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (tptp.subset A (tptp.cartesian_product2 (tptp.relation_dom A) (tptp.relation_rng A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (tptp.subset (tptp.fiber (tptp.relation_restriction C A) B) (tptp.fiber C B)))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (and (not (tptp.empty_carrier C)) (tptp.full_subnetstr C A B) (tptp.subnetstr C A B)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier B)) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.the_carrier B)) (forall ((F $$unsorted)) (=> (tptp.element F (tptp.the_carrier C)) (forall ((G $$unsorted)) (=> (tptp.element G (tptp.the_carrier C)) (=> (and (= D F) (= E G) (tptp.related B D E)) (tptp.related C F G)))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.relation_composition C B))) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in A (tptp.relation_dom _let_1)) (= (tptp.apply _let_1 A) (tptp.apply B (tptp.apply C A))))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (let ((_let_2 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_2)) (= (tptp.subset_difference _let_2 _let_1 (tptp.subset_difference _let_2 _let_1 B)) B))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2_as_subset C B A) (= (forall ((D $$unsorted)) (not (and (tptp.in D B) (forall ((E $$unsorted)) (not (tptp.in (tptp.ordered_pair D E) C)))))) (= (tptp.relation_dom_as_subset B A C) B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.reflexive B) (tptp.reflexive (tptp.relation_restriction B A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in A (tptp.relation_dom B)) (= (tptp.apply (tptp.relation_composition B C) A) (tptp.apply C (tptp.apply B A)))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_absorbing A) (tptp.latt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (tptp.below A (tptp.meet_commut A B C) B))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.ordinal B) (=> (tptp.in A B) (tptp.ordinal A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2_as_subset C A B) (= (forall ((D $$unsorted)) (not (and (tptp.in D B) (forall ((E $$unsorted)) (not (tptp.in (tptp.ordered_pair E D) C)))))) (= (tptp.relation_rng_as_subset A B C) B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.connected B) (tptp.connected (tptp.relation_restriction B A))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (not (and (not (tptp.in A B)) (not (= A B)) (not (tptp.in B A)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.transitive B) (tptp.transitive (tptp.relation_restriction B A))))) (forall ((A $$unsorted)) (=> (and (tptp.antisymmetric_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (=> (and (tptp.related A B C) (tptp.related A C B)) (= B C)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset A B) (and (tptp.subset (tptp.relation_dom A) (tptp.relation_dom B)) (tptp.subset (tptp.relation_rng A) (tptp.relation_rng B)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.antisymmetric B) (tptp.antisymmetric (tptp.relation_restriction B A))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_restriction B A))) (=> (tptp.relation B) (=> (tptp.well_orders B A) (and (= (tptp.relation_field _let_1) A) (tptp.well_ordering _let_1)))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.finite (tptp.relation_dom A)) (tptp.finite (tptp.relation_rng A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.join_commutative A) (tptp.join_semilatt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (=> (and (tptp.below A B C) (tptp.below A C B)) (= B C)))))))) (forall ((A $$unsorted)) (=> (and (tptp.transitive_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (and (tptp.related A B C) (tptp.related A C D)) (tptp.related A B D)))))))))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.relation B) (tptp.well_orders B A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (tptp.subset (tptp.set_intersection2 A C) (tptp.set_intersection2 B C)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.lattice B) (tptp.latt_str B)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier B)) (= (tptp.latt_set_smaller B C A) (tptp.relstr_element_smaller (tptp.poset_of_lattice B) A (tptp.cast_to_el_of_LattPOSet B C))))))) (forall ((A $$unsorted)) (=> (not (tptp.empty A)) (not (and (forall ((B $$unsorted)) (not (and (tptp.in B A) (= B tptp.empty_set)))) (forall ((B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (not (and (= (tptp.relation_dom B) A) (forall ((C $$unsorted)) (=> (tptp.in C A) (tptp.in (tptp.apply B C) C))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset A B) (= (tptp.set_intersection2 A B) A))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.is_eventually_in A B C) (tptp.is_often_in A B C))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.lattice B) (tptp.latt_str