TSTP Solution File: SEU336+1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SEU336+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n022.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 : Tue Sep 20 07:28:59 EDT 2022
% Result : Theorem 1.60s 1.30s
% Output : Proof 1.73s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU336+1 : TPTP v8.1.0. Released v3.3.0.
% 0.11/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.34 % Computer : n022.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sat Sep 3 12:17:09 EDT 2022
% 0.20/0.34 % CPUTime :
% 0.20/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.20/0.34 Usage: tptp [options] [-file:]file
% 0.20/0.34 -h, -? prints this message.
% 0.20/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.20/0.34 -m, -model generate model.
% 0.20/0.34 -p, -proof generate proof.
% 0.20/0.34 -c, -core generate unsat core of named formulas.
% 0.20/0.34 -st, -statistics display statistics.
% 0.20/0.34 -t:timeout set timeout (in second).
% 0.20/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.20/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.20/0.34 -<param>:<value> configuration parameter and value.
% 0.20/0.34 -o:<output-file> file to place output in.
% 1.60/1.30 % SZS status Theorem
% 1.60/1.30 % SZS output start Proof
% 1.60/1.30 tff(element_type, type, (
% 1.60/1.30 element: ( $i * $i ) > $o)).
% 1.60/1.30 tff(powerset_type, type, (
% 1.60/1.30 powerset: $i > $i)).
% 1.60/1.30 tff(the_carrier_type, type, (
% 1.60/1.30 the_carrier: $i > $i)).
% 1.60/1.30 tff(tptp_fun_A_9_type, type, (
% 1.60/1.30 tptp_fun_A_9: $i)).
% 1.60/1.30 tff(subset_complement_type, type, (
% 1.60/1.30 subset_complement: ( $i * $i ) > $i)).
% 1.60/1.30 tff(tptp_fun_C_1_type, type, (
% 1.60/1.30 tptp_fun_C_1: ( $i * $i ) > $i)).
% 1.60/1.30 tff(tptp_fun_B_10_type, type, (
% 1.60/1.30 tptp_fun_B_10: $i)).
% 1.60/1.30 tff(in_type, type, (
% 1.60/1.30 in: ( $i * $i ) > $o)).
% 1.60/1.30 tff(complements_of_subsets_type, type, (
% 1.60/1.30 complements_of_subsets: ( $i * $i ) > $i)).
% 1.60/1.30 tff(tptp_fun_D_2_type, type, (
% 1.60/1.30 tptp_fun_D_2: ( $i * $i * $i ) > $i)).
% 1.60/1.30 tff(open_subsets_type, type, (
% 1.60/1.30 open_subsets: ( $i * $i ) > $o)).
% 1.60/1.30 tff(closed_subsets_type, type, (
% 1.60/1.30 closed_subsets: ( $i * $i ) > $o)).
% 1.60/1.30 tff(top_str_type, type, (
% 1.60/1.30 top_str: $i > $o)).
% 1.60/1.30 tff(closed_subset_type, type, (
% 1.60/1.30 closed_subset: ( $i * $i ) > $o)).
% 1.60/1.30 tff(tptp_fun_C_0_type, type, (
% 1.60/1.30 tptp_fun_C_0: ( $i * $i ) > $i)).
% 1.60/1.30 tff(open_subset_type, type, (
% 1.60/1.30 open_subset: ( $i * $i ) > $o)).
% 1.60/1.30 tff(1,plain,
% 1.60/1.30 (((~(~top_str(A!9))) & (~((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (closed_subsets(B!10, A!9) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9))))) <=> (top_str(A!9) & (~((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (closed_subsets(B!10, A!9) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)))))),
% 1.60/1.30 inference(rewrite,[status(thm)],[])).
% 1.60/1.30 tff(2,plain,
% 1.60/1.30 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))) <=> (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A)))))),
% 1.60/1.30 inference(rewrite,[status(thm)],[])).
% 1.60/1.30 tff(3,plain,
% 1.60/1.30 ((~![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))) <=> (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A)))))),
% 1.60/1.30 inference(rewrite,[status(thm)],[])).
% 1.60/1.30 tff(4,axiom,(~![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t16_tops_2')).
% 1.60/1.30 tff(5,plain,
% 1.60/1.30 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[4, 3])).
% 1.60/1.30 tff(6,plain,
% 1.60/1.30 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[5, 2])).
% 1.60/1.30 tff(7,plain,
% 1.60/1.30 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[6, 2])).
% 1.60/1.30 tff(8,plain,
% 1.60/1.30 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[7, 2])).
% 1.60/1.30 tff(9,plain,
% 1.60/1.30 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[8, 2])).
% 1.60/1.30 tff(10,plain,
% 1.60/1.30 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[9, 2])).
% 1.60/1.30 tff(11,plain,
% 1.60/1.30 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> open_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[10, 2])).
% 1.60/1.30 tff(12,plain,
% 1.60/1.30 (top_str(A!9) & (~((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (closed_subsets(B!10, A!9) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[11, 1])).
% 1.60/1.30 tff(13,plain,
% 1.60/1.30 (~((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (closed_subsets(B!10, A!9) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)))),
% 1.60/1.30 inference(and_elim,[status(thm)],[12])).
% 1.60/1.30 tff(14,plain,
% 1.60/1.30 (element(B!10, powerset(powerset(the_carrier(A!9))))),
% 1.60/1.30 inference(or_elim,[status(thm)],[13])).
% 1.60/1.30 tff(15,plain,
% 1.60/1.30 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))))),
% 1.60/1.30 inference(bind,[status(th)],[])).
% 1.60/1.30 tff(16,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.60/1.30 inference(quant_intro,[status(thm)],[15])).
% 1.60/1.30 tff(17,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.60/1.30 inference(rewrite,[status(thm)],[])).
