%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : TOP021+1 : TPTP v9.0.0. Released v3.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% 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 : Sat Jun 21 05:42:37 AM UTC 2025
% Result : Theorem 0.20s 0.38s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : TOP021+1 : TPTP v9.0.0. Released v3.1.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n015.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 13:52:26 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.38 % SZS status Theorem
% 0.20/0.38 % SZS output start Proof
% 0.20/0.38 tff(an_open_function_from_onto_type, type, (
% 0.20/0.38 an_open_function_from_onto: ( $i * $i * $i ) > $o)).
% 0.20/0.38 tff(apply_type, type, (
% 0.20/0.38 apply: ( $i * $i ) > $i)).
% 0.20/0.38 tff(tptp_fun_A_2_type, type, (
% 0.20/0.38 tptp_fun_A_2: $i)).
% 0.20/0.38 tff(tptp_fun_X1_1_type, type, (
% 0.20/0.38 tptp_fun_X1_1: $i)).
% 0.20/0.38 tff(the_product_top_space_over_type, type, (
% 0.20/0.38 the_product_top_space_over: ( $i * $i ) > $i)).
% 0.20/0.38 tff(tptp_fun_A1_0_type, type, (
% 0.20/0.38 tptp_fun_A1_0: $i)).
% 0.20/0.38 tff(the_projection_function_type, type, (
% 0.20/0.38 the_projection_function: ( $i * $i * $i ) > $i)).
% 0.20/0.38 tff(a_continuous_function_from_onto_type, type, (
% 0.20/0.38 a_continuous_function_from_onto: ( $i * $i * $i ) > $o)).
% 0.20/0.38 tff(a_locally_compact_top_space_type, type, (
% 0.20/0.38 a_locally_compact_top_space: $i > $o)).
% 0.20/0.38 tff(1,plain,
% 0.20/0.38 (^[A: $i, X: $i, A1: $i, X1: $i] : refl(an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A)) <=> an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A)))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(2,plain,
% 0.20/0.38 (![A: $i, X: $i, A1: $i, X1: $i] : an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A)) <=> ![A: $i, X: $i, A1: $i, X1: $i] : an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.38 tff(3,plain,
% 0.20/0.38 (![A: $i, X: $i, A1: $i, X1: $i] : an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A)) <=> ![A: $i, X: $i, A1: $i, X1: $i] : an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(4,axiom,(![A: $i, X: $i, A1: $i, X1: $i] : an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','kelley_3_2')).
% 0.20/0.38 tff(5,plain,
% 0.20/0.38 (![A: $i, X: $i, A1: $i, X1: $i] : an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.38 tff(6,plain,(
% 0.20/0.38 ![A: $i, X: $i, A1: $i, X1: $i] : an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A))),
% 0.20/0.38 inference(skolemize,[status(sab)],[5])).
% 0.20/0.38 tff(7,plain,
% 0.20/0.38 (![A: $i, X: $i, A1: $i, X1: $i] : an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.38 tff(8,plain,
% 0.20/0.38 ((~![A: $i, X: $i, A1: $i, X1: $i] : an_open_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X1, A1), apply(X1, A))) | an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(9,plain,
% 0.20/0.38 (an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.20/0.38 tff(10,plain,
% 0.20/0.38 (^[A: $i, X: $i, A1: $i] : refl(a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A)) <=> a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A)))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(11,plain,
% 0.20/0.38 (![A: $i, X: $i, A1: $i] : a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A)) <=> ![A: $i, X: $i, A1: $i] : a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[10])).
% 0.20/0.38 tff(12,plain,
% 0.20/0.38 (![A: $i, X: $i, A1: $i] : a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A)) <=> ![A: $i, X: $i, A1: $i] : a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(13,axiom,(![A: $i, X: $i, A1: $i] : a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','kelley_p90a')).
% 0.20/0.38 tff(14,plain,
% 0.20/0.38 (![A: $i, X: $i, A1: $i] : a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.20/0.38 tff(15,plain,(
% 0.20/0.38 ![A: $i, X: $i, A1: $i] : a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A))),
% 0.20/0.38 inference(skolemize,[status(sab)],[14])).