B)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.poset_of_lattice B))) (=> (tptp.element C (tptp.the_carrier _let_1)) (= (tptp.relstr_element_smaller _let_1 A C) (tptp.latt_set_smaller B (tptp.cast_to_el_of_lattice B C) A))))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.closed_subset B A) (tptp.open_subset (tptp.subset_complement _let_1 B) A))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (=> (tptp.in C (tptp.lim_points_of_net A B)) (tptp.is_a_cluster_point_of_netstr A B C)))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.complete_latt_str A) (tptp.latt_str A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.poset_of_lattice A))) (and (= (tptp.join_of_latt_set A B) (tptp.join_on_relstr _let_1 B)) (= (tptp.meet_of_latt_set A B) (tptp.meet_on_relstr _let_1 B))))))) (forall ((A $$unsorted)) (= (tptp.set_intersection2 A tptp.empty_set) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.the_carrier (tptp.boole_lattice A))) (forall ((C $$unsorted)) (let ((_let_1 (tptp.boole_lattice A))) (=> (tptp.element C (tptp.the_carrier _let_1)) (= (tptp.below _let_1 B C) (tptp.subset B C))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element A B) (or (tptp.empty B) (tptp.in A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (forall ((C $$unsorted)) (= (tptp.in C A) (tptp.in C B))) (= A B))) (forall ((A $$unsorted)) (tptp.reflexive (tptp.inclusion_relation A))) (forall ((A $$unsorted)) (tptp.subset tptp.empty_set A)) (forall ((A $$unsorted)) (=> (not (tptp.empty A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (let ((_let_2 (tptp.powerset (tptp.the_carrier _let_1)))) (=> (and (not (tptp.empty B)) (tptp.filtered_subset B _let_1) (tptp.upper_relstr_subset B _let_1) (tptp.proper_element B _let_2) (tptp.element B _let_2)) (forall ((C $$unsorted)) (not (and (tptp.in C B) (tptp.empty C)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.the_carrier (tptp.boole_POSet A))) (forall ((C $$unsorted)) (let ((_let_1 (tptp.boole_POSet A))) (=> (tptp.element C (tptp.the_carrier _let_1)) (= (tptp.related_reflexive _let_1 B C) (tptp.subset B C))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.lattice B) (tptp.latt_str B)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier B)) (= (tptp.latt_element_smaller B C A) (tptp.relstr_set_smaller (tptp.poset_of_lattice B) A (tptp.cast_to_el_of_LattPOSet B C))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_field C))) (=> (tptp.relation C) (=> (tptp.in (tptp.ordered_pair A B) C) (and (tptp.in A _let_1) (tptp.in B _let_1)))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.open_subset B A) (tptp.closed_subset (tptp.subset_complement _let_1 B) A))))))) (forall ((A $$unsorted)) (=> (and (tptp.antisymmetric_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (let ((_let_1 (and (= B (tptp.join_on_relstr A C)) (tptp.ex_sup_of_relstr_set A C)))) (let ((_let_2 (tptp.relstr_set_smaller A C B))) (and (=> _let_1 (and _let_2 (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.relstr_set_smaller A C D) (tptp.related A B D)))))) (=> (and _let_2 (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.relstr_set_smaller A C D) (tptp.related A B D))))) _let_1))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.preimage_subnetstr A B C))) (=> (tptp.is_often_in A B C) (and (not (tptp.empty_carrier _let_1)) (tptp.directed_relstr _let_1))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.lattice B) (tptp.latt_str B)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.poset_of_lattice B))) (=> (tptp.element C (tptp.the_carrier _let_1)) (= (tptp.relstr_set_smaller _let_1 A C) (tptp.latt_element_smaller B (tptp.cast_to_el_of_lattice B C) A))))))) (forall ((A $$unsorted)) (=> (forall ((B $$unsorted)) (=> (tptp.in B A) (and (tptp.ordinal B) (tptp.subset B A)))) (tptp.ordinal A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.well_founded_relation B) (tptp.well_founded_relation (tptp.relation_restriction B A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.is_often_in A B C) (tptp.subnet (tptp.preimage_subnetstr A B C) A B))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element C _let_1) (= (tptp.in (tptp.ordered_pair_as_product_element _let_1 _let_1 B C) (tptp.relation_of_lattice A)) (tptp.below_refl A B C))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.ordinal B) (not (and (tptp.subset A B) (not (= A tptp.empty_set)) (forall ((C $$unsorted)) (=> (tptp.ordinal C) (not (and (tptp.in C A) (forall ((D $$unsorted)) (=> (tptp.ordinal D) (=> (tptp.in D A) (tptp.ordinal_subset C D)))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.well_ordering B) (tptp.well_ordering (tptp.relation_restriction B A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted) (D $$unsorted)) (=> (tptp.subnet D A B) (=> (= D (tptp.preimage_subnetstr A B C)) (tptp.is_eventually_in A D