TSTP Solution File: SEU318+1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SEU318+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n015.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 07:28:52 EDT 2022
% Result : Theorem 2.11s 1.56s
% Output : Proof 2.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SEU318+1 : TPTP v8.1.0. Released v3.3.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n015.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sat Sep 3 12:12:55 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 2.11/1.56 % SZS status Theorem
% 2.11/1.56 % SZS output start Proof
% 2.11/1.56 tff(subset_type, type, (
% 2.11/1.56 subset: ( $i * $i ) > $o)).
% 2.11/1.56 tff(tptp_fun_B_12_type, type, (
% 2.11/1.56 tptp_fun_B_12: $i)).
% 2.11/1.56 tff(in_type, type, (
% 2.11/1.56 in: ( $i * $i ) > $o)).
% 2.11/1.56 tff(tptp_fun_C_0_type, type, (
% 2.11/1.56 tptp_fun_C_0: ( $i * $i ) > $i)).
% 2.11/1.56 tff(topstr_closure_type, type, (
% 2.11/1.56 topstr_closure: ( $i * $i ) > $i)).
% 2.11/1.56 tff(tptp_fun_A_11_type, type, (
% 2.11/1.56 tptp_fun_A_11: $i)).
% 2.11/1.56 tff(closed_subset_type, type, (
% 2.11/1.56 closed_subset: ( $i * $i ) > $o)).
% 2.11/1.56 tff(element_type, type, (
% 2.11/1.56 element: ( $i * $i ) > $o)).
% 2.11/1.56 tff(powerset_type, type, (
% 2.11/1.56 powerset: $i > $i)).
% 2.11/1.56 tff(the_carrier_type, type, (
% 2.11/1.56 the_carrier: $i > $i)).
% 2.11/1.56 tff(tptp_fun_D_9_type, type, (
% 2.11/1.56 tptp_fun_D_9: ( $i * $i * $i ) > $i)).
% 2.11/1.56 tff(topological_space_type, type, (
% 2.11/1.56 topological_space: $i > $o)).
% 2.11/1.56 tff(tptp_fun_C_8_type, type, (
% 2.11/1.56 tptp_fun_C_8: ( $i * $i ) > $i)).
% 2.11/1.56 tff(tptp_fun_C_10_type, type, (
% 2.11/1.56 tptp_fun_C_10: ( $i * $i ) > $i)).
% 2.11/1.56 tff(meet_of_subsets_type, type, (
% 2.11/1.56 meet_of_subsets: ( $i * $i ) > $i)).
% 2.11/1.56 tff(top_str_type, type, (
% 2.11/1.56 top_str: $i > $o)).
% 2.11/1.56 tff(1,plain,
% 2.11/1.56 (^[A: $i, B: $i] : refl((~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))))),
% 2.11/1.56 inference(bind,[status(th)],[])).
% 2.11/1.56 tff(2,plain,
% 2.11/1.56 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 2.11/1.56 inference(quant_intro,[status(thm)],[1])).
% 2.11/1.56 tff(3,plain,
% 2.11/1.56 (^[A: $i, B: $i] : rewrite((~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))))),
% 2.11/1.56 inference(bind,[status(th)],[])).
% 2.11/1.56 tff(4,plain,
% 2.11/1.56 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 2.11/1.56 inference(quant_intro,[status(thm)],[3])).
% 2.11/1.56 tff(5,plain,
% 2.11/1.56 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 2.11/1.56 inference(transitivity,[status(thm)],[4, 2])).
% 2.11/1.56 tff(6,plain,
% 2.11/1.56 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) <=> ((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))), rewrite((subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))) <=> (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))), ((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))) <=> (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))), rewrite((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))), ((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))))),
% 2.11/1.56 inference(bind,[status(th)],[])).
% 2.11/1.56 tff(7,plain,
% 2.11/1.56 (![A: $i, B: $i] : (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 2.11/1.56 inference(quant_intro,[status(thm)],[6])).
% 2.11/1.56 tff(8,plain,
% 2.11/1.56 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 2.11/1.56 inference(rewrite,[status(thm)],[])).
% 2.11/1.56 tff(9,plain,
% 2.11/1.56 (^[A: $i, B: $i] : rewrite((subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B))) <=> (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B))))),
% 2.11/1.56 inference(bind,[status(th)],[])).
% 2.11/1.56 tff(10,plain,
% 2.11/1.56 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 2.11/1.56 inference(quant_intro,[status(thm)],[9])).
% 2.11/1.56 tff(11,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d3_tarski')).
% 2.11/1.56 tff(12,plain,
% 2.11/1.56 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[11, 10])).
% 2.11/1.56 tff(13,plain,
% 2.11/1.56 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[12, 8])).
% 2.11/1.56 tff(14,plain,(
% 2.11/1.56 ![A: $i, B: $i] : (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))),
% 2.11/1.56 inference(skolemize,[status(sab)],[13])).
% 2.11/1.56 tff(15,plain,
% 2.11/1.56 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[14, 7])).
% 2.11/1.56 tff(16,plain,
% 2.11/1.56 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[15, 5])).
% 2.11/1.56 tff(17,plain,
% 2.11/1.56 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))) | (~((~((~subset(topstr_closure(A!11, B!12), B!12)) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, B!12)))) | (~(subset(topstr_closure(A!11, B!12), B!12) | (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12)))))))),
% 2.11/1.56 inference(quant_inst,[status(thm)],[])).
% 2.11/1.56 tff(18,plain,
% 2.11/1.56 (~((~((~subset(topstr_closure(A!11, B!12), B!12)) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, B!12)))) | (~(subset(topstr_closure(A!11, B!12), B!12) | (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12))))))),
% 2.11/1.56 inference(unit_resolution,[status(thm)],[17, 16])).
% 2.11/1.56 tff(19,plain,
% 2.11/1.56 (((~((~subset(topstr_closure(A!11, B!12), B!12)) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, B!12)))) | (~(subset(topstr_closure(A!11, B!12), B!12) | (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12)))))) | (subset(topstr_closure(A!11, B!12), B!12) | (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12))))),
% 2.11/1.56 inference(tautology,[status(thm)],[])).
% 2.11/1.56 tff(20,plain,
% 2.11/1.56 (subset(topstr_closure(A!11, B!12), B!12) | (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12)))),
% 2.11/1.56 inference(unit_resolution,[status(thm)],[19, 18])).
% 2.11/1.56 tff(21,assumption,(~(closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))), introduced(assumption)).
% 2.11/1.56 tff(22,plain,
% 2.11/1.56 ((closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))) | (topstr_closure(A!11, B!12) = B!12)),
% 2.11/1.56 inference(tautology,[status(thm)],[])).
% 2.11/1.56 tff(23,plain,
% 2.11/1.56 (topstr_closure(A!11, B!12) = B!12),
% 2.11/1.56 inference(unit_resolution,[status(thm)],[22, 21])).
% 2.11/1.56 tff(24,plain,
% 2.11/1.56 ((closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))) | topological_space(A!11)),
% 2.11/1.56 inference(tautology,[status(thm)],[])).
% 2.11/1.56 tff(25,plain,
% 2.11/1.56 (topological_space(A!11)),
% 2.11/1.56 inference(unit_resolution,[status(thm)],[24, 21])).
% 2.11/1.56 tff(26,plain,
% 2.11/1.56 (((~(~top_str(A!11))) & (~((~element(B!12, powerset(the_carrier(A!11)))) | (((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11)))))) <=> (top_str(A!11) & (~((~element(B!12, powerset(the_carrier(A!11)))) | (((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11))))))),
% 2.11/1.56 inference(rewrite,[status(thm)],[])).
% 2.11/1.56 tff(27,plain,
% 2.11/1.56 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A)))))) <=> (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A))))))),
% 2.11/1.56 inference(rewrite,[status(thm)],[])).
% 2.11/1.56 tff(28,plain,
% 2.11/1.56 ((~![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ((closed_subset(B, A) => (topstr_closure(A, B) = B)) & ((topological_space(A) & (topstr_closure(A, B) = B)) => closed_subset(B, A)))))) <=> (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A))))))),
% 2.11/1.56 inference(rewrite,[status(thm)],[])).
% 2.11/1.56 tff(29,axiom,(~![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ((closed_subset(B, A) => (topstr_closure(A, B) = B)) & ((topological_space(A) & (topstr_closure(A, B) = B)) => closed_subset(B, A)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t52_pre_topc')).
% 2.11/1.56 tff(30,plain,
% 2.11/1.56 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A)))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[29, 28])).
% 2.11/1.56 tff(31,plain,
% 2.11/1.56 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A)))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[30, 27])).
% 2.11/1.56 tff(32,plain,
% 2.11/1.56 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A)))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[31, 27])).
% 2.11/1.56 tff(33,plain,
% 2.11/1.56 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A)))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[32, 27])).
% 2.11/1.56 tff(34,plain,
% 2.11/1.56 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A)))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[33, 27])).
% 2.11/1.56 tff(35,plain,
% 2.11/1.56 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A)))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[34, 27])).
% 2.11/1.56 tff(36,plain,
% 2.11/1.56 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (((~closed_subset(B, A)) | (topstr_closure(A, B) = B)) & ((~(topological_space(A) & (topstr_closure(A, B) = B))) | closed_subset(B, A)))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[35, 27])).
% 2.11/1.56 tff(37,plain,
% 2.11/1.56 (top_str(A!11) & (~((~element(B!12, powerset(the_carrier(A!11)))) | (((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11)))))),
% 2.11/1.56 inference(modus_ponens,[status(thm)],[36, 26])).
