%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET731+4 : TPTP v9.0.0. Bugfixed v2.2.1.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:26:14 AM UTC 2025
% Result : Theorem 0.21s 0.40s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET731+4 : TPTP v9.0.0. Bugfixed v2.2.1.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n001.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 05:43:05 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.21/0.40 % SZS status Theorem
% 0.21/0.40 % SZS output start Proof
% 0.21/0.40 tff(apply_type, type, (
% 0.21/0.40 apply: ( $i * $i * $i ) > $o)).
% 0.21/0.40 tff(tptp_fun_Y_20_type, type, (
% 0.21/0.40 tptp_fun_Y_20: ( $i * $i * $i ) > $i)).
% 0.21/0.40 tff(tptp_fun_G_41_type, type, (
% 0.21/0.40 tptp_fun_G_41: $i)).
% 0.21/0.40 tff(tptp_fun_A_40_type, type, (
% 0.21/0.40 tptp_fun_A_40: $i)).
% 0.21/0.40 tff(image2_type, type, (
% 0.21/0.40 image2: ( $i * $i ) > $i)).
% 0.21/0.40 tff(tptp_fun_F_42_type, type, (
% 0.21/0.40 tptp_fun_F_42: $i)).
% 0.21/0.40 tff(tptp_fun_X_23_type, type, (
% 0.21/0.40 tptp_fun_X_23: ( $i * $i * $i ) > $i)).
% 0.21/0.40 tff(member_type, type, (
% 0.21/0.40 member: ( $i * $i ) > $o)).
% 0.21/0.40 tff(surjective_type, type, (
% 0.21/0.40 surjective: ( $i * $i * $i ) > $o)).
% 0.21/0.40 tff(tptp_fun_E_19_type, type, (
% 0.21/0.40 tptp_fun_E_19: ( $i * $i * $i ) > $i)).
% 0.21/0.40 tff(maps_type, type, (
% 0.21/0.40 maps: ( $i * $i * $i ) > $o)).
% 0.21/0.40 tff(tptp_fun_B_39_type, type, (
% 0.21/0.40 tptp_fun_B_39: $i)).
% 0.21/0.40 tff(subset_type, type, (
% 0.21/0.40 subset: ( $i * $i ) > $o)).
% 0.21/0.40 tff(1,plain,
% 0.21/0.40 (^[F: $i, A: $i, Y: $i] : refl((~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))))) <=> (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(2,plain,
% 0.21/0.40 (![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))))) <=> ![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.40 tff(3,plain,
% 0.21/0.40 (^[F: $i, A: $i, Y: $i] : rewrite((~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))))) <=> (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(4,plain,
% 0.21/0.40 (![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))))) <=> ![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[3])).
% 0.21/0.40 tff(5,plain,
% 0.21/0.40 (![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))))) <=> ![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))),
% 0.21/0.40 inference(transitivity,[status(thm)],[4, 2])).
% 0.21/0.40 tff(6,plain,
% 0.21/0.40 (^[F: $i, A: $i, Y: $i] : trans(monotonicity(rewrite(((~member(Y, image2(F, A))) | (member(tptp_fun_X_23(Y, A, F), A) & apply(F, tptp_fun_X_23(Y, A, F), Y))) <=> ((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))), rewrite((member(Y, image2(F, A)) | ![X: $i] : (~(member(X, A) & apply(F, X, Y)))) <=> (member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))), ((((~member(Y, image2(F, A))) | (member(tptp_fun_X_23(Y, A, F), A) & apply(F, tptp_fun_X_23(Y, A, F), Y))) & (member(Y, image2(F, A)) | ![X: $i] : (~(member(X, A) & apply(F, X, Y))))) <=> (((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y))))) & (member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))))), rewrite((((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y))))) & (member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y))))) <=> (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))), ((((~member(Y, image2(F, A))) | (member(tptp_fun_X_23(Y, A, F), A) & apply(F, tptp_fun_X_23(Y, A, F), Y))) & (member(Y, image2(F, A)) | ![X: $i] : (~(member(X, A) & apply(F, X, Y))))) <=> (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(7,plain,
% 0.21/0.40 (![F: $i, A: $i, Y: $i] : (((~member(Y, image2(F, A))) | (member(tptp_fun_X_23(Y, A, F), A) & apply(F, tptp_fun_X_23(Y, A, F), Y))) & (member(Y, image2(F, A)) | ![X: $i] : (~(member(X, A) & apply(F, X, Y))))) <=> ![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[6])).