C)))))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (= (tptp.in A B) (tptp.ordinal_subset (tptp.succ A) B)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (tptp.subset (tptp.set_difference A C) (tptp.set_difference B C)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (= (tptp.ordered_pair A B) (tptp.ordered_pair C D)) (and (= A C) (= B D)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (= (= B (tptp.identity_relation A)) (and (= (tptp.relation_dom B) A) (forall ((C $$unsorted)) (=> (tptp.in C A) (= (tptp.apply B C) C))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.complete_latt_str A) (tptp.latt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (= (= B (tptp.meet_of_latt_set A C)) (and (tptp.latt_set_smaller A B C) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (=> (tptp.latt_set_smaller A D C) (tptp.below_refl A D B))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in B A) (= (tptp.apply (tptp.identity_relation A) B) B))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset (tptp.set_difference A B) A)) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_inverse A))) (=> (tptp.relation A) (and (= (tptp.relation_rng A) (tptp.relation_dom _let_1)) (= (tptp.relation_dom A) (tptp.relation_rng _let_1)))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= (tptp.set_difference A B) tptp.empty_set) (tptp.subset A B))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.subset (tptp.singleton A) B) (tptp.in A B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (tptp.subset (tptp.unordered_pair A B) C) (and (tptp.in A C) (tptp.in B C)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (and (tptp.well_ordering B) (tptp.subset A (tptp.relation_field B))) (= (tptp.relation_field (tptp.relation_restriction B A)) A)))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_union2 A (tptp.set_difference B A)) (tptp.set_union2 A B))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.singleton B))) (= (tptp.subset A _let_1) (or (= A tptp.empty_set) (= A _let_1))))) (forall ((A $$unsorted)) (= (tptp.set_difference A tptp.empty_set) A)) (forall ((A $$unsorted)) (let ((_let_1 (tptp.boole_lattice A))) (and (tptp.lower_bounded_semilattstr _let_1) (= (tptp.bottom_of_semilattstr _let_1) tptp.empty_set)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (not (and (tptp.in A B) (tptp.in B C) (tptp.in C A)))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.element A (tptp.powerset B)) (tptp.subset A B))) (forall ((A $$unsorted)) (tptp.transitive (tptp.inclusion_relation A))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.disjoint A B))) (and (not (and (not _let_1) (forall ((C $$unsorted)) (not (and (tptp.in C A) (tptp.in C B)))))) (not (and (exists ((C $$unsorted)) (and (tptp.in C A) (tptp.in C B))) _let_1))))) (forall ((A $$unsorted)) (=> (tptp.subset A tptp.empty_set) (= A tptp.empty_set))) _let_1 (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_difference (tptp.set_union2 A B) B) (tptp.set_difference A B))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (= (tptp.being_limit_ordinal A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (=> (tptp.in B A) (tptp.in (tptp.succ B) A))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.transitive_relstr B) (tptp.directed_relstr B) (tptp.net_str B A)) (forall ((C $$unsorted)) (=> (tptp.subnet C A B) (tptp.subset (tptp.lim_points_of_net A B) (tptp.lim_points_of_net A C)))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.being_limit_ordinal A))) (=> (tptp.ordinal A) (and (not (and (not _let_1) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (not (= A (tptp.succ B))))))) (not (and (exists ((B $$unsorted)) (and (tptp.ordinal B) (= A (tptp.succ B)))) _let_1)))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.antisymmetric_relstr A) (tptp.lower_bounded_relstr A) (tptp.rel_str A)) (and (tptp.ex_sup_of_relstr_set A tptp.empty_set) (tptp.ex_inf_of_relstr_set A (tptp.the_carrier A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset A)) (= (tptp.disjoint B C) (tptp.subset B (tptp.subset_complement A C))))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset (tptp.powerset _let_1))) (=> (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.the_carrier A))) (=> (tptp.in C B) (tptp.closed_subset C A)))) (tptp.closed_subset (tptp.meet_of_subsets _let_1 B) A))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_dom (tptp.relation_composition A B)) (tptp.relation_dom A)))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.subset (tptp.interior A B) B))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.antisymmetric_relstr A) (tptp.lower_bounded_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (tptp.related A (tptp.bottom_of_relstr A) B))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (forall ((C $$unsorted)) (=> (tptp.in C (tptp.the_carrier