% 1.60/1.30 tff(18,plain,
% 1.60/1.30 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))))),
% 1.60/1.30 inference(bind,[status(th)],[])).
% 1.60/1.30 tff(19,plain,
% 1.60/1.30 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.60/1.30 inference(quant_intro,[status(thm)],[18])).
% 1.60/1.30 tff(20,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k7_setfam_1')).
% 1.60/1.30 tff(21,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[20, 19])).
% 1.60/1.30 tff(22,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[21, 17])).
% 1.60/1.30 tff(23,plain,(
% 1.60/1.30 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.60/1.30 inference(skolemize,[status(sab)],[22])).
% 1.60/1.30 tff(24,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[23, 16])).
% 1.60/1.30 tff(25,plain,
% 1.60/1.30 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9)))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9)))))),
% 1.60/1.30 inference(rewrite,[status(thm)],[])).
% 1.60/1.30 tff(26,plain,
% 1.60/1.30 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9)))))),
% 1.60/1.30 inference(quant_inst,[status(thm)],[])).
% 1.60/1.30 tff(27,plain,
% 1.60/1.30 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))),
% 1.60/1.30 inference(modus_ponens,[status(thm)],[26, 25])).
% 1.60/1.30 tff(28,plain,
% 1.60/1.30 (element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))),
% 1.60/1.30 inference(unit_resolution,[status(thm)],[27, 24, 14])).
% 1.60/1.30 tff(29,plain,
% 1.60/1.30 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))))),
% 1.60/1.30 inference(bind,[status(th)],[])).
% 1.60/1.30 tff(30,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.60/1.30 inference(quant_intro,[status(thm)],[29])).
% 1.60/1.30 tff(31,plain,
% 1.60/1.30 (^[A: $i, B: $i] : rewrite(((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))))),
% 1.60/1.30 inference(bind,[status(th)],[])).
% 1.60/1.30 tff(32,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.60/1.30 inference(quant_intro,[status(thm)],[31])).
% 1.60/1.30 tff(33,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.60/1.30 inference(transitivity,[status(thm)],[32, 30])).
% 1.60/1.30 tff(34,plain,
% 1.60/1.30 (^[A: $i, B: $i] : rewrite(((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))) & ((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))))),
% 1.60/1.30 inference(bind,[status(th)],[])).
% 1.60/1.30 tff(35,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))) & ((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.60/1.30 inference(quant_intro,[status(thm)],[34])).
% 1.60/1.30 tff(36,plain,
% 1.60/1.30 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))),
% 1.60/1.30 inference(rewrite,[status(thm)],[])).
% 1.60/1.30 tff(37,plain,
% 1.60/1.30 (^[A: $i, B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(rewrite(((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B)))) <=> ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))), ((element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B))))) <=> (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))), rewrite((element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) <=> ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))), ((element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B))))) <=> ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))))), (![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B))))) <=> ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))), ((element(B, powerset(powerset(A))) => ![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> (element(B, powerset(powerset(A))) => ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))))), rewrite((element(B, powerset(powerset(A))) => ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))), ((element(B, powerset(powerset(A))) => ![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))))),
% 1.60/1.31 inference(bind,[status(th)],[])).
% 1.60/1.31 tff(38,plain,
% 1.60/1.31 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))),
% 1.60/1.31 inference(quant_intro,[status(thm)],[37])).
% 1.60/1.31 tff(39,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d8_setfam_1')).
% 1.60/1.31 tff(40,plain,
% 1.60/1.31 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[39, 38])).
% 1.60/1.31 tff(41,plain,
% 1.60/1.31 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[40, 36])).
% 1.60/1.31 tff(42,plain,(
% 1.60/1.31 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))) & ((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))),
% 1.60/1.31 inference(skolemize,[status(sab)],[41])).
% 1.60/1.31 tff(43,plain,
% 1.60/1.31 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[42, 35])).
% 1.60/1.31 tff(44,plain,
% 1.60/1.31 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[43, 33])).
% 1.60/1.31 tff(45,plain,
% 1.60/1.31 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10)))))))))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10)))))))))))),
% 1.60/1.31 inference(rewrite,[status(thm)],[])).
% 1.60/1.31 tff(46,plain,
% 1.60/1.31 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10)))))))))))),
% 1.60/1.31 inference(quant_inst,[status(thm)],[])).
% 1.60/1.31 tff(47,plain,
% 1.60/1.31 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[46, 45])).
% 1.60/1.31 tff(48,plain,
% 1.60/1.31 (![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))),
% 1.60/1.31 inference(unit_resolution,[status(thm)],[47, 44, 28])).
% 1.60/1.31 tff(49,plain,
% 1.60/1.31 (((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))) <=> ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))),
% 1.60/1.31 inference(rewrite,[status(thm)],[])).
% 1.60/1.31 tff(50,plain,
% 1.60/1.31 ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))),
% 1.60/1.31 inference(quant_inst,[status(thm)],[])).
% 1.60/1.31 tff(51,plain,
% 1.60/1.31 ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10)))))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[50, 49])).
% 1.60/1.31 tff(52,plain,
% 1.60/1.31 (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))),
% 1.60/1.31 inference(unit_resolution,[status(thm)],[51, 14, 48])).
% 1.60/1.31 tff(53,plain,
% 1.60/1.31 (((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10)))))))) | ((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))),
% 1.60/1.31 inference(tautology,[status(thm)],[])).
% 1.60/1.31 tff(54,plain,
% 1.60/1.31 ((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.60/1.31 inference(unit_resolution,[status(thm)],[53, 52])).
% 1.60/1.31 tff(55,plain,
% 1.60/1.31 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)))),
% 1.60/1.31 inference(bind,[status(th)],[])).