% 2.11/1.56 tff(38,plain,
% 2.11/1.56 (top_str(A!11)),
% 2.11/1.56 inference(and_elim,[status(thm)],[37])).
% 2.11/1.56 tff(39,plain,
% 2.11/1.56 (^[A: $i] : rewrite(((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))) <=> ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))))))))),
% 2.11/1.56 inference(bind,[status(th)],[])).
% 2.11/1.56 tff(40,plain,
% 2.11/1.56 (![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))) <=> ![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D)))))))))))),
% 2.11/1.56 inference(quant_intro,[status(thm)],[39])).
% 2.11/1.56 tff(41,plain,
% 2.11/1.56 (^[A: $i] : refl(((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))) <=> ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))))),
% 2.11/1.56 inference(bind,[status(th)],[])).
% 2.11/1.56 tff(42,plain,
% 2.11/1.56 (![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))) <=> ![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))),
% 2.11/1.56 inference(quant_intro,[status(thm)],[41])).
% 2.11/1.56 tff(43,plain,
% 2.11/1.56 (^[A: $i] : rewrite(((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))) <=> ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))))),
% 2.11/1.56 inference(bind,[status(th)],[])).
% 2.11/1.56 tff(44,plain,
% 2.11/1.56 (![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))) <=> ![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))),
% 2.11/1.56 inference(quant_intro,[status(thm)],[43])).
% 2.11/1.56 tff(45,plain,
% 2.11/1.56 (![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))) <=> ![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))),
% 2.11/1.56 inference(transitivity,[status(thm)],[44, 42])).
% 2.11/1.56 tff(46,plain,
% 2.11/1.56 (^[A: $i] : trans(monotonicity(trans(monotonicity(rewrite((topological_space(A) & top_str(A)) <=> (~((~top_str(A)) | (~topological_space(A))))), ((~(topological_space(A) & top_str(A))) <=> (~(~((~top_str(A)) | (~topological_space(A))))))), rewrite((~(~((~top_str(A)) | (~topological_space(A))))) <=> ((~top_str(A)) | (~topological_space(A)))), ((~(topological_space(A) & top_str(A))) <=> ((~top_str(A)) | (~topological_space(A))))), quant_intro(proof_bind(^[B: $i] : rewrite(((~element(B, powerset(the_carrier(A)))) | (element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))) <=> ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))), (![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))), (((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))) <=> (((~top_str(A)) | (~topological_space(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))))), rewrite((((~top_str(A)) | (~topological_space(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))))) <=> ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))), (((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))) <=> ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))))),
% 2.11/1.56 inference(bind,[status(th)],[])).
% 2.11/1.56 tff(47,plain,
% 2.11/1.56 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))))) <=> ![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))),
% 2.11/1.56 inference(quant_intro,[status(thm)],[46])).
% 2.11/1.57 tff(48,plain,
% 2.11/1.57 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))) <=> ![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))),
% 2.11/1.57 inference(rewrite,[status(thm)],[])).
% 2.11/1.57 tff(49,plain,
% 2.11/1.57 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(rewrite((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) <=> (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))))), (((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))) <=> ((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))), rewrite(((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))) <=> (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))), (((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))) <=> (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))), (?[C: $i] : ((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))) <=> ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))), ((element(B, powerset(the_carrier(A))) => ?[C: $i] : ((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))) <=> (element(B, powerset(the_carrier(A))) => ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))), rewrite((element(B, powerset(the_carrier(A))) => ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))) <=> ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))), ((element(B, powerset(the_carrier(A))) => ?[C: $i] : ((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))) <=> ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => ?[C: $i] : ((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))), (((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ?[C: $i] : ((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))) <=> ((topological_space(A) & top_str(A)) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))))), rewrite(((topological_space(A) & top_str(A)) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))) <=> ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))), (((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ?[C: $i] : ((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))) <=> ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))))),
% 2.11/1.57 inference(bind,[status(th)],[])).
% 2.11/1.57 tff(50,plain,
% 2.11/1.57 (![A: $i] : ((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ?[C: $i] : ((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C))))) <=> ![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))),
% 2.11/1.57 inference(quant_intro,[status(thm)],[49])).
% 2.11/1.57 tff(51,axiom,(![A: $i] : ((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ?[C: $i] : ((element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : (element(D, powerset(the_carrier(A))) => (in(D, C) <=> (closed_subset(D, A) & subset(B, D))))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t46_pre_topc')).
% 2.11/1.57 tff(52,plain,
% 2.11/1.57 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))),
% 2.11/1.57 inference(modus_ponens,[status(thm)],[51, 50])).
% 2.11/1.57 tff(53,plain,
% 2.11/1.57 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ?[C: $i] : (element(C, powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, C) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), C)))))),
% 2.11/1.57 inference(modus_ponens,[status(thm)],[52, 48])).
% 2.11/1.57 tff(54,plain,(
% 2.11/1.57 ![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A)))) & ![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (closed_subset(D, A) & subset(B, D)))) & (topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))),
% 2.11/1.57 inference(skolemize,[status(sab)],[53])).
% 2.11/1.57 tff(55,plain,
% 2.11/1.57 (![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))),
% 2.11/1.57 inference(modus_ponens,[status(thm)],[54, 47])).
% 2.11/1.57 tff(56,plain,
% 2.11/1.57 (![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D))))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A))))))))),
% 2.11/1.57 inference(modus_ponens,[status(thm)],[55, 45])).
% 2.11/1.57 tff(57,plain,
% 2.11/1.57 (![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D)))))))))))),
% 2.11/1.57 inference(modus_ponens,[status(thm)],[56, 40])).
% 2.11/1.57 tff(58,plain,
% 2.11/1.57 (((~![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D)))))))))))) | ((~top_str(A!11)) | (~topological_space(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D)))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D)))))))))))) | (~top_str(A!11)) | (~topological_space(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D)))))))))))),
% 2.11/1.57 inference(rewrite,[status(thm)],[])).
% 2.11/1.57 tff(59,plain,
% 2.11/1.57 ((~![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D)))))))))))) | ((~top_str(A!11)) | (~topological_space(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D)))))))))))),
% 2.11/1.57 inference(quant_inst,[status(thm)],[])).
% 2.11/1.57 tff(60,plain,
% 2.11/1.57 ((~![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (~((~element(tptp_fun_C_10(B, A), powerset(powerset(the_carrier(A))))) | (~(topstr_closure(A, B) = meet_of_subsets(the_carrier(A), tptp_fun_C_10(B, A)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A)))) | (in(D, tptp_fun_C_10(B, A)) <=> (~((~closed_subset(D, A)) | (~subset(B, D)))))))))))) | (~top_str(A!11)) | (~topological_space(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D))))))))))),
% 2.11/1.57 inference(modus_ponens,[status(thm)],[59, 58])).
% 2.11/1.57 tff(61,plain,
% 2.11/1.57 ((~topological_space(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D))))))))))),
% 2.11/1.57 inference(unit_resolution,[status(thm)],[60, 57, 38])).
% 2.11/1.57 tff(62,plain,
% 2.11/1.57 (![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D))))))))))),
% 2.11/1.57 inference(unit_resolution,[status(thm)],[61, 25])).
% 2.11/1.57 tff(63,plain,
% 2.11/1.57 (~((~element(B!12, powerset(the_carrier(A!11)))) | (((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11))))),
% 2.11/1.57 inference(and_elim,[status(thm)],[37])).
% 2.11/1.57 tff(64,plain,
% 2.11/1.57 (element(B!12, powerset(the_carrier(A!11)))),
% 2.11/1.57 inference(or_elim,[status(thm)],[63])).
% 2.11/1.57 tff(65,plain,
% 2.11/1.57 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D))))))))))) | ((~element(B!12, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D))))))))))) | (~element(B!12, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))))))),
% 2.11/1.57 inference(rewrite,[status(thm)],[])).
% 2.11/1.57 tff(66,plain,
% 2.11/1.57 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D))))))))))) | ((~element(B!12, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))))))),
% 2.11/1.57 inference(quant_inst,[status(thm)],[])).
% 2.11/1.57 tff(67,plain,
% 2.11/1.57 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B, D))))))))))) | (~element(B!12, powerset(the_carrier(A!11)))) | (~((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D)))))))))),
% 2.11/1.57 inference(modus_ponens,[status(thm)],[66, 65])).
% 2.11/1.57 tff(68,plain,
% 2.11/1.57 (~((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))))),
% 2.11/1.57 inference(unit_resolution,[status(thm)],[67, 64, 62])).
% 2.11/1.57 tff(69,plain,
% 2.11/1.57 (((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D)))))))) | (topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)))),
% 2.11/1.57 inference(tautology,[status(thm)],[])).
% 2.11/1.57 tff(70,plain,
% 2.11/1.57 (topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11))),
% 2.11/1.57 inference(unit_resolution,[status(thm)],[69, 68])).
% 2.11/1.57 tff(71,plain,
% 2.11/1.58 (meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)) = topstr_closure(A!11, B!12)),
% 2.11/1.58 inference(symmetry,[status(thm)],[70])).
% 2.11/1.58 tff(72,plain,
% 2.11/1.58 (meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)) = B!12),
% 2.11/1.58 inference(transitivity,[status(thm)],[71, 23])).
% 2.11/1.58 tff(73,plain,
% 2.11/1.58 (closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) <=> closed_subset(B!12, A!11)),
% 2.11/1.58 inference(monotonicity,[status(thm)],[72])).
% 2.11/1.58 tff(74,plain,
% 2.11/1.58 (closed_subset(B!12, A!11) <=> closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11)),
% 2.11/1.58 inference(symmetry,[status(thm)],[73])).