A)) (= (tptp.in C (tptp.topstr_closure A B)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier A))) (=> (and (tptp.closed_subset D A) (tptp.subset B D)) (tptp.in C D))))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_composition A B)) (tptp.relation_rng B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset A B) (= B (tptp.set_union2 A (tptp.set_difference B A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.function D) (tptp.quasi_total D A B) (tptp.relation_of2_as_subset D A B)) (=> (not (= B tptp.empty_set)) (forall ((E $$unsorted)) (= (tptp.in E (tptp.relation_inverse_image D C)) (and (tptp.in E A) (tptp.in (tptp.apply D E) C))))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (exists ((C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (and (tptp.element C (tptp.powerset (tptp.powerset _let_1))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier A))) (= (tptp.in D C) (and (tptp.closed_subset D A) (tptp.subset B D))))) (= (tptp.topstr_closure A B) (tptp.meet_of_subsets _let_1 C))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset (tptp.relation_rng A) (tptp.relation_dom B)) (= (tptp.relation_dom (tptp.relation_composition A B)) (tptp.relation_dom A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (not (and (not (= B tptp.empty_set)) (= (tptp.complements_of_subsets A B) tptp.empty_set))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (= (tptp.set_union2 (tptp.singleton A) B) B))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset (tptp.relation_dom A) (tptp.relation_rng B)) (= (tptp.relation_rng (tptp.relation_composition B A)) (tptp.relation_rng A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (=> (not (= B tptp.empty_set)) (= (tptp.subset_difference A (tptp.cast_to_subset A) (tptp.union_of_subsets A B)) (tptp.meet_of_subsets A (tptp.complements_of_subsets A B)))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.subset B (tptp.topstr_closure A B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (=> (not (= B tptp.empty_set)) (= (tptp.union_of_subsets A (tptp.complements_of_subsets A B)) (tptp.subset_difference A (tptp.cast_to_subset A) (tptp.meet_of_subsets A B)))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_difference A (tptp.set_difference A B)) (tptp.set_intersection2 A B))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.relation_isomorphism A B C) (tptp.relation_isomorphism B A (tptp.function_inverse C))))))))) (forall ((A $$unsorted)) (= (tptp.set_difference tptp.empty_set A) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.in A B) (tptp.element B (tptp.powerset C))) (tptp.element A C))) (forall ((A $$unsorted)) (= (tptp.the_carrier (tptp.boole_POSet A)) (tptp.powerset A))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (tptp.connected (tptp.inclusion_relation A)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.disjoint A B))) (and (not (and (not _let_1) (forall ((C $$unsorted)) (not (tptp.in C (tptp.set_intersection2 A B)))))) (not (and (exists ((C $$unsorted)) (tptp.in C (tptp.set_intersection2 A B))) _let_1))))) (forall ((A $$unsorted)) (= (tptp.boole_POSet A) (tptp.incl_POSet (tptp.powerset A)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.latt_str A))) (let ((_let_2 (tptp.lattice A))) (let ((_let_3 (not (tptp.empty_carrier A)))) (=> (and _let_3 _let_2 (tptp.complete_latt_str A) _let_1) (and _let_3 _let_2 (tptp.lower_bounded_semilattstr A) _let_1 (= (tptp.bottom_of_semilattstr A) (tptp.join_of_latt_set A tptp.empty_set)))))))) (forall ((A $$unsorted)) (=> (not (= A tptp.empty_set)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.element C A) (=> (not (tptp.in C B)) (tptp.in C (tptp.subset_complement A B))))))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.open_subset (tptp.interior A B) A))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.closed_subset B A))) (let ((_let_2 (= (tptp.topstr_closure A B) B))) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (and (=> _let_1 _let_2) (=> (and (tptp.topological_space A) _let_2) _let_1)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.relation_isomorphism A B C) (and (=> (tptp.reflexive A) (tptp.reflexive B)) (=> (tptp.transitive A) (tptp.transitive B)) (=> (tptp.connected A) (tptp.connected B)) (=> (tptp.antisymmetric A) (tptp.antisymmetric B)) (=> (tptp.well_founded_relation A) (tptp.well_founded_relation B)))))))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (forall ((B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (= (= B (tptp.function_inverse A)) (and (= (tptp.relation_dom B) (tptp.relation_rng A)) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (and (tptp.in C (tptp.relation_rng A)) (= D (tptp.apply B C))))) (let ((_let_2 (and (tptp.in D (tptp.relation_dom A)) (= C (tptp.apply A D))))) (and (=> _let_1 _let_2) (=> _let_2 _let_1)))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.element C (tptp.powerset A)) (not (and (tptp.in B (tptp.subset_complement A C)) (tptp.in B C))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (and (tptp.well_ordering A) (tptp.relation_isomorphism A B C)) (tptp.well_ordering B)))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.function_inverse A))) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (and (= (tptp.relation_rng A) (tptp.relation_dom _let_1)) (= (tptp.relation_dom A) (tptp.relation_rng _let_1))))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.top_str B) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.the_carrier A))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier B))) (and (=> (tptp.open_subset D B) (= (tptp.interior B D) D)) (=> (= (tptp.interior A C) C) (tptp.open_subset C A))))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (=> (forall ((B $$unsorted) (C $$unsorted)) (not (tptp.in (tptp.ordered_pair B C) A))) (= A tptp.empty_set)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.function_inverse B))) (=> (and (tptp.relation B) (tptp.function B)) (=> (and (tptp.one_to_one B) (tptp.in A (tptp.relation_rng B))) (and (= A (tptp.apply B (tptp.apply _let_1 A))) (= A (tptp.apply (tptp.relation_composition _let_1 B) A))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (=> (and (tptp.open_subset B A) (tptp.in C B)) (tptp.point_neighbourhood B A C)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (not (and (tptp.in A B) (tptp.element B (tptp.powerset C)) (tptp.empty C)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B _let_1) (= (tptp.proper_element B _let_1) (not (= B A)))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (not (and (tptp.is_a_cover_of_carrier A B) (= B tptp.empty_set))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.well_founded_relation A) (tptp.is_well_founded_in A (tptp.relation_field A))))) (forall ((A $$unsorted)) (tptp.antisymmetric (tptp.inclusion_relation A))) (and (= (tptp.relation_dom tptp.empty_set) tptp.empty_set) (= (tptp.relation_rng tptp.empty_set) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.subset A B) (tptp.proper_subset B A)))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (=> (tptp.subrelstr B A) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.the_carrier B)) (forall ((F $$unsorted)) (=> (tptp.element F (tptp.the_carrier B)) (=> (and (= E C) (= F D) (tptp.related B E F)) (tptp.related A C D)))))))))))))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (=> (and (tptp.full_subrelstr B A) (tptp.subrelstr B A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.the_carrier A)) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.the_carrier B)) (forall ((F $$unsorted)) (let ((_let_1 (tptp.the_carrier B))) (=> (tptp.element F _let_1) (=> (and (= E C) (= F D) (tptp.related A C D) (tptp.in E _let_1) (tptp.in F _let_1)) (tptp.related B E F))))))))))))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (tptp.one_to_one (tptp.function_inverse A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.disjoint B C)) (tptp.disjoint A C))) (forall ((A $$unsorted)) (=> (tptp.relation A) (=> (or (= (tptp.relation_dom A) tptp.empty_set) (= (tptp.relation_rng A) tptp.empty_set)) (= A tptp.empty_set)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (= (tptp.relation_dom A) tptp.empty_set) (= (tptp.relation_rng A) tptp.empty_set)))) (forall ((A $$unsorted) (B $$unsorted)) (= (= (tptp.set_difference A (tptp.singleton B)) A) (not (tptp.in B A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (= B (tptp.relation_dom_restriction C A)) (and (= (tptp.relation_dom B) (tptp.set_intersection2 (tptp.relation_dom C) A)) (forall ((D $$unsorted)) (=> (tptp.in D (tptp.relation_dom B)) (= (tptp.apply B D) (tptp.apply C D)))))))))) (forall ((A $$unsorted)) (= (tptp.unordered_pair A A) (tptp.singleton A))) (forall ((A $$unsorted)) (=> (tptp.empty A) (= A tptp.empty_set))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.function D) (tptp.quasi_total D A B) (tptp.relation_of2_as_subset D A B)) (=> (tptp.in C A) (or (= B tptp.empty_set) (tptp.in (tptp.apply D C) (tptp.relation_rng D)))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (tptp.well_founded_relation (tptp.inclusion_relation A)))) (forall ((A $$unsorted)) (=> (tptp.rel_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (and (tptp.relstr_set_smaller A tptp.empty_set B) (tptp.relstr_element_smaller A tptp.empty_set B)))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.in C (tptp.topstr_closure A B)) (forall ((D $$unsorted)) (=> (tptp.point_neighbourhood D A C) (not (tptp.disjoint D B))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset (tptp.singleton A) (tptp.singleton B)) (= A B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_dom_restriction C