% 1.60/1.31 tff(56,plain,
% 1.60/1.31 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.60/1.31 inference(quant_intro,[status(thm)],[55])).
% 1.60/1.31 tff(57,plain,
% 1.60/1.31 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.60/1.31 inference(rewrite,[status(thm)],[])).
% 1.60/1.31 tff(58,plain,
% 1.60/1.31 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)))),
% 1.60/1.31 inference(bind,[status(th)],[])).
% 1.60/1.31 tff(59,plain,
% 1.60/1.31 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.60/1.31 inference(quant_intro,[status(thm)],[58])).
% 1.60/1.31 tff(60,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','involutiveness_k7_setfam_1')).
% 1.60/1.31 tff(61,plain,
% 1.60/1.31 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[60, 59])).
% 1.60/1.31 tff(62,plain,
% 1.60/1.31 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[61, 57])).
% 1.60/1.31 tff(63,plain,(
% 1.60/1.31 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.60/1.31 inference(skolemize,[status(sab)],[62])).
% 1.60/1.31 tff(64,plain,
% 1.60/1.31 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[63, 56])).
% 1.60/1.31 tff(65,plain,
% 1.60/1.31 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10))),
% 1.60/1.31 inference(rewrite,[status(thm)],[])).
% 1.60/1.31 tff(66,plain,
% 1.60/1.31 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10))),
% 1.60/1.31 inference(quant_inst,[status(thm)],[])).
% 1.60/1.31 tff(67,plain,
% 1.60/1.31 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10)),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[66, 65])).
% 1.60/1.31 tff(68,plain,
% 1.60/1.31 (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10),
% 1.60/1.31 inference(unit_resolution,[status(thm)],[67, 64, 14])).
% 1.60/1.31 tff(69,plain,
% 1.60/1.31 (B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.60/1.31 inference(symmetry,[status(thm)],[68])).
% 1.60/1.31 tff(70,plain,
% 1.60/1.31 ((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.60/1.31 inference(tautology,[status(thm)],[])).
% 1.60/1.31 tff(71,plain,
% 1.60/1.31 ((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.60/1.31 inference(unit_resolution,[status(thm)],[70, 69])).
% 1.60/1.31 tff(72,plain,
% 1.60/1.31 (![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.60/1.31 inference(unit_resolution,[status(thm)],[71, 54])).
% 1.60/1.31 tff(73,assumption,(~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)), introduced(assumption)).
% 1.60/1.31 tff(74,plain,
% 1.60/1.31 (top_str(A!9)),
% 1.60/1.31 inference(and_elim,[status(thm)],[12])).
% 1.60/1.31 tff(75,plain,
% 1.60/1.31 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))))),
% 1.60/1.31 inference(bind,[status(th)],[])).
% 1.60/1.31 tff(76,plain,
% 1.60/1.31 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.60/1.31 inference(quant_intro,[status(thm)],[75])).
% 1.60/1.31 tff(77,plain,
% 1.60/1.31 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))))),
% 1.60/1.31 inference(bind,[status(th)],[])).
% 1.60/1.31 tff(78,plain,
% 1.60/1.31 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.60/1.31 inference(quant_intro,[status(thm)],[77])).
% 1.60/1.31 tff(79,plain,
% 1.60/1.31 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.60/1.31 inference(transitivity,[status(thm)],[78, 76])).
% 1.60/1.31 tff(80,plain,
% 1.60/1.31 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))))),
% 1.60/1.31 inference(bind,[status(th)],[])).
% 1.60/1.31 tff(81,plain,
% 1.60/1.31 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.60/1.31 inference(quant_intro,[status(thm)],[80])).
% 1.60/1.31 tff(82,plain,
% 1.60/1.31 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.60/1.31 inference(rewrite,[status(thm)],[])).
% 1.60/1.31 tff(83,plain,
% 1.60/1.31 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(rewrite((open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A)))) <=> (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))), ((element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A))))) <=> (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), rewrite((element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) <=> ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))), ((element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A))))) <=> ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))))), (![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A))))) <=> ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A)))))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A)))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))))),
% 1.60/1.31 inference(bind,[status(th)],[])).
% 1.60/1.31 tff(84,plain,
% 1.60/1.31 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A)))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.60/1.31 inference(quant_intro,[status(thm)],[83])).
% 1.60/1.31 tff(85,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_tops_2')).
% 1.60/1.31 tff(86,plain,
% 1.60/1.31 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[85, 84])).
% 1.60/1.31 tff(87,plain,
% 1.60/1.31 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[86, 82])).
% 1.60/1.31 tff(88,plain,(
% 1.60/1.31 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))),
% 1.60/1.31 inference(skolemize,[status(sab)],[87])).
% 1.60/1.31 tff(89,plain,
% 1.60/1.31 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[88, 81])).
% 1.60/1.31 tff(90,plain,
% 1.60/1.31 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.60/1.31 inference(modus_ponens,[status(thm)],[89, 79])).
% 1.60/1.31 tff(91,plain,
% 1.60/1.31 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.60/1.31 inference(rewrite,[status(thm)],[])).
% 1.60/1.31 tff(92,plain,
% 1.60/1.31 (((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.60/1.32 inference(rewrite,[status(thm)],[])).
% 1.60/1.32 tff(93,plain,
% 1.60/1.32 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))))),
% 1.60/1.32 inference(monotonicity,[status(thm)],[92])).
% 1.60/1.32 tff(94,plain,
% 1.60/1.32 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.60/1.32 inference(transitivity,[status(thm)],[93, 91])).
% 1.60/1.32 tff(95,plain,
% 1.60/1.32 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.60/1.32 inference(quant_inst,[status(thm)],[])).
% 1.60/1.32 tff(96,plain,
% 1.60/1.32 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))),
% 1.60/1.32 inference(modus_ponens,[status(thm)],[95, 94])).