% 2.11/1.58 tff(75,plain,
% 2.11/1.58 ((~closed_subset(B!12, A!11)) <=> (~closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11))),
% 2.11/1.58 inference(monotonicity,[status(thm)],[74])).
% 2.11/1.58 tff(76,plain,
% 2.11/1.58 ((closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))) | (~closed_subset(B!12, A!11))),
% 2.11/1.58 inference(tautology,[status(thm)],[])).
% 2.11/1.58 tff(77,plain,
% 2.11/1.58 (~closed_subset(B!12, A!11)),
% 2.11/1.58 inference(unit_resolution,[status(thm)],[76, 21])).
% 2.11/1.58 tff(78,plain,
% 2.11/1.58 (~closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11)),
% 2.11/1.58 inference(modus_ponens,[status(thm)],[77, 75])).
% 2.11/1.58 tff(79,plain,
% 2.11/1.58 (((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D)))))))) | element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))),
% 2.11/1.58 inference(tautology,[status(thm)],[])).
% 2.11/1.58 tff(80,plain,
% 2.11/1.58 (element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))),
% 2.11/1.58 inference(unit_resolution,[status(thm)],[79, 68])).
% 2.11/1.58 tff(81,plain,
% 2.11/1.58 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A))) <=> ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))))))),
% 2.11/1.58 inference(bind,[status(th)],[])).
% 2.11/1.58 tff(82,plain,
% 2.11/1.58 (![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A))) <=> ![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))))),
% 2.11/1.58 inference(quant_intro,[status(thm)],[81])).
% 2.11/1.58 tff(83,plain,
% 2.11/1.58 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A))) <=> ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A))))),
% 2.11/1.58 inference(bind,[status(th)],[])).
% 2.11/1.58 tff(84,plain,
% 2.11/1.58 (![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A)))),
% 2.11/1.58 inference(quant_intro,[status(thm)],[83])).
% 2.11/1.58 tff(85,plain,
% 2.11/1.58 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A))) <=> ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A))))),
% 2.11/1.58 inference(bind,[status(th)],[])).
% 2.11/1.58 tff(86,plain,
% 2.11/1.58 (![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A)))),
% 2.11/1.58 inference(quant_intro,[status(thm)],[85])).
% 2.11/1.58 tff(87,plain,
% 2.11/1.58 (![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A)))),
% 2.11/1.58 inference(transitivity,[status(thm)],[86, 84])).
% 2.11/1.58 tff(88,plain,
% 2.11/1.58 (^[A: $i] : trans(monotonicity(trans(monotonicity(rewrite((topological_space(A) & top_str(A)) <=> (~((~top_str(A)) | (~topological_space(A))))), ((~(topological_space(A) & top_str(A))) <=> (~(~((~top_str(A)) | (~topological_space(A))))))), rewrite((~(~((~top_str(A)) | (~topological_space(A))))) <=> ((~top_str(A)) | (~topological_space(A)))), ((~(topological_space(A) & top_str(A))) <=> ((~top_str(A)) | (~topological_space(A))))), (((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))))) <=> (((~top_str(A)) | (~topological_space(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))))))), rewrite((((~top_str(A)) | (~topological_space(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))))) <=> ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A)))), (((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))))) <=> ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A)))))),
% 2.11/1.58 inference(bind,[status(th)],[])).
% 2.11/1.58 tff(89,plain,
% 2.11/1.58 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A)))),
% 2.11/1.58 inference(quant_intro,[status(thm)],[88])).
% 2.11/1.58 tff(90,plain,
% 2.11/1.58 (^[A: $i] : rewrite(((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A))))))) <=> ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))))))),
% 2.11/1.58 inference(bind,[status(th)],[])).
% 2.11/1.58 tff(91,plain,
% 2.11/1.58 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A))))))) <=> ![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))))),
% 2.11/1.58 inference(quant_intro,[status(thm)],[90])).
% 2.11/1.58 tff(92,plain,
% 2.11/1.58 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A))))))) <=> ![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A)))))))),
% 2.11/1.58 inference(rewrite,[status(thm)],[])).
% 2.11/1.58 tff(93,plain,
% 2.11/1.58 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(rewrite((in(C, B) => closed_subset(C, A)) <=> ((~in(C, B)) | closed_subset(C, A))), ((element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) <=> (element(C, powerset(the_carrier(A))) => ((~in(C, B)) | closed_subset(C, A))))), rewrite((element(C, powerset(the_carrier(A))) => ((~in(C, B)) | closed_subset(C, A))) <=> ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))), ((element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) <=> ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))), (![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))), ((![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) => closed_subset(meet_of_subsets(the_carrier(A), B), A)) <=> (![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))) => closed_subset(meet_of_subsets(the_carrier(A), B), A)))), rewrite((![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))) => closed_subset(meet_of_subsets(the_carrier(A), B), A)) <=> ((~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | closed_subset(meet_of_subsets(the_carrier(A), B), A))), ((![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) => closed_subset(meet_of_subsets(the_carrier(A), B), A)) <=> ((~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | closed_subset(meet_of_subsets(the_carrier(A), B), A)))), ((element(B, powerset(powerset(the_carrier(A)))) => (![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) => closed_subset(meet_of_subsets(the_carrier(A), B), A))) <=> (element(B, powerset(powerset(the_carrier(A)))) => ((~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | closed_subset(meet_of_subsets(the_carrier(A), B), A))))), rewrite((element(B, powerset(powerset(the_carrier(A)))) => ((~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | closed_subset(meet_of_subsets(the_carrier(A), B), A))) <=> (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A))))))), ((element(B, powerset(powerset(the_carrier(A)))) => (![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) => closed_subset(meet_of_subsets(the_carrier(A), B), A))) <=> (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A))))))))), (![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) => closed_subset(meet_of_subsets(the_carrier(A), B), A))) <=> ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A)))))))), (((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) => closed_subset(meet_of_subsets(the_carrier(A), B), A)))) <=> ((topological_space(A) & top_str(A)) => ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A))))))))), rewrite(((topological_space(A) & top_str(A)) => ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A))))))) <=> ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A)))))))), (((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) => closed_subset(meet_of_subsets(the_carrier(A), B), A)))) <=> ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A)))))))))),
% 2.11/1.58 inference(bind,[status(th)],[])).
% 2.11/1.58 tff(94,plain,
% 2.11/1.58 (![A: $i] : ((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) => closed_subset(meet_of_subsets(the_carrier(A), B), A)))) <=> ![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A)))))))),
% 2.11/1.58 inference(quant_intro,[status(thm)],[93])).
% 2.11/1.58 tff(95,axiom,(![A: $i] : ((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))) => closed_subset(meet_of_subsets(the_carrier(A), B), A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t44_pre_topc')).
% 2.11/1.58 tff(96,plain,
% 2.11/1.58 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A)))))))),
% 2.11/1.58 inference(modus_ponens,[status(thm)],[95, 94])).
% 2.11/1.58 tff(97,plain,
% 2.11/1.58 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A)))))))),
% 2.11/1.58 inference(modus_ponens,[status(thm)],[96, 92])).
% 2.11/1.58 tff(98,plain,(
% 2.11/1.58 ![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A)))))) | (~element(B, powerset(powerset(the_carrier(A)))))))),
% 2.11/1.58 inference(skolemize,[status(sab)],[97])).
% 2.11/1.58 tff(99,plain,
% 2.11/1.58 (![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))))),
% 2.11/1.58 inference(modus_ponens,[status(thm)],[98, 91])).
% 2.11/1.58 tff(100,plain,
% 2.11/1.58 (![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A)))),
% 2.11/1.58 inference(modus_ponens,[status(thm)],[99, 89])).
% 2.11/1.58 tff(101,plain,
% 2.11/1.58 (![A: $i] : ((~top_str(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))) | (~topological_space(A)))),
% 2.11/1.58 inference(modus_ponens,[status(thm)],[100, 87])).
% 2.11/1.58 tff(102,plain,
% 2.11/1.58 (![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))))),
% 2.11/1.58 inference(modus_ponens,[status(thm)],[101, 82])).
% 2.11/1.58 tff(103,plain,
% 2.11/1.58 (((~![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))))) | ((~top_str(A!11)) | (~topological_space(A!11)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11))))))))) <=> ((~![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))))) | (~top_str(A!11)) | (~topological_space(A!11)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11))))))))),
% 2.11/1.58 inference(rewrite,[status(thm)],[])).
% 2.11/1.58 tff(104,plain,
% 2.11/1.58 ((~![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))))) | ((~top_str(A!11)) | (~topological_space(A!11)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11))))))))),
% 2.11/1.58 inference(quant_inst,[status(thm)],[])).
% 2.11/1.58 tff(105,plain,
% 2.11/1.58 ((~![A: $i] : ((~top_str(A)) | (~topological_space(A)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A), B), A) | (~element(B, powerset(powerset(the_carrier(A))))) | (~((~in(tptp_fun_C_8(B, A), B)) | closed_subset(tptp_fun_C_8(B, A), A) | (~element(tptp_fun_C_8(B, A), powerset(the_carrier(A))))))))) | (~top_str(A!11)) | (~topological_space(A!11)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))),
% 2.11/1.58 inference(modus_ponens,[status(thm)],[104, 103])).
% 2.11/1.58 tff(106,plain,
% 2.11/1.58 ((~topological_space(A!11)) | ![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))),
% 2.11/1.58 inference(unit_resolution,[status(thm)],[105, 102, 38])).
% 2.11/1.58 tff(107,plain,
% 2.11/1.58 (![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))),
% 2.11/1.58 inference(unit_resolution,[status(thm)],[106, 25])).