% 0.20/0.38 tff(16,plain,
% 0.20/0.38 (![A: $i, X: $i, A1: $i] : a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.20/0.38 tff(17,plain,
% 0.20/0.38 ((~![A: $i, X: $i, A1: $i] : a_continuous_function_from_onto(the_projection_function(A, X, A1), the_product_top_space_over(X, A1), apply(X, A))) | a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(18,plain,
% 0.20/0.38 (a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.20/0.38 tff(19,plain,
% 0.20/0.38 ((~![X1: $i, A1: $i] : ((~a_locally_compact_top_space(the_product_top_space_over(X1, A1))) | ![A: $i] : a_locally_compact_top_space(apply(X1, A)))) <=> (~![X1: $i, A1: $i] : ((~a_locally_compact_top_space(the_product_top_space_over(X1, A1))) | ![A: $i] : a_locally_compact_top_space(apply(X1, A))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(20,plain,
% 0.20/0.38 ((~![X1: $i, A1: $i] : (a_locally_compact_top_space(the_product_top_space_over(X1, A1)) => ![A: $i] : a_locally_compact_top_space(apply(X1, A)))) <=> (~![X1: $i, A1: $i] : ((~a_locally_compact_top_space(the_product_top_space_over(X1, A1))) | ![A: $i] : a_locally_compact_top_space(apply(X1, A))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(21,axiom,(~![X1: $i, A1: $i] : (a_locally_compact_top_space(the_product_top_space_over(X1, A1)) => ![A: $i] : a_locally_compact_top_space(apply(X1, A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','kelley_5_19a')).
% 0.20/0.38 tff(22,plain,
% 0.20/0.38 (~![X1: $i, A1: $i] : ((~a_locally_compact_top_space(the_product_top_space_over(X1, A1))) | ![A: $i] : a_locally_compact_top_space(apply(X1, A)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.20/0.38 tff(23,plain,
% 0.20/0.38 (~![X1: $i, A1: $i] : ((~a_locally_compact_top_space(the_product_top_space_over(X1, A1))) | ![A: $i] : a_locally_compact_top_space(apply(X1, A)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[22, 19])).
% 0.20/0.38 tff(24,plain,
% 0.20/0.38 (~![X1: $i, A1: $i] : ((~a_locally_compact_top_space(the_product_top_space_over(X1, A1))) | ![A: $i] : a_locally_compact_top_space(apply(X1, A)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[23, 19])).
% 0.20/0.38 tff(25,plain,
% 0.20/0.38 (~![X1: $i, A1: $i] : ((~a_locally_compact_top_space(the_product_top_space_over(X1, A1))) | ![A: $i] : a_locally_compact_top_space(apply(X1, A)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[24, 19])).
% 0.20/0.38 tff(26,plain,
% 0.20/0.38 (~a_locally_compact_top_space(apply(X1!1, A!2))),
% 0.20/0.38 inference(and_elim,[status(thm)],[25])).
% 0.20/0.38 tff(27,plain,
% 0.20/0.38 (a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))),
% 0.20/0.38 inference(and_elim,[status(thm)],[25])).
% 0.20/0.38 tff(28,plain,
% 0.20/0.38 (^[F: $i, A: $i, B: $i] : refl((a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A))) <=> (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(29,plain,
% 0.20/0.38 (![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A))) <=> ![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[28])).
% 0.20/0.38 tff(30,plain,
% 0.20/0.38 (^[F: $i, A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A)) <=> (~((~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A))))), ((~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))) <=> (~(~((~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A))))))), rewrite((~(~((~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A))))) <=> ((~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))), ((~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))) <=> ((~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A))))), ((a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A)))) <=> (a_locally_compact_top_space(B) | ((~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))))), rewrite((a_locally_compact_top_space(B) | ((~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))) <=> (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))), ((a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A)))) <=> (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(31,plain,
% 0.20/0.38 (![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A)))) <=> ![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[30])).