% 1.60/1.32 tff(97,plain,
% 1.60/1.32 (![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))),
% 1.60/1.32 inference(unit_resolution,[status(thm)],[96, 90, 74])).
% 1.60/1.32 tff(98,plain,
% 1.60/1.32 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))),
% 1.60/1.32 inference(rewrite,[status(thm)],[])).
% 1.60/1.32 tff(99,plain,
% 1.60/1.32 (((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~in(C, complements_of_subsets(the_carrier(A!9), B!10))) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))))))) <=> ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))),
% 1.60/1.32 inference(rewrite,[status(thm)],[])).
% 1.60/1.32 tff(100,plain,
% 1.60/1.32 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~in(C, complements_of_subsets(the_carrier(A!9), B!10))) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))))))))),
% 1.60/1.32 inference(monotonicity,[status(thm)],[99])).
% 1.60/1.32 tff(101,plain,
% 1.60/1.32 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~in(C, complements_of_subsets(the_carrier(A!9), B!10))) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))),
% 1.60/1.32 inference(transitivity,[status(thm)],[100, 98])).
% 1.60/1.32 tff(102,plain,
% 1.60/1.32 ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~in(C, complements_of_subsets(the_carrier(A!9), B!10))) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))),
% 1.60/1.32 inference(quant_inst,[status(thm)],[])).
% 1.60/1.32 tff(103,plain,
% 1.60/1.32 ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))))))),
% 1.73/1.32 inference(modus_ponens,[status(thm)],[102, 101])).
% 1.73/1.32 tff(104,plain,
% 1.73/1.32 (~((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))),
% 1.73/1.32 inference(unit_resolution,[status(thm)],[103, 97, 28])).
% 1.73/1.32 tff(105,plain,
% 1.73/1.32 (((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))))) | (open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))),
% 1.73/1.32 inference(tautology,[status(thm)],[])).
% 1.73/1.32 tff(106,plain,
% 1.73/1.32 (open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))),
% 1.73/1.32 inference(unit_resolution,[status(thm)],[105, 104])).
% 1.73/1.32 tff(107,plain,
% 1.73/1.32 ((~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))) | open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))),
% 1.73/1.32 inference(tautology,[status(thm)],[])).
% 1.73/1.32 tff(108,plain,
% 1.73/1.32 (open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))),
% 1.73/1.32 inference(unit_resolution,[status(thm)],[107, 106])).
% 1.73/1.32 tff(109,plain,
% 1.73/1.32 (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))),
% 1.73/1.32 inference(unit_resolution,[status(thm)],[108, 73])).
% 1.73/1.32 tff(110,plain,
% 1.73/1.32 (((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))) | element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))),
% 1.73/1.32 inference(tautology,[status(thm)],[])).
% 1.73/1.32 tff(111,plain,
% 1.73/1.32 (element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))),
% 1.73/1.32 inference(unit_resolution,[status(thm)],[110, 109])).
% 1.73/1.32 tff(112,plain,
% 1.73/1.32 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10))))))))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10))))))))))),
% 1.73/1.32 inference(rewrite,[status(thm)],[])).
% 1.73/1.32 tff(113,plain,
% 1.73/1.32 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10))))))))))),
% 1.73/1.32 inference(quant_inst,[status(thm)],[])).
% 1.73/1.32 tff(114,plain,
% 1.73/1.32 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))),
% 1.73/1.32 inference(modus_ponens,[status(thm)],[113, 112])).
% 1.73/1.32 tff(115,plain,
% 1.73/1.32 (![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))),
% 1.73/1.32 inference(unit_resolution,[status(thm)],[114, 44, 14])).
% 1.73/1.32 tff(116,plain,
% 1.73/1.32 (((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) <=> ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))),
% 1.73/1.32 inference(rewrite,[status(thm)],[])).
% 1.73/1.32 tff(117,plain,
% 1.73/1.32 (((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~(complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9))), B!10))))))))) <=> ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))),
% 1.73/1.33 inference(rewrite,[status(thm)],[])).
% 1.73/1.33 tff(118,plain,
% 1.73/1.33 (((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~(complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9))), B!10)))))))))) <=> ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))))),
% 1.73/1.33 inference(monotonicity,[status(thm)],[117])).
% 1.73/1.33 tff(119,plain,
% 1.73/1.33 (((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~(complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9))), B!10)))))))))) <=> ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))),
% 1.73/1.33 inference(transitivity,[status(thm)],[118, 116])).
% 1.73/1.33 tff(120,plain,
% 1.73/1.33 ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~(complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9))), B!10)))))))))),
% 1.73/1.33 inference(quant_inst,[status(thm)],[])).
% 1.73/1.33 tff(121,plain,
% 1.73/1.33 ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[120, 119])).
% 1.73/1.33 tff(122,plain,
% 1.73/1.33 (![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[121, 115, 28])).
% 1.73/1.33 tff(123,plain,
% 1.73/1.33 (((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))) | ((~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)))) <=> ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)))),
% 1.73/1.33 inference(rewrite,[status(thm)],[])).
% 1.73/1.33 tff(124,plain,
% 1.73/1.33 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))) | ((~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)))),
% 1.73/1.33 inference(quant_inst,[status(thm)],[])).
% 1.73/1.33 tff(125,plain,
% 1.73/1.33 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[124, 123])).
% 1.73/1.33 tff(126,plain,
% 1.73/1.33 (in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[125, 122, 111])).
% 1.73/1.33 tff(127,plain,
% 1.73/1.33 (((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))) | in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.73/1.33 inference(tautology,[status(thm)],[])).
% 1.73/1.33 tff(128,plain,
% 1.73/1.33 (in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[127, 109])).