% 2.11/1.58 tff(108,plain,
% 2.11/1.58 (((~![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))) | ((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))))) <=> ((~![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))))),
% 2.11/1.59 inference(rewrite,[status(thm)],[])).
% 2.11/1.59 tff(109,plain,
% 2.11/1.59 ((closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))))) <=> ((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))))),
% 2.11/1.59 inference(rewrite,[status(thm)],[])).
% 2.11/1.59 tff(110,plain,
% 2.11/1.59 ((~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))) <=> (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))))),
% 2.11/1.59 inference(rewrite,[status(thm)],[])).
% 2.11/1.59 tff(111,plain,
% 2.11/1.59 ((closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))))) <=> (closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))))),
% 2.11/1.59 inference(monotonicity,[status(thm)],[110])).
% 2.11/1.59 tff(112,plain,
% 2.11/1.59 ((closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))))) <=> ((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))))),
% 2.11/1.59 inference(transitivity,[status(thm)],[111, 109])).
% 2.11/1.59 tff(113,plain,
% 2.11/1.59 (((~![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))) | (closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))))) <=> ((~![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))) | ((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))))))),
% 2.11/1.59 inference(monotonicity,[status(thm)],[112])).
% 2.11/1.59 tff(114,plain,
% 2.11/1.59 (((~![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))) | (closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))))) <=> ((~![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))))),
% 2.11/1.59 inference(transitivity,[status(thm)],[113, 108])).
% 2.11/1.59 tff(115,plain,
% 2.11/1.59 ((~![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))) | (closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))))),
% 2.11/1.59 inference(quant_inst,[status(thm)],[])).
% 2.11/1.59 tff(116,plain,
% 2.11/1.59 ((~![B: $i] : (closed_subset(meet_of_subsets(the_carrier(A!11), B), A!11) | (~element(B, powerset(powerset(the_carrier(A!11))))) | (~((~in(tptp_fun_C_8(B, A!11), B)) | closed_subset(tptp_fun_C_8(B, A!11), A!11) | (~element(tptp_fun_C_8(B, A!11), powerset(the_carrier(A!11)))))))) | (~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))))),
% 2.11/1.59 inference(modus_ponens,[status(thm)],[115, 114])).
% 2.11/1.59 tff(117,plain,
% 2.11/1.59 (closed_subset(meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)), A!11) | (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))))),
% 2.11/1.59 inference(unit_resolution,[status(thm)],[116, 107, 80])).
% 2.11/1.59 tff(118,plain,
% 2.11/1.59 (~((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))))),
% 2.11/1.59 inference(unit_resolution,[status(thm)],[117, 78])).
% 2.11/1.59 tff(119,plain,
% 2.11/1.59 (((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))) | (~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11))),
% 2.11/1.59 inference(tautology,[status(thm)],[])).
% 2.11/1.59 tff(120,plain,
% 2.11/1.59 (~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)),
% 2.11/1.59 inference(unit_resolution,[status(thm)],[119, 118])).
% 2.11/1.59 tff(121,plain,
% 2.11/1.59 (((~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)) | (~subset(B!12, tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11)))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)),
% 2.11/1.59 inference(tautology,[status(thm)],[])).
% 2.11/1.59 tff(122,plain,
% 2.11/1.59 ((~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)) | (~subset(B!12, tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11)))),
% 2.11/1.59 inference(unit_resolution,[status(thm)],[121, 120])).
% 2.11/1.59 tff(123,plain,
% 2.11/1.59 (((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))) | in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))),
% 2.11/1.59 inference(tautology,[status(thm)],[])).
% 2.11/1.59 tff(124,plain,
% 2.11/1.59 (in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))),
% 2.11/1.59 inference(unit_resolution,[status(thm)],[123, 118])).
% 2.11/1.59 tff(125,plain,
% 2.11/1.59 ((~(in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)) | (~subset(B!12, tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11))))))) | (~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | (~((~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)) | (~subset(B!12, tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11)))))),
% 2.11/1.59 inference(tautology,[status(thm)],[])).
% 2.11/1.59 tff(126,plain,
% 2.11/1.59 (~(in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)) | (~subset(B!12, tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11))))))),
% 2.11/1.59 inference(unit_resolution,[status(thm)],[125, 124, 122])).
% 2.11/1.59 tff(127,plain,
% 2.11/1.59 (((~in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11))) | closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11))))) | element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))),
% 2.11/1.59 inference(tautology,[status(thm)],[])).
% 2.11/1.59 tff(128,plain,
% 2.11/1.59 (element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))),
% 2.11/1.59 inference(unit_resolution,[status(thm)],[127, 118])).
% 2.11/1.59 tff(129,plain,
% 2.11/1.59 (((~element(tptp_fun_C_10(B!12, A!11), powerset(powerset(the_carrier(A!11))))) | (~(topstr_closure(A!11, B!12) = meet_of_subsets(the_carrier(A!11), tptp_fun_C_10(B!12, A!11)))) | (~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D)))))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))),
% 2.11/1.60 inference(tautology,[status(thm)],[])).
% 2.11/1.60 tff(130,plain,
% 2.11/1.60 (![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))),
% 2.11/1.60 inference(unit_resolution,[status(thm)],[129, 68])).
% 2.11/1.60 tff(131,plain,
% 2.11/1.60 (((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))) | ((~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))) | (in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)) | (~subset(B!12, tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11)))))))) <=> ((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))) | (in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)) | (~subset(B!12, tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11)))))))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(132,plain,
% 2.11/1.60 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))) | ((~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))) | (in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)) | (~subset(B!12, tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11)))))))),
% 2.11/1.60 inference(quant_inst,[status(thm)],[])).
% 2.11/1.60 tff(133,plain,
% 2.11/1.60 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (in(D, tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(D, A!11)) | (~subset(B!12, D))))))) | (~element(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), powerset(the_carrier(A!11)))) | (in(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), tptp_fun_C_10(B!12, A!11)) <=> (~((~closed_subset(tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11), A!11)) | (~subset(B!12, tptp_fun_C_8(tptp_fun_C_10(B!12, A!11), A!11))))))),
% 2.11/1.60 inference(modus_ponens,[status(thm)],[132, 131])).
% 2.11/1.60 tff(134,plain,
% 2.11/1.60 ($false),
% 2.11/1.60 inference(unit_resolution,[status(thm)],[133, 130, 128, 126])).
% 2.11/1.60 tff(135,plain,(closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))), inference(lemma,lemma(discharge,[]))).
% 2.11/1.60 tff(136,plain,
% 2.11/1.60 ((~(~((~((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12))) | (~(closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))))))) <=> ((~((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12))) | (~(closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(137,plain,
% 2.11/1.60 ((((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & (closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))) <=> (~((~((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12))) | (~(closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))))))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(138,plain,
% 2.11/1.60 ((((~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11)) <=> (closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(139,plain,
% 2.11/1.60 ((~(~((~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))))) <=> ((~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(140,plain,
% 2.11/1.60 ((topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12)) <=> (~((~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(141,plain,
% 2.11/1.60 ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) <=> (~(~((~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))))),
% 2.11/1.60 inference(monotonicity,[status(thm)],[140])).
% 2.11/1.60 tff(142,plain,
% 2.11/1.60 ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) <=> ((~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))),
% 2.11/1.60 inference(transitivity,[status(thm)],[141, 139])).
% 2.11/1.60 tff(143,plain,
% 2.11/1.60 (((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11)) <=> (((~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11))),
% 2.11/1.60 inference(monotonicity,[status(thm)],[142])).
% 2.11/1.60 tff(144,plain,
% 2.11/1.60 (((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11)) <=> (closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))),
% 2.11/1.60 inference(transitivity,[status(thm)],[143, 138])).
% 2.11/1.60 tff(145,plain,
% 2.11/1.60 ((((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11))) <=> (((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & (closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))))),
% 2.11/1.60 inference(monotonicity,[status(thm)],[144])).
% 2.11/1.60 tff(146,plain,
% 2.11/1.60 ((((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11))) <=> (~((~((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12))) | (~(closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))))))),
% 2.11/1.60 inference(transitivity,[status(thm)],[145, 137])).
% 2.11/1.60 tff(147,plain,
% 2.11/1.60 ((~(((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11)))) <=> (~(~((~((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12))) | (~(closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))))))),
% 2.11/1.60 inference(monotonicity,[status(thm)],[146])).
% 2.11/1.60 tff(148,plain,
% 2.11/1.60 ((~(((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11)))) <=> ((~((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12))) | (~(closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12)))))),
% 2.11/1.60 inference(transitivity,[status(thm)],[147, 136])).
% 2.11/1.60 tff(149,plain,
% 2.11/1.60 (~(((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) & ((~(topological_space(A!11) & (topstr_closure(A!11, B!12) = B!12))) | closed_subset(B!12, A!11)))),
% 2.11/1.60 inference(or_elim,[status(thm)],[63])).
% 2.11/1.60 tff(150,plain,
% 2.11/1.60 ((~((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12))) | (~(closed_subset(B!12, A!11) | (~topological_space(A!11)) | (~(topstr_closure(A!11, B!12) = B!12))))),
% 2.11/1.60 inference(modus_ponens,[status(thm)],[149, 148])).
% 2.11/1.60 tff(151,plain,
% 2.11/1.60 (~((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12))),
% 2.11/1.60 inference(unit_resolution,[status(thm)],[150, 135])).
% 2.11/1.60 tff(152,plain,
% 2.11/1.60 (((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) | (~(topstr_closure(A!11, B!12) = B!12))),
% 2.11/1.60 inference(tautology,[status(thm)],[])).