% 0.21/0.40 tff(8,plain,
% 0.21/0.40 (![F: $i, A: $i, Y: $i] : (member(Y, image2(F, A)) <=> ?[X: $i] : (member(X, A) & apply(F, X, Y))) <=> ![F: $i, A: $i, Y: $i] : (member(Y, image2(F, A)) <=> ?[X: $i] : (member(X, A) & apply(F, X, Y)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(9,axiom,(![F: $i, A: $i, Y: $i] : (member(Y, image2(F, A)) <=> ?[X: $i] : (member(X, A) & apply(F, X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+1.ax','image2')).
% 0.21/0.40 tff(10,plain,
% 0.21/0.40 (![F: $i, A: $i, Y: $i] : (member(Y, image2(F, A)) <=> ?[X: $i] : (member(X, A) & apply(F, X, Y)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[9, 8])).
% 0.21/0.40 tff(11,plain,(
% 0.21/0.40 ![F: $i, A: $i, Y: $i] : (((~member(Y, image2(F, A))) | (member(tptp_fun_X_23(Y, A, F), A) & apply(F, tptp_fun_X_23(Y, A, F), Y))) & (member(Y, image2(F, A)) | ![X: $i] : (~(member(X, A) & apply(F, X, Y)))))),
% 0.21/0.40 inference(skolemize,[status(sab)],[10])).
% 0.21/0.40 tff(12,plain,
% 0.21/0.40 (![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[11, 7])).
% 0.21/0.40 tff(13,plain,
% 0.21/0.40 (![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[12, 5])).
% 0.21/0.40 tff(14,plain,
% 0.21/0.40 (((~![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))) | (~((~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))))) <=> ((~![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))) | (~((~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(15,plain,
% 0.21/0.40 ((~((~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))) | (~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))) <=> (~((~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(16,plain,
% 0.21/0.40 (((~![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))) | (~((~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))) | (~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))) <=> ((~![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))) | (~((~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))))),
% 0.21/0.40 inference(monotonicity,[status(thm)],[15])).
% 0.21/0.40 tff(17,plain,
% 0.21/0.40 (((~![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))) | (~((~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))) | (~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))) <=> ((~![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))) | (~((~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))))),
% 0.21/0.41 inference(transitivity,[status(thm)],[16, 14])).
% 0.21/0.41 tff(18,plain,
% 0.21/0.41 ((~![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))) | (~((~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))) | (~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(19,plain,
% 0.21/0.41 ((~![F: $i, A: $i, Y: $i] : (~((~((~member(Y, image2(F, A))) | (~((~member(tptp_fun_X_23(Y, A, F), A)) | (~apply(F, tptp_fun_X_23(Y, A, F), Y)))))) | (~(member(Y, image2(F, A)) | ![X: $i] : ((~member(X, A)) | (~apply(F, X, Y)))))))) | (~((~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.21/0.41 tff(20,plain,
% 0.21/0.41 (~((~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[19, 13])).
% 0.21/0.41 tff(21,plain,
% 0.21/0.41 (((~(member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)) | ![X: $i] : ((~member(X, A!40)) | (~apply(F!42, X, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))) | ((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(22,plain,
% 0.21/0.41 ((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[21, 20])).
% 0.21/0.41 tff(23,plain,
% 0.21/0.41 (^[F: $i, A: $i, B: $i] : refl((~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(24,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> ![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[23])).