% 1.73/1.33 tff(129,plain,
% 1.73/1.33 ((~(in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10))) | (~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)),
% 1.73/1.33 inference(tautology,[status(thm)],[])).
% 1.73/1.33 tff(130,plain,
% 1.73/1.33 ((~(in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10))) | in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[129, 128])).
% 1.73/1.33 tff(131,plain,
% 1.73/1.33 (in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[130, 126])).
% 1.73/1.33 tff(132,plain,
% 1.73/1.33 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))))),
% 1.73/1.33 inference(bind,[status(th)],[])).
% 1.73/1.33 tff(133,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.33 inference(quant_intro,[status(thm)],[132])).
% 1.73/1.33 tff(134,plain,
% 1.73/1.33 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))))),
% 1.73/1.33 inference(bind,[status(th)],[])).
% 1.73/1.33 tff(135,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.33 inference(quant_intro,[status(thm)],[134])).
% 1.73/1.33 tff(136,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.33 inference(transitivity,[status(thm)],[135, 133])).
% 1.73/1.33 tff(137,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.33 inference(rewrite,[status(thm)],[])).
% 1.73/1.33 tff(138,plain,
% 1.73/1.33 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))) <=> ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))))),
% 1.73/1.33 inference(bind,[status(th)],[])).
% 1.73/1.33 tff(139,plain,
% 1.73/1.33 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.33 inference(quant_intro,[status(thm)],[138])).
% 1.73/1.33 tff(140,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t30_tops_1')).
% 1.73/1.33 tff(141,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[140, 139])).
% 1.73/1.33 tff(142,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[141, 137])).
% 1.73/1.33 tff(143,plain,(
% 1.73/1.33 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.33 inference(skolemize,[status(sab)],[142])).
% 1.73/1.33 tff(144,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[143, 136])).
% 1.73/1.33 tff(145,plain,
% 1.73/1.33 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9))))),
% 1.73/1.33 inference(rewrite,[status(thm)],[])).
% 1.73/1.33 tff(146,plain,
% 1.73/1.33 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9))))),
% 1.73/1.33 inference(quant_inst,[status(thm)],[])).
% 1.73/1.33 tff(147,plain,
% 1.73/1.33 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[146, 145])).
% 1.73/1.33 tff(148,plain,
% 1.73/1.33 (![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[147, 74, 144])).
% 1.73/1.33 tff(149,plain,
% 1.73/1.33 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))) | ((~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)))),
% 1.73/1.33 inference(rewrite,[status(thm)],[])).
% 1.73/1.33 tff(150,plain,
% 1.73/1.33 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))) | ((~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)))),
% 1.73/1.33 inference(quant_inst,[status(thm)],[])).
% 1.73/1.33 tff(151,plain,
% 1.73/1.33 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[150, 149])).
% 1.73/1.33 tff(152,plain,
% 1.73/1.33 (open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[151, 148, 111])).
% 1.73/1.33 tff(153,plain,
% 1.73/1.33 (((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))) | (~open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9))),
% 1.73/1.33 inference(tautology,[status(thm)],[])).
% 1.73/1.33 tff(154,plain,
% 1.73/1.33 (~open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9)),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[153, 109])).
% 1.73/1.33 tff(155,plain,
% 1.73/1.33 ((~(open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))),
% 1.73/1.33 inference(tautology,[status(thm)],[])).
% 1.73/1.33 tff(156,plain,
% 1.73/1.33 ((~(open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))) | (~closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[155, 154])).
% 1.73/1.33 tff(157,plain,
% 1.73/1.33 (~closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[156, 152])).
% 1.73/1.33 tff(158,plain,
% 1.73/1.33 ((~(closed_subsets(B!10, A!9) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9))) <=> ((~closed_subsets(B!10, A!9)) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9))),
% 1.73/1.33 inference(rewrite,[status(thm)],[])).
% 1.73/1.33 tff(159,plain,
% 1.73/1.33 (~(closed_subsets(B!10, A!9) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9))),
% 1.73/1.33 inference(or_elim,[status(thm)],[13])).
% 1.73/1.33 tff(160,plain,
% 1.73/1.33 ((~closed_subsets(B!10, A!9)) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[159, 158])).
% 1.73/1.33 tff(161,plain,
% 1.73/1.33 (closed_subsets(B!10, A!9) | open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~closed_subsets(B!10, A!9)) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)))),
% 1.73/1.33 inference(tautology,[status(thm)],[])).
% 1.73/1.33 tff(162,plain,
% 1.73/1.33 (closed_subsets(B!10, A!9) | open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[161, 160])).
% 1.73/1.33 tff(163,plain,
% 1.73/1.33 (closed_subsets(B!10, A!9)),
% 1.73/1.33 inference(unit_resolution,[status(thm)],[162, 73])).
% 1.73/1.33 tff(164,plain,
% 1.73/1.33 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))))),
% 1.73/1.33 inference(bind,[status(th)],[])).
% 1.73/1.33 tff(165,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.73/1.33 inference(quant_intro,[status(thm)],[164])).
% 1.73/1.33 tff(166,plain,
% 1.73/1.33 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))))),
% 1.73/1.33 inference(bind,[status(th)],[])).
% 1.73/1.33 tff(167,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.73/1.33 inference(quant_intro,[status(thm)],[166])).
% 1.73/1.33 tff(168,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.73/1.33 inference(transitivity,[status(thm)],[167, 165])).
% 1.73/1.33 tff(169,plain,
% 1.73/1.33 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))))),
% 1.73/1.33 inference(bind,[status(th)],[])).
% 1.73/1.33 tff(170,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.73/1.33 inference(quant_intro,[status(thm)],[169])).
% 1.73/1.33 tff(171,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.73/1.33 inference(rewrite,[status(thm)],[])).