% 2.11/1.60 tff(153,plain,
% 2.11/1.60 (~(topstr_closure(A!11, B!12) = B!12)),
% 2.11/1.60 inference(unit_resolution,[status(thm)],[152, 151])).
% 2.11/1.60 tff(154,plain,
% 2.11/1.60 (^[A: $i, B: $i] : refl(((A = B) <=> (~((~subset(A, B)) | (~subset(B, A))))) <=> ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A))))))),
% 2.11/1.60 inference(bind,[status(th)],[])).
% 2.11/1.60 tff(155,plain,
% 2.11/1.60 (![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A))))) <=> ![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))),
% 2.11/1.60 inference(quant_intro,[status(thm)],[154])).
% 2.11/1.60 tff(156,plain,
% 2.11/1.60 (^[A: $i, B: $i] : rewrite(((A = B) <=> (subset(A, B) & subset(B, A))) <=> ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A))))))),
% 2.11/1.60 inference(bind,[status(th)],[])).
% 2.11/1.60 tff(157,plain,
% 2.11/1.60 (![A: $i, B: $i] : ((A = B) <=> (subset(A, B) & subset(B, A))) <=> ![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))),
% 2.11/1.60 inference(quant_intro,[status(thm)],[156])).
% 2.11/1.60 tff(158,plain,
% 2.11/1.60 (![A: $i, B: $i] : ((A = B) <=> (subset(A, B) & subset(B, A))) <=> ![A: $i, B: $i] : ((A = B) <=> (subset(A, B) & subset(B, A)))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(159,axiom,(![A: $i, B: $i] : ((A = B) <=> (subset(A, B) & subset(B, A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d10_xboole_0')).
% 2.11/1.60 tff(160,plain,
% 2.11/1.60 (![A: $i, B: $i] : ((A = B) <=> (subset(A, B) & subset(B, A)))),
% 2.11/1.60 inference(modus_ponens,[status(thm)],[159, 158])).
% 2.11/1.60 tff(161,plain,(
% 2.11/1.60 ![A: $i, B: $i] : ((A = B) <=> (subset(A, B) & subset(B, A)))),
% 2.11/1.60 inference(skolemize,[status(sab)],[160])).
% 2.11/1.60 tff(162,plain,
% 2.11/1.60 (![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))),
% 2.11/1.60 inference(modus_ponens,[status(thm)],[161, 157])).
% 2.11/1.60 tff(163,plain,
% 2.11/1.60 (![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))),
% 2.11/1.60 inference(modus_ponens,[status(thm)],[162, 155])).
% 2.11/1.60 tff(164,plain,
% 2.11/1.60 (((~![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))) | ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12)))))) <=> ((~![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))) | ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12))))))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(165,plain,
% 2.11/1.60 (((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(topstr_closure(A!11, B!12), B!12)) | (~subset(B!12, topstr_closure(A!11, B!12)))))) <=> ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12)))))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(166,plain,
% 2.11/1.60 (((~![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))) | ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(topstr_closure(A!11, B!12), B!12)) | (~subset(B!12, topstr_closure(A!11, B!12))))))) <=> ((~![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))) | ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12))))))),
% 2.11/1.60 inference(monotonicity,[status(thm)],[165])).
% 2.11/1.60 tff(167,plain,
% 2.11/1.60 (((~![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))) | ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(topstr_closure(A!11, B!12), B!12)) | (~subset(B!12, topstr_closure(A!11, B!12))))))) <=> ((~![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))) | ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12))))))),
% 2.11/1.60 inference(transitivity,[status(thm)],[166, 164])).
% 2.11/1.60 tff(168,plain,
% 2.11/1.60 ((~![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))) | ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(topstr_closure(A!11, B!12), B!12)) | (~subset(B!12, topstr_closure(A!11, B!12))))))),
% 2.11/1.60 inference(quant_inst,[status(thm)],[])).
% 2.11/1.60 tff(169,plain,
% 2.11/1.60 ((~![A: $i, B: $i] : ((A = B) <=> (~((~subset(A, B)) | (~subset(B, A)))))) | ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12)))))),
% 2.11/1.60 inference(modus_ponens,[status(thm)],[168, 167])).
% 2.11/1.60 tff(170,plain,
% 2.11/1.60 ((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12))))),
% 2.11/1.60 inference(unit_resolution,[status(thm)],[169, 163])).
% 2.11/1.60 tff(171,plain,
% 2.11/1.60 ((~((topstr_closure(A!11, B!12) = B!12) <=> (~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12)))))) | (topstr_closure(A!11, B!12) = B!12) | ((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12)))),
% 2.11/1.60 inference(tautology,[status(thm)],[])).
% 2.11/1.60 tff(172,plain,
% 2.11/1.60 ((topstr_closure(A!11, B!12) = B!12) | ((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12)))),
% 2.11/1.60 inference(unit_resolution,[status(thm)],[171, 170])).
% 2.11/1.60 tff(173,plain,
% 2.11/1.60 ((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12))),
% 2.11/1.60 inference(unit_resolution,[status(thm)],[172, 153])).
% 2.11/1.60 tff(174,plain,
% 2.11/1.60 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))))),
% 2.11/1.60 inference(bind,[status(th)],[])).
% 2.11/1.60 tff(175,plain,
% 2.11/1.60 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))),
% 2.11/1.60 inference(quant_intro,[status(thm)],[174])).
% 2.11/1.60 tff(176,plain,
% 2.11/1.60 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))))),
% 2.11/1.60 inference(bind,[status(th)],[])).
% 2.11/1.60 tff(177,plain,
% 2.11/1.60 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))),
% 2.11/1.60 inference(quant_intro,[status(thm)],[176])).
% 2.11/1.60 tff(178,plain,
% 2.11/1.60 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))),
% 2.11/1.60 inference(transitivity,[status(thm)],[177, 175])).
% 2.11/1.60 tff(179,plain,
% 2.11/1.60 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))),
% 2.11/1.60 inference(rewrite,[status(thm)],[])).
% 2.11/1.60 tff(180,plain,
% 2.11/1.60 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => subset(B, topstr_closure(A, B))) <=> ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => subset(B, topstr_closure(A, B))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => subset(B, topstr_closure(A, B)))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => subset(B, topstr_closure(A, B)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))))),
% 2.11/1.60 inference(bind,[status(th)],[])).
% 2.11/1.60 tff(181,plain,
% 2.11/1.60 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => subset(B, topstr_closure(A, B)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))),
% 2.11/1.60 inference(quant_intro,[status(thm)],[180])).
% 2.11/1.60 tff(182,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => subset(B, topstr_closure(A, B))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t48_pre_topc')).
% 2.11/1.61 tff(183,plain,
% 2.11/1.61 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[182, 181])).
% 2.11/1.61 tff(184,plain,
% 2.11/1.61 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[183, 179])).
% 2.11/1.61 tff(185,plain,(
% 2.11/1.61 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))),
% 2.11/1.61 inference(skolemize,[status(sab)],[184])).
% 2.11/1.61 tff(186,plain,
% 2.11/1.61 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[185, 178])).
% 2.11/1.61 tff(187,plain,
% 2.11/1.61 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))) | ((~top_str(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | subset(B, topstr_closure(A!11, B))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))) | (~top_str(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | subset(B, topstr_closure(A!11, B))))),
% 2.11/1.61 inference(rewrite,[status(thm)],[])).
% 2.11/1.61 tff(188,plain,
% 2.11/1.61 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))) | ((~top_str(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | subset(B, topstr_closure(A!11, B))))),
% 2.11/1.61 inference(quant_inst,[status(thm)],[])).
% 2.11/1.61 tff(189,plain,
% 2.11/1.61 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | subset(B, topstr_closure(A, B))))) | (~top_str(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | subset(B, topstr_closure(A!11, B)))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[188, 187])).
% 2.11/1.61 tff(190,plain,
% 2.11/1.61 (![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | subset(B, topstr_closure(A!11, B)))),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[189, 186, 38])).
% 2.11/1.61 tff(191,plain,
% 2.11/1.61 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | subset(B, topstr_closure(A!11, B)))) | ((~element(B!12, powerset(the_carrier(A!11)))) | subset(B!12, topstr_closure(A!11, B!12)))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | subset(B, topstr_closure(A!11, B)))) | (~element(B!12, powerset(the_carrier(A!11)))) | subset(B!12, topstr_closure(A!11, B!12)))),
% 2.11/1.61 inference(rewrite,[status(thm)],[])).
% 2.11/1.61 tff(192,plain,
% 2.11/1.61 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | subset(B, topstr_closure(A!11, B)))) | ((~element(B!12, powerset(the_carrier(A!11)))) | subset(B!12, topstr_closure(A!11, B!12)))),
% 2.11/1.61 inference(quant_inst,[status(thm)],[])).
% 2.11/1.61 tff(193,plain,
% 2.11/1.61 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | subset(B, topstr_closure(A!11, B)))) | (~element(B!12, powerset(the_carrier(A!11)))) | subset(B!12, topstr_closure(A!11, B!12))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[192, 191])).
% 2.11/1.61 tff(194,plain,
% 2.11/1.61 (subset(B!12, topstr_closure(A!11, B!12))),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[193, 64, 190])).
% 2.11/1.61 tff(195,plain,
% 2.11/1.61 ((~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12)))) | (~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12))),
% 2.11/1.61 inference(tautology,[status(thm)],[])).
% 2.11/1.61 tff(196,plain,
% 2.11/1.61 ((~((~subset(B!12, topstr_closure(A!11, B!12))) | (~subset(topstr_closure(A!11, B!12), B!12)))) | (~subset(topstr_closure(A!11, B!12), B!12))),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[195, 194])).