% 0.21/0.41 tff(25,plain,
% 0.21/0.41 (^[F: $i, A: $i, B: $i] : rewrite((~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(26,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> ![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[25])).
% 0.21/0.41 tff(27,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> ![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.41 inference(transitivity,[status(thm)],[26, 24])).
% 0.21/0.41 tff(28,plain,
% 0.21/0.41 (^[F: $i, A: $i, B: $i] : trans(monotonicity(rewrite(((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) <=> ((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))), trans(monotonicity(trans(monotonicity(quant_intro(proof_bind(^[E: $i] : trans(monotonicity(rewrite((member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))) <=> (~((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))), ((~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))) <=> (~(~((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))), rewrite((~(~((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))) <=> ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))), ((~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))) <=> ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))), (![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))) <=> ![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))), ((member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))) <=> (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))), rewrite((member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))) <=> (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))), ((member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))) <=> (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))), ((surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))))) <=> (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))), rewrite((surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))) <=> (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))), ((surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))))) <=> (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))), ((((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) & (surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))))) <=> (((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y)))))) & (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))), rewrite((((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y)))))) & (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))) <=> (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))), ((((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) & (surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))))) <=> (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(29,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) & (surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))))) <=> ![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[28])).
% 0.21/0.41 tff(30,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y)))) <=> ![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(31,plain,
% 0.21/0.41 (^[F: $i, A: $i, B: $i] : rewrite((surjective(F, A, B) <=> ![Y: $i] : (member(Y, B) => ?[E: $i] : (member(E, A) & apply(F, E, Y)))) <=> (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y)))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(32,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : (member(Y, B) => ?[E: $i] : (member(E, A) & apply(F, E, Y)))) <=> ![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[31])).
% 0.21/0.41 tff(33,axiom,(![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : (member(Y, B) => ?[E: $i] : (member(E, A) & apply(F, E, Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+1.ax','surjective')).
% 0.21/0.41 tff(34,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[33, 32])).
% 0.21/0.41 tff(35,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[34, 30])).
% 0.21/0.41 tff(36,plain,(
% 0.21/0.41 ![F: $i, A: $i, B: $i] : (((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) & (surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))))))),
% 0.21/0.41 inference(skolemize,[status(sab)],[35])).
% 0.21/0.41 tff(37,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[36, 29])).
% 0.21/0.41 tff(38,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[37, 27])).
% 0.21/0.41 tff(39,plain,
% 0.21/0.41 ((~![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))) | (~((~((~surjective(G!41, A!40, image2(F!42, A!40))) | ![Y: $i] : ((~member(Y, image2(F!42, A!40))) | (~((~member(tptp_fun_E_19(Y, A!40, G!41), A!40)) | (~apply(G!41, tptp_fun_E_19(Y, A!40, G!41), Y))))))) | (~(surjective(G!41, A!40, image2(F!42, A!40)) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(40,plain,
% 0.21/0.41 (~((~((~surjective(G!41, A!40, image2(F!42, A!40))) | ![Y: $i] : ((~member(Y, image2(F!42, A!40))) | (~((~member(tptp_fun_E_19(Y, A!40, G!41), A!40)) | (~apply(G!41, tptp_fun_E_19(Y, A!40, G!41), Y))))))) | (~(surjective(G!41, A!40, image2(F!42, A!40)) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[39, 38])).
% 0.21/0.42 tff(41,plain,
% 0.21/0.42 (((~((~surjective(G!41, A!40, image2(F!42, A!40))) | ![Y: $i] : ((~member(Y, image2(F!42, A!40))) | (~((~member(tptp_fun_E_19(Y, A!40, G!41), A!40)) | (~apply(G!41, tptp_fun_E_19(Y, A!40, G!41), Y))))))) | (~(surjective(G!41, A!40, image2(F!42, A!40)) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))))) | (surjective(G!41, A!40, image2(F!42, A!40)) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(42,plain,
% 0.21/0.42 (surjective(G!41, A!40, image2(F!42, A!40)) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[41, 40])).