% 1.73/1.33 tff(172,plain,
% 1.73/1.33 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(rewrite((closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A)))) <=> (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))), ((element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))))) <=> (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), rewrite((element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) <=> ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))), ((element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))))) <=> ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))))), (![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))))) <=> ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A)))))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A)))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))))),
% 1.73/1.33 inference(bind,[status(th)],[])).
% 1.73/1.33 tff(173,plain,
% 1.73/1.33 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A)))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.73/1.33 inference(quant_intro,[status(thm)],[172])).
% 1.73/1.33 tff(174,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d2_tops_2')).
% 1.73/1.33 tff(175,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[174, 173])).
% 1.73/1.33 tff(176,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[175, 171])).
% 1.73/1.33 tff(177,plain,(
% 1.73/1.33 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))),
% 1.73/1.33 inference(skolemize,[status(sab)],[176])).
% 1.73/1.33 tff(178,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[177, 170])).
% 1.73/1.33 tff(179,plain,
% 1.73/1.33 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.73/1.33 inference(modus_ponens,[status(thm)],[178, 168])).
% 1.73/1.33 tff(180,plain,
% 1.73/1.33 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.73/1.33 inference(rewrite,[status(thm)],[])).
% 1.73/1.33 tff(181,plain,
% 1.73/1.33 (((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.73/1.33 inference(rewrite,[status(thm)],[])).
% 1.73/1.33 tff(182,plain,
% 1.73/1.33 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))))))))))))),
% 1.73/1.34 inference(monotonicity,[status(thm)],[181])).
% 1.73/1.34 tff(183,plain,
% 1.73/1.34 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.73/1.34 inference(transitivity,[status(thm)],[182, 180])).
% 1.73/1.34 tff(184,plain,
% 1.73/1.34 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.73/1.34 inference(quant_inst,[status(thm)],[])).
% 1.73/1.34 tff(185,plain,
% 1.73/1.34 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))))))))))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[184, 183])).
% 1.73/1.34 tff(186,plain,
% 1.73/1.34 (![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))))))))))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[185, 179, 74])).
% 1.73/1.34 tff(187,plain,
% 1.73/1.34 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))))))))))),
% 1.73/1.34 inference(rewrite,[status(thm)],[])).
% 1.73/1.34 tff(188,plain,
% 1.73/1.34 ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))))))))))),
% 1.73/1.34 inference(quant_inst,[status(thm)],[])).
% 1.73/1.34 tff(189,plain,
% 1.73/1.34 ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9))))))))))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[188, 187])).
% 1.73/1.34 tff(190,plain,
% 1.73/1.34 (~((~((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))))))))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[189, 14, 186])).
% 1.73/1.34 tff(191,plain,
% 1.73/1.34 (((~((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9))))))))) | ((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))),
% 1.73/1.34 inference(tautology,[status(thm)],[])).
% 1.73/1.34 tff(192,plain,
% 1.73/1.34 ((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[191, 190])).
% 1.73/1.34 tff(193,plain,
% 1.73/1.34 ((~((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))),
% 1.73/1.34 inference(tautology,[status(thm)],[])).
% 1.73/1.34 tff(194,plain,
% 1.73/1.34 ((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[193, 192])).
% 1.73/1.34 tff(195,plain,
% 1.73/1.34 (![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[194, 163])).
% 1.73/1.34 tff(196,plain,
% 1.73/1.34 (((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))) | (closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))))) <=> ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))))),
% 1.73/1.34 inference(rewrite,[status(thm)],[])).
% 1.73/1.34 tff(197,plain,
% 1.73/1.34 (((~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)) <=> (closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))))),
% 1.73/1.34 inference(rewrite,[status(thm)],[])).
% 1.73/1.34 tff(198,plain,
% 1.73/1.34 (((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))) | ((~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))) <=> ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))) | (closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))))),
% 1.73/1.34 inference(monotonicity,[status(thm)],[197])).
% 1.73/1.34 tff(199,plain,
% 1.73/1.34 (((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))) | ((~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))) <=> ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))))),
% 1.73/1.34 inference(transitivity,[status(thm)],[198, 196])).
% 1.73/1.34 tff(200,plain,
% 1.73/1.34 ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))) | ((~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))),
% 1.73/1.34 inference(quant_inst,[status(thm)],[])).
% 1.73/1.34 tff(201,plain,
% 1.73/1.34 ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[200, 199])).
% 1.73/1.34 tff(202,plain,
% 1.73/1.34 (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[201, 195, 157, 131])).
% 1.73/1.34 tff(203,plain,
% 1.73/1.34 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))))),
% 1.73/1.34 inference(bind,[status(th)],[])).
% 1.73/1.34 tff(204,plain,
% 1.73/1.34 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.73/1.34 inference(quant_intro,[status(thm)],[203])).
% 1.73/1.34 tff(205,plain,
% 1.73/1.34 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.73/1.34 inference(rewrite,[status(thm)],[])).
% 1.73/1.34 tff(206,plain,
% 1.73/1.34 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => element(subset_complement(A, B), powerset(A))) <=> ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))))),
% 1.73/1.34 inference(bind,[status(th)],[])).
% 1.73/1.34 tff(207,plain,
% 1.73/1.34 (![A: $i, B: $i] : (element(B, powerset(A)) => element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.73/1.34 inference(quant_intro,[status(thm)],[206])).
% 1.73/1.34 tff(208,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => element(subset_complement(A, B), powerset(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k3_subset_1')).
% 1.73/1.34 tff(209,plain,
% 1.73/1.34 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[208, 207])).
% 1.73/1.34 tff(210,plain,
% 1.73/1.34 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[209, 205])).
% 1.73/1.34 tff(211,plain,(
% 1.73/1.34 ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.73/1.34 inference(skolemize,[status(sab)],[210])).