% 2.11/1.61 tff(197,plain,
% 2.11/1.61 (~subset(topstr_closure(A!11, B!12), B!12)),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[196, 173])).
% 2.11/1.61 tff(198,plain,
% 2.11/1.61 ((~(subset(topstr_closure(A!11, B!12), B!12) | (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12))))) | subset(topstr_closure(A!11, B!12), B!12) | (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12)))),
% 2.11/1.61 inference(tautology,[status(thm)],[])).
% 2.11/1.61 tff(199,plain,
% 2.11/1.61 ((~(subset(topstr_closure(A!11, B!12), B!12) | (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12))))) | (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12)))),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[198, 197])).
% 2.11/1.61 tff(200,plain,
% 2.11/1.61 (~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12))),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[199, 20])).
% 2.11/1.61 tff(201,plain,
% 2.11/1.61 (((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))),
% 2.11/1.61 inference(tautology,[status(thm)],[])).
% 2.11/1.61 tff(202,plain,
% 2.11/1.61 (in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[201, 200])).
% 2.11/1.61 tff(203,plain,
% 2.11/1.61 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))) | (~((~((~subset(topstr_closure(A!11, B!12), the_carrier(A!11))) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11))))) | (~(subset(topstr_closure(A!11, B!12), the_carrier(A!11)) | (~((~in(tptp_fun_C_0(the_carrier(A!11), topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(the_carrier(A!11), topstr_closure(A!11, B!12)), the_carrier(A!11))))))))),
% 2.11/1.61 inference(quant_inst,[status(thm)],[])).
% 2.11/1.61 tff(204,plain,
% 2.11/1.61 (~((~((~subset(topstr_closure(A!11, B!12), the_carrier(A!11))) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11))))) | (~(subset(topstr_closure(A!11, B!12), the_carrier(A!11)) | (~((~in(tptp_fun_C_0(the_carrier(A!11), topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(the_carrier(A!11), topstr_closure(A!11, B!12)), the_carrier(A!11)))))))),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[203, 16])).
% 2.11/1.61 tff(205,plain,
% 2.11/1.61 (((~((~subset(topstr_closure(A!11, B!12), the_carrier(A!11))) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11))))) | (~(subset(topstr_closure(A!11, B!12), the_carrier(A!11)) | (~((~in(tptp_fun_C_0(the_carrier(A!11), topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(the_carrier(A!11), topstr_closure(A!11, B!12)), the_carrier(A!11))))))) | ((~subset(topstr_closure(A!11, B!12), the_carrier(A!11))) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11))))),
% 2.11/1.61 inference(tautology,[status(thm)],[])).
% 2.11/1.61 tff(206,plain,
% 2.11/1.61 ((~subset(topstr_closure(A!11, B!12), the_carrier(A!11))) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11)))),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[205, 204])).
% 2.11/1.61 tff(207,plain,
% 2.11/1.61 (^[A: $i, B: $i] : refl((element(A, powerset(B)) <=> subset(A, B)) <=> (element(A, powerset(B)) <=> subset(A, B)))),
% 2.11/1.61 inference(bind,[status(th)],[])).
% 2.11/1.61 tff(208,plain,
% 2.11/1.61 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B)) <=> ![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 2.11/1.61 inference(quant_intro,[status(thm)],[207])).
% 2.11/1.61 tff(209,plain,
% 2.11/1.61 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B)) <=> ![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 2.11/1.61 inference(rewrite,[status(thm)],[])).
% 2.11/1.61 tff(210,axiom,(![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t3_subset')).
% 2.11/1.61 tff(211,plain,
% 2.11/1.61 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[210, 209])).
% 2.11/1.61 tff(212,plain,(
% 2.11/1.61 ![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 2.11/1.61 inference(skolemize,[status(sab)],[211])).
% 2.11/1.61 tff(213,plain,
% 2.11/1.61 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[212, 208])).
% 2.11/1.61 tff(214,plain,
% 2.11/1.61 ((~![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))) | (element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))) <=> subset(topstr_closure(A!11, B!12), the_carrier(A!11)))),
% 2.11/1.61 inference(quant_inst,[status(thm)],[])).
% 2.11/1.61 tff(215,plain,
% 2.11/1.61 (element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))) <=> subset(topstr_closure(A!11, B!12), the_carrier(A!11))),
% 2.11/1.61 inference(unit_resolution,[status(thm)],[214, 213])).
% 2.11/1.61 tff(216,plain,
% 2.11/1.61 (^[A: $i, B: $i] : refl((element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A))) <=> (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A))))),
% 2.11/1.61 inference(bind,[status(th)],[])).
% 2.11/1.61 tff(217,plain,
% 2.11/1.61 (![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A))) <=> ![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))),
% 2.11/1.61 inference(quant_intro,[status(thm)],[216])).
% 2.11/1.61 tff(218,plain,
% 2.11/1.61 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((top_str(A) & element(B, powerset(the_carrier(A)))) <=> (~((~top_str(A)) | (~element(B, powerset(the_carrier(A))))))), ((~(top_str(A) & element(B, powerset(the_carrier(A))))) <=> (~(~((~top_str(A)) | (~element(B, powerset(the_carrier(A))))))))), rewrite((~(~((~top_str(A)) | (~element(B, powerset(the_carrier(A))))))) <=> ((~top_str(A)) | (~element(B, powerset(the_carrier(A)))))), ((~(top_str(A) & element(B, powerset(the_carrier(A))))) <=> ((~top_str(A)) | (~element(B, powerset(the_carrier(A))))))), (((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> (((~top_str(A)) | (~element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))))), rewrite((((~top_str(A)) | (~element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))), (((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))))),
% 2.11/1.61 inference(bind,[status(th)],[])).
% 2.11/1.61 tff(219,plain,
% 2.11/1.61 (![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> ![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))),
% 2.11/1.61 inference(quant_intro,[status(thm)],[218])).
% 2.11/1.61 tff(220,plain,
% 2.11/1.61 (![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> ![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 2.11/1.61 inference(rewrite,[status(thm)],[])).
% 2.11/1.61 tff(221,plain,
% 2.11/1.61 (^[A: $i, B: $i] : rewrite(((top_str(A) & element(B, powerset(the_carrier(A)))) => element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))))),
% 2.11/1.61 inference(bind,[status(th)],[])).
% 2.11/1.61 tff(222,plain,
% 2.11/1.61 (![A: $i, B: $i] : ((top_str(A) & element(B, powerset(the_carrier(A)))) => element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> ![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 2.11/1.61 inference(quant_intro,[status(thm)],[221])).
% 2.11/1.61 tff(223,axiom,(![A: $i, B: $i] : ((top_str(A) & element(B, powerset(the_carrier(A)))) => element(topstr_closure(A, B), powerset(the_carrier(A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k6_pre_topc')).
% 2.11/1.61 tff(224,plain,
% 2.11/1.61 (![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[223, 222])).
% 2.11/1.61 tff(225,plain,
% 2.11/1.61 (![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[224, 220])).
% 2.11/1.61 tff(226,plain,(
% 2.11/1.61 ![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 2.11/1.61 inference(skolemize,[status(sab)],[225])).
% 2.11/1.61 tff(227,plain,
% 2.11/1.61 (![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[226, 219])).
% 2.11/1.61 tff(228,plain,
% 2.11/1.61 (![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))),
% 2.11/1.61 inference(modus_ponens,[status(thm)],[227, 217])).
% 2.11/1.61 tff(229,plain,
% 2.11/1.61 (((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | ((~top_str(A!11)) | (~element(B!12, powerset(the_carrier(A!11)))) | element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))))) <=> ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (~top_str(A!11)) | (~element(B!12, powerset(the_carrier(A!11)))) | element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))))),
% 2.11/1.61 inference(rewrite,[status(thm)],[])).
% 2.11/1.61 tff(230,plain,
% 2.11/1.61 ((element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))) | (~element(B!12, powerset(the_carrier(A!11)))) | (~top_str(A!11))) <=> ((~top_str(A!11)) | (~element(B!12, powerset(the_carrier(A!11)))) | element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))))),
% 2.11/1.61 inference(rewrite,[status(thm)],[])).
% 2.11/1.61 tff(231,plain,
% 2.11/1.61 (((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))) | (~element(B!12, powerset(the_carrier(A!11)))) | (~top_str(A!11)))) <=> ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | ((~top_str(A!11)) | (~element(B!12, powerset(the_carrier(A!11)))) | element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11)))))),
% 2.11/1.61 inference(monotonicity,[status(thm)],[230])).
% 2.11/1.61 tff(232,plain,
% 2.11/1.61 (((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))) | (~element(B!12, powerset(the_carrier(A!11)))) | (~top_str(A!11)))) <=> ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (~top_str(A!11)) | (~element(B!12, powerset(the_carrier(A!11)))) | element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))))),
% 2.11/1.61 inference(transitivity,[status(thm)],[231, 229])).
% 2.11/1.61 tff(233,plain,
% 2.11/1.61 ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))) | (~element(B!12, powerset(the_carrier(A!11)))) | (~top_str(A!11)))),
% 2.11/1.61 inference(quant_inst,[status(thm)],[])).
% 2.11/1.61 tff(234,plain,
% 2.11/1.61 ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (~top_str(A!11)) | (~element(B!12, powerset(the_carrier(A!11)))) | element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11)))),
% 2.11/1.62 inference(modus_ponens,[status(thm)],[233, 232])).
% 2.11/1.62 tff(235,plain,
% 2.11/1.62 (element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11)))),
% 2.11/1.62 inference(unit_resolution,[status(thm)],[234, 228, 38, 64])).