% 0.20/0.38 tff(32,plain,
% 0.20/0.38 (![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A)))) <=> ![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(33,plain,
% 0.20/0.38 (^[F: $i, A: $i, B: $i] : trans(monotonicity(rewrite(((an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B)) & a_locally_compact_top_space(A)) <=> (an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))), ((((an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B)) & a_locally_compact_top_space(A)) => a_locally_compact_top_space(B)) <=> ((an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A)) => a_locally_compact_top_space(B)))), rewrite(((an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A)) => a_locally_compact_top_space(B)) <=> (a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))))), ((((an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B)) & a_locally_compact_top_space(A)) => a_locally_compact_top_space(B)) <=> (a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(34,plain,
% 0.20/0.38 (![F: $i, A: $i, B: $i] : (((an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B)) & a_locally_compact_top_space(A)) => a_locally_compact_top_space(B)) <=> ![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[33])).
% 0.20/0.38 tff(35,axiom,(![F: $i, A: $i, B: $i] : (((an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B)) & a_locally_compact_top_space(A)) => a_locally_compact_top_space(B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','kelley_p_147e')).
% 0.20/0.38 tff(36,plain,
% 0.20/0.38 (![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.20/0.38 tff(37,plain,
% 0.20/0.38 (![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.20/0.38 tff(38,plain,(
% 0.20/0.38 ![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~(an_open_function_from_onto(F, A, B) & a_continuous_function_from_onto(F, A, B) & a_locally_compact_top_space(A))))),
% 0.20/0.38 inference(skolemize,[status(sab)],[37])).
% 0.20/0.38 tff(39,plain,
% 0.20/0.38 (![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.20/0.38 tff(40,plain,
% 0.20/0.38 (![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[39, 29])).
% 0.20/0.38 tff(41,plain,
% 0.20/0.38 (((~![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))) | ((~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))) | a_locally_compact_top_space(apply(X1!1, A!2)) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))))) <=> ((~![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))) | (~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))) | a_locally_compact_top_space(apply(X1!1, A!2)) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(42,plain,
% 0.20/0.38 ((a_locally_compact_top_space(apply(X1!1, A!2)) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0)))) <=> ((~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))) | a_locally_compact_top_space(apply(X1!1, A!2)) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(43,plain,
% 0.20/0.38 (((~![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))) | (a_locally_compact_top_space(apply(X1!1, A!2)) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))))) <=> ((~![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))) | ((~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))) | a_locally_compact_top_space(apply(X1!1, A!2)) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2)))))),
% 0.20/0.38 inference(monotonicity,[status(thm)],[42])).
% 0.20/0.38 tff(44,plain,
% 0.20/0.38 (((~![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))) | (a_locally_compact_top_space(apply(X1!1, A!2)) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))))) <=> ((~![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))) | (~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))) | a_locally_compact_top_space(apply(X1!1, A!2)) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))))),
% 0.20/0.38 inference(transitivity,[status(thm)],[43, 41])).
% 0.20/0.38 tff(45,plain,
% 0.20/0.38 ((~![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))) | (a_locally_compact_top_space(apply(X1!1, A!2)) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(46,plain,
% 0.20/0.38 ((~![F: $i, A: $i, B: $i] : (a_locally_compact_top_space(B) | (~an_open_function_from_onto(F, A, B)) | (~a_continuous_function_from_onto(F, A, B)) | (~a_locally_compact_top_space(A)))) | (~a_locally_compact_top_space(the_product_top_space_over(X1!1, A1!0))) | a_locally_compact_top_space(apply(X1!1, A!2)) | (~a_continuous_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2))) | (~an_open_function_from_onto(the_projection_function(A!2, X1!1, A1!0), the_product_top_space_over(X1!1, A1!0), apply(X1!1, A!2)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.20/0.39 tff(47,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[46, 40, 27, 26, 18, 9])).
% 0.20/0.39 % SZS output end Proof
% 0.20/0.39 % E exiting
%------------------------------------------------------------------------------