% 1.73/1.34 tff(212,plain,
% 1.73/1.34 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[211, 204])).
% 1.73/1.34 tff(213,plain,
% 1.73/1.34 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))),
% 1.73/1.34 inference(rewrite,[status(thm)],[])).
% 1.73/1.34 tff(214,plain,
% 1.73/1.34 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))),
% 1.73/1.34 inference(quant_inst,[status(thm)],[])).
% 1.73/1.34 tff(215,plain,
% 1.73/1.34 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[214, 213])).
% 1.73/1.34 tff(216,plain,
% 1.73/1.34 ($false),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[215, 212, 111, 202])).
% 1.73/1.34 tff(217,plain,(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)), inference(lemma,lemma(discharge,[]))).
% 1.73/1.34 tff(218,plain,
% 1.73/1.34 ((~closed_subsets(B!10, A!9)) | (~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | (~((~closed_subsets(B!10, A!9)) <=> open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)))),
% 1.73/1.34 inference(tautology,[status(thm)],[])).
% 1.73/1.34 tff(219,plain,
% 1.73/1.34 ((~closed_subsets(B!10, A!9)) | (~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[218, 160])).
% 1.73/1.34 tff(220,plain,
% 1.73/1.34 (~closed_subsets(B!10, A!9)),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[219, 217])).
% 1.73/1.34 tff(221,plain,
% 1.73/1.34 (((~((~closed_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9))))))))) | (closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))))))),
% 1.73/1.34 inference(tautology,[status(thm)],[])).
% 1.73/1.34 tff(222,plain,
% 1.73/1.34 (closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9))))))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[221, 190])).
% 1.73/1.34 tff(223,plain,
% 1.73/1.34 ((~(closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))))))) | closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9))))))),
% 1.73/1.34 inference(tautology,[status(thm)],[])).
% 1.73/1.34 tff(224,plain,
% 1.73/1.34 (closed_subsets(B!10, A!9) | (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9))))))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[223, 222])).
% 1.73/1.34 tff(225,plain,
% 1.73/1.34 (~((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[224, 220])).
% 1.73/1.34 tff(226,plain,
% 1.73/1.34 (((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9))))) | element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))),
% 1.73/1.34 inference(tautology,[status(thm)],[])).
% 1.73/1.34 tff(227,plain,
% 1.73/1.34 (element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[226, 225])).
% 1.73/1.34 tff(228,plain,
% 1.73/1.34 (((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_1(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))) <=> ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_1(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.73/1.34 inference(rewrite,[status(thm)],[])).
% 1.73/1.34 tff(229,plain,
% 1.73/1.34 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_1(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.73/1.34 inference(quant_inst,[status(thm)],[])).
% 1.73/1.34 tff(230,plain,
% 1.73/1.34 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_1(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10)))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[229, 228])).
% 1.73/1.34 tff(231,plain,
% 1.73/1.34 (in(tptp_fun_C_1(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[230, 227, 72])).
% 1.73/1.34 tff(232,plain,
% 1.73/1.34 (((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9))))) | in(tptp_fun_C_1(B!10, A!9), B!10)),
% 1.73/1.34 inference(tautology,[status(thm)],[])).
% 1.73/1.34 tff(233,plain,
% 1.73/1.34 (in(tptp_fun_C_1(B!10, A!9), B!10)),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[232, 225])).
% 1.73/1.34 tff(234,plain,
% 1.73/1.34 ((~(in(tptp_fun_C_1(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10)))) | (~in(tptp_fun_C_1(B!10, A!9), B!10)) | in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.73/1.34 inference(tautology,[status(thm)],[])).
% 1.73/1.34 tff(235,plain,
% 1.73/1.34 ((~(in(tptp_fun_C_1(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10)))) | in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[234, 233])).
% 1.73/1.34 tff(236,plain,
% 1.73/1.34 (in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[235, 231])).
% 1.73/1.34 tff(237,plain,
% 1.73/1.34 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))))),
% 1.73/1.34 inference(bind,[status(th)],[])).
% 1.73/1.34 tff(238,plain,
% 1.73/1.34 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.34 inference(quant_intro,[status(thm)],[237])).
% 1.73/1.34 tff(239,plain,
% 1.73/1.34 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))))),
% 1.73/1.34 inference(bind,[status(th)],[])).
% 1.73/1.34 tff(240,plain,
% 1.73/1.34 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.34 inference(quant_intro,[status(thm)],[239])).
% 1.73/1.34 tff(241,plain,
% 1.73/1.34 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.34 inference(transitivity,[status(thm)],[240, 238])).
% 1.73/1.34 tff(242,plain,
% 1.73/1.34 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.34 inference(rewrite,[status(thm)],[])).
% 1.73/1.34 tff(243,plain,
% 1.73/1.34 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))) <=> ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))))),
% 1.73/1.34 inference(bind,[status(th)],[])).
% 1.73/1.34 tff(244,plain,
% 1.73/1.34 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.34 inference(quant_intro,[status(thm)],[243])).
% 1.73/1.34 tff(245,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t29_tops_1')).
% 1.73/1.34 tff(246,plain,
% 1.73/1.34 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[245, 244])).
% 1.73/1.34 tff(247,plain,
% 1.73/1.34 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[246, 242])).
% 1.73/1.34 tff(248,plain,(
% 1.73/1.34 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.34 inference(skolemize,[status(sab)],[247])).
% 1.73/1.34 tff(249,plain,
% 1.73/1.34 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[248, 241])).
% 1.73/1.34 tff(250,plain,
% 1.73/1.34 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9))))),
% 1.73/1.34 inference(rewrite,[status(thm)],[])).
% 1.73/1.34 tff(251,plain,
% 1.73/1.34 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9))))),
% 1.73/1.34 inference(quant_inst,[status(thm)],[])).