A))) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in B (tptp.relation_dom _let_1)) (= (tptp.apply _let_1 B) (tptp.apply C B)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.identity_relation A))) (and (= (tptp.relation_dom _let_1) A) (= (tptp.relation_rng _let_1) A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in B A) (= (tptp.apply (tptp.relation_dom_restriction C A) B) (tptp.apply C B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.ordered_pair A B))) (=> (tptp.relation D) (= (tptp.in _let_1 (tptp.relation_composition (tptp.identity_relation C) D)) (and (tptp.in A C) (tptp.in _let_1 D)))))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.in A B) (tptp.empty B)))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.lattice A) (tptp.latt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.below_refl A B C) (tptp.related_reflexive (tptp.poset_of_lattice A) (tptp.cast_to_el_of_LattPOSet A B) (tptp.cast_to_el_of_LattPOSet A C))))))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.ordered_pair A B))) (and (= (tptp.pair_first _let_1) A) (= (tptp.pair_second _let_1) B)))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.in A B) (forall ((C $$unsorted)) (not (and (tptp.in C B) (forall ((D $$unsorted)) (not (and (tptp.in D B) (tptp.in D C)))))))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (tptp.well_ordering (tptp.inclusion_relation A)))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset A (tptp.set_union2 A B))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.disjoint A B) (= (tptp.set_difference A B) A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_dom (tptp.relation_dom_restriction C B))) (and (tptp.in A B) (tptp.in A (tptp.relation_dom C)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_dom_restriction B A) B))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.empty A) (not (= A B)) (tptp.empty B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.in (tptp.ordered_pair A B) C) (and (tptp.in A (tptp.relation_dom C)) (= B (tptp.apply C A)))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (and (not (tptp.empty_carrier B)) (tptp.net_str B A)) (forall ((C $$unsorted) (D $$unsorted)) (=> (tptp.subset C D) (and (=> (tptp.is_eventually_in A B C) (tptp.is_eventually_in A B D)) (=> (tptp.is_often_in A B C) (tptp.is_often_in A B D))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.reflexive_relstr A) (tptp.transitive_relstr A) (tptp.antisymmetric_relstr A) (tptp.lower_bounded_relstr A) (tptp.rel_str A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.powerset (tptp.the_carrier A)))) (=> (and (not (tptp.empty B)) (tptp.filtered_subset B A) (tptp.upper_relstr_subset B A) (tptp.element B _let_1)) (= (tptp.proper_element B _let_1) (not (tptp.in (tptp.bottom_of_relstr A) B)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.well_orders A (tptp.relation_field A)) (tptp.well_ordering A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset C B)) (tptp.subset (tptp.set_union2 A C) B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (= (tptp.singleton A) (tptp.unordered_pair B C)) (= A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_dom (tptp.relation_dom_restriction B A)) (tptp.set_intersection2 (tptp.relation_dom B) A)))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B _let_1) (= (tptp.apply_as_element _let_1 _let_1 (tptp.identity_on_carrier A) B) B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (tptp.subset A (tptp.union B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_dom_restriction B A) (tptp.relation_composition (tptp.identity_relation A) B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_dom_restriction B A)) (tptp.relation_rng B)))) (forall ((A $$unsorted)) (= (tptp.union (tptp.powerset A)) A)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.function D))) (=> (and _let_1 (tptp.quasi_total D A B) (tptp.relation_of2_as_subset D A B)) (=> (tptp.subset B C) (or (and (= B tptp.empty_set) (not (= A tptp.empty_set))) (and _let_1 (tptp.quasi_total D A C) (tptp.relation_of2_as_subset D A C))))))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.in A B) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C B) (tptp.subset D C)) (tptp.in D B))) (forall ((C $$unsorted)) (not (and (tptp.in C B) (forall ((D $$unsorted)) (not (and (tptp.in D B) (forall ((E $$unsorted)) (=> (tptp.subset E C) (tptp.in E D))))))))) (forall ((C $$unsorted)) (not (and (tptp.subset C B) (not (tptp.are_equipotent C B)) (not (tptp.in C B)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (= (tptp.singleton A) (tptp.unordered_pair B C)) (= B C))) true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 1.85/2.17  )
% 1.85/2.17  % SZS output end Proof for SEU388+2
% 1.85/2.17  % cvc5---1.0.5 exiting
% 1.85/2.17  % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------