% 2.11/1.62 tff(236,plain,
% 2.11/1.62 ((~(element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))) <=> subset(topstr_closure(A!11, B!12), the_carrier(A!11)))) | (~element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11)))) | subset(topstr_closure(A!11, B!12), the_carrier(A!11))),
% 2.11/1.62 inference(tautology,[status(thm)],[])).
% 2.11/1.62 tff(237,plain,
% 2.11/1.62 ((~(element(topstr_closure(A!11, B!12), powerset(the_carrier(A!11))) <=> subset(topstr_closure(A!11, B!12), the_carrier(A!11)))) | subset(topstr_closure(A!11, B!12), the_carrier(A!11))),
% 2.11/1.62 inference(unit_resolution,[status(thm)],[236, 235])).
% 2.11/1.62 tff(238,plain,
% 2.11/1.62 (subset(topstr_closure(A!11, B!12), the_carrier(A!11))),
% 2.11/1.62 inference(unit_resolution,[status(thm)],[237, 215])).
% 2.11/1.62 tff(239,plain,
% 2.11/1.62 ((~((~subset(topstr_closure(A!11, B!12), the_carrier(A!11))) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11))))) | (~subset(topstr_closure(A!11, B!12), the_carrier(A!11))) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11)))),
% 2.11/1.62 inference(tautology,[status(thm)],[])).
% 2.11/1.62 tff(240,plain,
% 2.11/1.62 ((~((~subset(topstr_closure(A!11, B!12), the_carrier(A!11))) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11))))) | ![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11)))),
% 2.11/1.62 inference(unit_resolution,[status(thm)],[239, 238])).
% 2.11/1.62 tff(241,plain,
% 2.11/1.62 (![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11)))),
% 2.11/1.62 inference(unit_resolution,[status(thm)],[240, 206])).
% 2.11/1.62 tff(242,plain,
% 2.11/1.62 (((~![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11)))) | ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11)))) <=> ((~![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11)))) | (~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11)))),
% 2.11/1.62 inference(rewrite,[status(thm)],[])).
% 2.11/1.62 tff(243,plain,
% 2.11/1.62 ((~![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11)))) | ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11)))),
% 2.11/1.62 inference(quant_inst,[status(thm)],[])).
% 2.11/1.62 tff(244,plain,
% 2.11/1.62 ((~![C: $i] : ((~in(C, topstr_closure(A!11, B!12))) | in(C, the_carrier(A!11)))) | (~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))),
% 2.11/1.62 inference(modus_ponens,[status(thm)],[243, 242])).
% 2.11/1.62 tff(245,plain,
% 2.11/1.62 (in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))),
% 2.11/1.62 inference(unit_resolution,[status(thm)],[244, 241, 202])).
% 2.11/1.62 tff(246,plain,
% 2.11/1.62 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A)))))))))))))),
% 2.11/1.62 inference(bind,[status(th)],[])).
% 2.11/1.62 tff(247,plain,
% 2.11/1.62 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))),
% 2.11/1.62 inference(quant_intro,[status(thm)],[246])).
% 2.11/1.62 tff(248,plain,
% 2.11/1.62 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A)))))))))))))),
% 2.11/1.62 inference(bind,[status(th)],[])).
% 2.11/1.62 tff(249,plain,
% 2.11/1.62 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))),
% 2.11/1.62 inference(quant_intro,[status(thm)],[248])).
% 2.11/1.62 tff(250,plain,
% 2.11/1.62 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))),
% 2.11/1.62 inference(transitivity,[status(thm)],[249, 247])).
% 2.11/1.62 tff(251,plain,
% 2.11/1.62 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))) & (in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~(closed_subset(tptp_fun_D_9(C, B, A), A) & subset(B, tptp_fun_D_9(C, B, A)))) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A)))))))))))))),
% 2.11/1.62 inference(bind,[status(th)],[])).
% 2.11/1.62 tff(252,plain,
% 2.11/1.62 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))) & (in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~(closed_subset(tptp_fun_D_9(C, B, A), A) & subset(B, tptp_fun_D_9(C, B, A)))) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))),
% 2.11/1.62 inference(quant_intro,[status(thm)],[251])).
% 2.11/1.62 tff(253,plain,
% 2.11/1.62 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))))),
% 2.11/1.62 inference(rewrite,[status(thm)],[])).
% 2.11/1.62 tff(254,plain,
% 2.11/1.62 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(rewrite((in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D)))) <=> (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A))))))), ((in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D))))) <=> (in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A))))))))), rewrite((in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A))))))) <=> ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))), ((in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D))))) <=> ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))))), (![C: $i] : (in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D))))) <=> ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A))))))))), ((element(B, powerset(the_carrier(A))) => ![C: $i] : (in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D)))))) <=> (element(B, powerset(the_carrier(A))) => ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))))), rewrite((element(B, powerset(the_carrier(A))) => ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))) <=> ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A))))))))), ((element(B, powerset(the_carrier(A))) => ![C: $i] : (in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D)))))) <=> ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A))))))))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => ![C: $i] : (in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D)))))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ![C: $i] : (in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D))))))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A))))))))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ![C: $i] : (in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))))))),
% 2.11/1.62 inference(bind,[status(th)],[])).
% 2.11/1.62 tff(255,plain,
% 2.11/1.62 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ![C: $i] : (in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))))),
% 2.11/1.62 inference(quant_intro,[status(thm)],[254])).
% 2.11/1.62 tff(256,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => ![C: $i] : (in(C, the_carrier(A)) => (in(C, topstr_closure(A, B)) <=> ![D: $i] : (element(D, powerset(the_carrier(A))) => ((closed_subset(D, A) & subset(B, D)) => in(C, D)))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t45_pre_topc')).
% 2.11/1.62 tff(257,plain,
% 2.11/1.62 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))))),
% 2.11/1.62 inference(modus_ponens,[status(thm)],[256, 255])).
% 2.11/1.62 tff(258,plain,
% 2.11/1.62 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (in(C, topstr_closure(A, B)) <=> ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))))))),
% 2.11/1.62 inference(modus_ponens,[status(thm)],[257, 253])).
% 2.11/1.62 tff(259,plain,(
% 2.11/1.62 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~(closed_subset(D, A) & subset(B, D))) | (~element(D, powerset(the_carrier(A)))))) & (in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~(closed_subset(tptp_fun_D_9(C, B, A), A) & subset(B, tptp_fun_D_9(C, B, A)))) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))))))))))),
% 2.11/1.62 inference(skolemize,[status(sab)],[258])).
% 2.11/1.62 tff(260,plain,
% 2.11/1.62 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))),
% 2.11/1.62 inference(modus_ponens,[status(thm)],[259, 252])).
% 2.11/1.62 tff(261,plain,
% 2.11/1.62 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))),
% 2.11/1.62 inference(modus_ponens,[status(thm)],[260, 250])).
% 2.11/1.62 tff(262,plain,
% 2.11/1.62 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))) | ((~top_str(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A!11, B)) | (~(in(C, tptp_fun_D_9(C, B, A!11)) | (~element(tptp_fun_D_9(C, B, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B, A!11), A!11)) | (~subset(B, tptp_fun_D_9(C, B, A!11))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))) | (~top_str(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A!11, B)) | (~(in(C, tptp_fun_D_9(C, B, A!11)) | (~element(tptp_fun_D_9(C, B, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B, A!11), A!11)) | (~subset(B, tptp_fun_D_9(C, B, A!11))))))))))))),
% 2.11/1.63 inference(rewrite,[status(thm)],[])).
% 2.11/1.63 tff(263,plain,
% 2.11/1.63 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))) | ((~top_str(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A!11, B)) | (~(in(C, tptp_fun_D_9(C, B, A!11)) | (~element(tptp_fun_D_9(C, B, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B, A!11), A!11)) | (~subset(B, tptp_fun_D_9(C, B, A!11))))))))))))),
% 2.11/1.63 inference(quant_inst,[status(thm)],[])).
% 2.11/1.63 tff(264,plain,
% 2.11/1.63 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | ![C: $i] : ((~in(C, the_carrier(A))) | (~((~((~in(C, topstr_closure(A, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A)))) | (~closed_subset(D, A)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A, B)) | (~(in(C, tptp_fun_D_9(C, B, A)) | (~element(tptp_fun_D_9(C, B, A), powerset(the_carrier(A)))) | (~closed_subset(tptp_fun_D_9(C, B, A), A)) | (~subset(B, tptp_fun_D_9(C, B, A))))))))))))) | (~top_str(A!11)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A!11, B)) | (~(in(C, tptp_fun_D_9(C, B, A!11)) | (~element(tptp_fun_D_9(C, B, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B, A!11), A!11)) | (~subset(B, tptp_fun_D_9(C, B, A!11)))))))))))),
% 2.11/1.63 inference(modus_ponens,[status(thm)],[263, 262])).
% 2.11/1.63 tff(265,plain,
% 2.11/1.63 (![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A!11, B)) | (~(in(C, tptp_fun_D_9(C, B, A!11)) | (~element(tptp_fun_D_9(C, B, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B, A!11), A!11)) | (~subset(B, tptp_fun_D_9(C, B, A!11)))))))))))),
% 2.11/1.63 inference(unit_resolution,[status(thm)],[264, 261, 38])).
% 2.11/1.63 tff(266,plain,
% 2.11/1.63 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A!11, B)) | (~(in(C, tptp_fun_D_9(C, B, A!11)) | (~element(tptp_fun_D_9(C, B, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B, A!11), A!11)) | (~subset(B, tptp_fun_D_9(C, B, A!11)))))))))))) | ((~element(B!12, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11)))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A!11, B)) | (~(in(C, tptp_fun_D_9(C, B, A!11)) | (~element(tptp_fun_D_9(C, B, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B, A!11), A!11)) | (~subset(B, tptp_fun_D_9(C, B, A!11)))))))))))) | (~element(B!12, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11)))))))))))),
% 2.19/1.63 inference(rewrite,[status(thm)],[])).