% 1.73/1.34 tff(252,plain,
% 1.73/1.34 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))),
% 1.73/1.34 inference(modus_ponens,[status(thm)],[251, 250])).
% 1.73/1.34 tff(253,plain,
% 1.73/1.34 (![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))),
% 1.73/1.34 inference(unit_resolution,[status(thm)],[252, 74, 249])).
% 1.73/1.34 tff(254,plain,
% 1.73/1.34 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))) | ((~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | (closed_subset(tptp_fun_C_1(B!10, A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9)))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | (closed_subset(tptp_fun_C_1(B!10, A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9)))),
% 1.73/1.34 inference(rewrite,[status(thm)],[])).
% 1.73/1.34 tff(255,plain,
% 1.73/1.34 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))) | ((~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | (closed_subset(tptp_fun_C_1(B!10, A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9)))),
% 1.73/1.35 inference(quant_inst,[status(thm)],[])).
% 1.73/1.35 tff(256,plain,
% 1.73/1.35 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | (closed_subset(tptp_fun_C_1(B!10, A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9))),
% 1.73/1.35 inference(modus_ponens,[status(thm)],[255, 254])).
% 1.73/1.35 tff(257,plain,
% 1.73/1.35 (closed_subset(tptp_fun_C_1(B!10, A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9)),
% 1.73/1.35 inference(unit_resolution,[status(thm)],[256, 253, 227])).
% 1.73/1.35 tff(258,plain,
% 1.73/1.35 (((~in(tptp_fun_C_1(B!10, A!9), B!10)) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9))))) | (~closed_subset(tptp_fun_C_1(B!10, A!9), A!9))),
% 1.73/1.35 inference(tautology,[status(thm)],[])).
% 1.73/1.35 tff(259,plain,
% 1.73/1.35 (~closed_subset(tptp_fun_C_1(B!10, A!9), A!9)),
% 1.73/1.35 inference(unit_resolution,[status(thm)],[258, 225])).
% 1.73/1.35 tff(260,plain,
% 1.73/1.35 ((~(closed_subset(tptp_fun_C_1(B!10, A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9))) | closed_subset(tptp_fun_C_1(B!10, A!9), A!9) | (~open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9))),
% 1.73/1.35 inference(tautology,[status(thm)],[])).
% 1.73/1.35 tff(261,plain,
% 1.73/1.35 ((~(closed_subset(tptp_fun_C_1(B!10, A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9))) | (~open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9))),
% 1.73/1.35 inference(unit_resolution,[status(thm)],[260, 259])).
% 1.73/1.35 tff(262,plain,
% 1.73/1.35 (~open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9)),
% 1.73/1.35 inference(unit_resolution,[status(thm)],[261, 257])).
% 1.73/1.35 tff(263,plain,
% 1.73/1.35 (((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | open_subset(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_0(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))))) | ((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))),
% 1.73/1.35 inference(tautology,[status(thm)],[])).
% 1.73/1.35 tff(264,plain,
% 1.73/1.35 ((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.73/1.35 inference(unit_resolution,[status(thm)],[263, 104])).
% 1.73/1.35 tff(265,plain,
% 1.73/1.35 ((~((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.73/1.35 inference(tautology,[status(thm)],[])).
% 1.73/1.35 tff(266,plain,
% 1.73/1.35 ((~open_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.73/1.35 inference(unit_resolution,[status(thm)],[265, 264])).
% 1.73/1.35 tff(267,plain,
% 1.73/1.35 (![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.73/1.35 inference(unit_resolution,[status(thm)],[266, 217])).
% 1.73/1.35 tff(268,plain,
% 1.73/1.35 (((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | (open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))))) <=> ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))))),
% 1.73/1.35 inference(rewrite,[status(thm)],[])).
% 1.73/1.35 tff(269,plain,
% 1.73/1.35 (((~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10)))) <=> (open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))))),
% 1.73/1.35 inference(rewrite,[status(thm)],[])).
% 1.73/1.35 tff(270,plain,
% 1.73/1.35 (((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))) <=> ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | (open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9))))))),
% 1.73/1.35 inference(monotonicity,[status(thm)],[269])).
% 1.73/1.35 tff(271,plain,
% 1.73/1.35 (((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))) <=> ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))))),
% 1.73/1.35 inference(transitivity,[status(thm)],[270, 268])).
% 1.73/1.35 tff(272,plain,
% 1.73/1.35 ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.73/1.35 inference(quant_inst,[status(thm)],[])).
% 1.73/1.35 tff(273,plain,
% 1.73/1.35 ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9))))),
% 1.73/1.36 inference(modus_ponens,[status(thm)],[272, 271])).
% 1.73/1.36 tff(274,plain,
% 1.73/1.36 (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))),
% 1.73/1.36 inference(unit_resolution,[status(thm)],[273, 267, 262, 236])).
% 1.73/1.36 tff(275,plain,
% 1.73/1.36 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9))))),
% 1.73/1.36 inference(rewrite,[status(thm)],[])).
% 1.73/1.36 tff(276,plain,
% 1.73/1.36 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9))))),
% 1.73/1.36 inference(quant_inst,[status(thm)],[])).
% 1.73/1.36 tff(277,plain,
% 1.73/1.36 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(tptp_fun_C_1(B!10, A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_1(B!10, A!9)), powerset(the_carrier(A!9)))),
% 1.73/1.36 inference(modus_ponens,[status(thm)],[276, 275])).
% 1.73/1.36 tff(278,plain,
% 1.73/1.36 ($false),
% 1.73/1.36 inference(unit_resolution,[status(thm)],[277, 212, 227, 274])).
% 1.73/1.36 % SZS output end Proof
%------------------------------------------------------------------------------