% 2.19/1.63 tff(267,plain,
% 2.19/1.63 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A!11, B)) | (~(in(C, tptp_fun_D_9(C, B, A!11)) | (~element(tptp_fun_D_9(C, B, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B, A!11), A!11)) | (~subset(B, tptp_fun_D_9(C, B, A!11)))))))))))) | ((~element(B!12, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11)))))))))))),
% 2.19/1.63 inference(quant_inst,[status(thm)],[])).
% 2.19/1.63 tff(268,plain,
% 2.19/1.63 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B, D))))) | (~(in(C, topstr_closure(A!11, B)) | (~(in(C, tptp_fun_D_9(C, B, A!11)) | (~element(tptp_fun_D_9(C, B, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B, A!11), A!11)) | (~subset(B, tptp_fun_D_9(C, B, A!11)))))))))))) | (~element(B!12, powerset(the_carrier(A!11)))) | ![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))),
% 2.19/1.63 inference(modus_ponens,[status(thm)],[267, 266])).
% 2.19/1.63 tff(269,plain,
% 2.19/1.63 (![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))),
% 2.19/1.63 inference(unit_resolution,[status(thm)],[268, 64, 265])).
% 2.19/1.63 tff(270,plain,
% 2.19/1.63 (((~![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))) | ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11))))))))))) <=> ((~![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))) | (~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11))))))))))),
% 2.19/1.63 inference(rewrite,[status(thm)],[])).
% 2.19/1.63 tff(271,plain,
% 2.19/1.63 (((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : (in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)))))))))) <=> ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11))))))))))),
% 2.19/1.64 inference(rewrite,[status(thm)],[])).
% 2.19/1.64 tff(272,plain,
% 2.19/1.64 (((~![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))) | ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : (in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11))))))))))) <=> ((~![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))) | ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)))))))))))),
% 2.19/1.64 inference(monotonicity,[status(thm)],[271])).
% 2.19/1.64 tff(273,plain,
% 2.19/1.64 (((~![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))) | ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : (in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11))))))))))) <=> ((~![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))) | (~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11))))))))))),
% 2.19/1.64 inference(transitivity,[status(thm)],[272, 270])).
% 2.19/1.64 tff(274,plain,
% 2.19/1.64 ((~![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))) | ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : (in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11))))))))))),
% 2.19/1.64 inference(quant_inst,[status(thm)],[])).
% 2.19/1.64 tff(275,plain,
% 2.19/1.64 ((~![C: $i] : ((~in(C, the_carrier(A!11))) | (~((~((~in(C, topstr_closure(A!11, B!12))) | ![D: $i] : (in(C, D) | (~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D))))) | (~(in(C, topstr_closure(A!11, B!12)) | (~(in(C, tptp_fun_D_9(C, B!12, A!11)) | (~element(tptp_fun_D_9(C, B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(C, B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(C, B!12, A!11))))))))))) | (~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), the_carrier(A!11))) | (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)))))))))),
% 2.19/1.64 inference(modus_ponens,[status(thm)],[274, 273])).
% 2.19/1.64 tff(276,plain,
% 2.19/1.64 (~((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11))))))))),
% 2.19/1.64 inference(unit_resolution,[status(thm)],[275, 269, 245])).
% 2.19/1.64 tff(277,plain,
% 2.19/1.64 (((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12)) | (~(in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)) | (~element(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), powerset(the_carrier(A!11)))) | (~closed_subset(tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11), A!11)) | (~subset(B!12, tptp_fun_D_9(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12, A!11)))))))) | ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))),
% 2.19/1.64 inference(tautology,[status(thm)],[])).
% 2.19/1.64 tff(278,plain,
% 2.19/1.64 ((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))),
% 2.19/1.64 inference(unit_resolution,[status(thm)],[277, 276])).
% 2.19/1.64 tff(279,plain,
% 2.19/1.64 ((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | (~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))),
% 2.19/1.64 inference(tautology,[status(thm)],[])).
% 2.19/1.64 tff(280,plain,
% 2.19/1.64 ((~((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))),
% 2.19/1.65 inference(unit_resolution,[status(thm)],[279, 202])).
% 2.19/1.65 tff(281,plain,
% 2.19/1.65 (![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))),
% 2.19/1.65 inference(unit_resolution,[status(thm)],[280, 278])).
% 2.19/1.65 tff(282,plain,
% 2.19/1.65 (((~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), topstr_closure(A!11, B!12))) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12)) | (~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12))),
% 2.19/1.65 inference(tautology,[status(thm)],[])).
% 2.19/1.65 tff(283,plain,
% 2.19/1.65 (~in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12)),
% 2.19/1.65 inference(unit_resolution,[status(thm)],[282, 200])).
% 2.19/1.65 tff(284,plain,
% 2.19/1.65 (((~closed_subset(B!12, A!11)) | (topstr_closure(A!11, B!12) = B!12)) | closed_subset(B!12, A!11)),
% 2.19/1.65 inference(tautology,[status(thm)],[])).
% 2.19/1.65 tff(285,plain,
% 2.19/1.65 (closed_subset(B!12, A!11)),
% 2.19/1.65 inference(unit_resolution,[status(thm)],[284, 151])).
% 2.19/1.65 tff(286,plain,
% 2.19/1.65 (((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))) | ((~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12) | (~subset(B!12, B!12)))) <=> ((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))) | (~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12) | (~subset(B!12, B!12)))),
% 2.19/1.65 inference(rewrite,[status(thm)],[])).
% 2.19/1.65 tff(287,plain,
% 2.19/1.65 (((~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | (~subset(B!12, B!12)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12)) <=> ((~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12) | (~subset(B!12, B!12)))),
% 2.19/1.65 inference(rewrite,[status(thm)],[])).
% 2.19/1.65 tff(288,plain,
% 2.19/1.65 (((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))) | ((~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | (~subset(B!12, B!12)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12))) <=> ((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))) | ((~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12) | (~subset(B!12, B!12))))),
% 2.19/1.65 inference(monotonicity,[status(thm)],[287])).
% 2.19/1.65 tff(289,plain,
% 2.19/1.65 (((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))) | ((~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | (~subset(B!12, B!12)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12))) <=> ((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))) | (~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12) | (~subset(B!12, B!12)))),
% 2.19/1.65 inference(transitivity,[status(thm)],[288, 286])).
% 2.19/1.65 tff(290,plain,
% 2.19/1.65 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))) | ((~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | (~subset(B!12, B!12)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12))),
% 2.19/1.65 inference(quant_inst,[status(thm)],[])).
% 2.19/1.65 tff(291,plain,
% 2.19/1.65 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!11)))) | (~closed_subset(D, A!11)) | (~subset(B!12, D)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), D))) | (~element(B!12, powerset(the_carrier(A!11)))) | (~closed_subset(B!12, A!11)) | in(tptp_fun_C_0(B!12, topstr_closure(A!11, B!12)), B!12) | (~subset(B!12, B!12))),
% 2.19/1.65 inference(modus_ponens,[status(thm)],[290, 289])).
% 2.19/1.65 tff(292,plain,
% 2.19/1.65 (~subset(B!12, B!12)),
% 2.19/1.65 inference(unit_resolution,[status(thm)],[291, 64, 285, 283, 281])).
% 2.19/1.65 tff(293,plain,
% 2.19/1.65 (^[A: $i] : refl(subset(A, A) <=> subset(A, A))),
% 2.19/1.65 inference(bind,[status(th)],[])).
% 2.19/1.65 tff(294,plain,
% 2.19/1.65 (![A: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 2.19/1.65 inference(quant_intro,[status(thm)],[293])).
% 2.19/1.65 tff(295,plain,
% 2.19/1.65 (![A: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 2.19/1.65 inference(rewrite,[status(thm)],[])).
% 2.19/1.65 tff(296,plain,
% 2.19/1.65 (![A: $i, B: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 2.19/1.65 inference(elim_unused_vars,[status(thm)],[])).
% 2.19/1.65 tff(297,axiom,(![A: $i, B: $i] : subset(A, A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','reflexivity_r1_tarski')).
% 2.19/1.65 tff(298,plain,
% 2.19/1.65 (![A: $i] : subset(A, A)),
% 2.19/1.65 inference(modus_ponens,[status(thm)],[297, 296])).
% 2.19/1.65 tff(299,plain,
% 2.19/1.65 (![A: $i] : subset(A, A)),
% 2.19/1.65 inference(modus_ponens,[status(thm)],[298, 295])).
% 2.19/1.65 tff(300,plain,(
% 2.19/1.65 ![A: $i] : subset(A, A)),
% 2.19/1.65 inference(skolemize,[status(sab)],[299])).
% 2.19/1.65 tff(301,plain,
% 2.19/1.65 (![A: $i] : subset(A, A)),
% 2.19/1.65 inference(modus_ponens,[status(thm)],[300, 294])).
% 2.19/1.65 tff(302,plain,
% 2.19/1.65 ((~![A: $i] : subset(A, A)) | subset(B!12, B!12)),
% 2.19/1.65 inference(quant_inst,[status(thm)],[])).
% 2.19/1.65 tff(303,plain,
% 2.19/1.65 ($false),
% 2.19/1.65 inference(unit_resolution,[status(thm)],[302, 301, 292])).
% 2.19/1.65 % SZS output end Proof
%------------------------------------------------------------------------------