% 0.21/0.42 tff(43,plain,
% 0.21/0.42 ((subset(image2(F!42, A!40), B!39) & maps(F!42, A!40, B!39) & ![X: $i, Y: $i] : ((~(member(X, A!40) & member(Y, image2(F!42, A!40)))) | (apply(G!41, X, Y) <=> apply(F!42, X, Y))) & (~surjective(G!41, A!40, image2(F!42, A!40)))) <=> (subset(image2(F!42, A!40), B!39) & maps(F!42, A!40, B!39) & ![X: $i, Y: $i] : ((~(member(X, A!40) & member(Y, image2(F!42, A!40)))) | (apply(G!41, X, Y) <=> apply(F!42, X, Y))) & (~surjective(G!41, A!40, image2(F!42, A!40))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(44,plain,
% 0.21/0.42 ((~![F: $i, G: $i, A: $i, B: $i] : ((~subset(image2(F, A), B)) | (~maps(F, A, B)) | (~![X: $i, Y: $i] : ((~(member(X, A) & member(Y, image2(F, A)))) | (apply(G, X, Y) <=> apply(F, X, Y)))) | surjective(G, A, image2(F, A)))) <=> (~![F: $i, G: $i, A: $i, B: $i] : ((~subset(image2(F, A), B)) | (~maps(F, A, B)) | (~![X: $i, Y: $i] : ((~(member(X, A) & member(Y, image2(F, A)))) | (apply(G, X, Y) <=> apply(F, X, Y)))) | surjective(G, A, image2(F, A))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(45,plain,
% 0.21/0.42 ((~![F: $i, G: $i, A: $i, B: $i, C: $i] : ((((maps(F, A, B) & subset(C, B)) & (image2(F, A) = C)) & ![X: $i, Y: $i] : ((member(X, A) & member(Y, C)) => (apply(G, X, Y) <=> apply(F, X, Y)))) => surjective(G, A, C))) <=> (~![F: $i, G: $i, A: $i, B: $i] : ((~subset(image2(F, A), B)) | (~maps(F, A, B)) | (~![X: $i, Y: $i] : ((~(member(X, A) & member(Y, image2(F, A)))) | (apply(G, X, Y) <=> apply(F, X, Y)))) | surjective(G, A, image2(F, A))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(46,axiom,(~![F: $i, G: $i, A: $i, B: $i, C: $i] : ((((maps(F, A, B) & subset(C, B)) & (image2(F, A) = C)) & ![X: $i, Y: $i] : ((member(X, A) & member(Y, C)) => (apply(G, X, Y) <=> apply(F, X, Y)))) => surjective(G, A, C))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thII22')).
% 0.21/0.42 tff(47,plain,
% 0.21/0.42 (~![F: $i, G: $i, A: $i, B: $i] : ((~subset(image2(F, A), B)) | (~maps(F, A, B)) | (~![X: $i, Y: $i] : ((~(member(X, A) & member(Y, image2(F, A)))) | (apply(G, X, Y) <=> apply(F, X, Y)))) | surjective(G, A, image2(F, A)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.21/0.42 tff(48,plain,
% 0.21/0.42 (~![F: $i, G: $i, A: $i, B: $i] : ((~subset(image2(F, A), B)) | (~maps(F, A, B)) | (~![X: $i, Y: $i] : ((~(member(X, A) & member(Y, image2(F, A)))) | (apply(G, X, Y) <=> apply(F, X, Y)))) | surjective(G, A, image2(F, A)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[47, 44])).
% 0.21/0.42 tff(49,plain,
% 0.21/0.42 (~![F: $i, G: $i, A: $i, B: $i] : ((~subset(image2(F, A), B)) | (~maps(F, A, B)) | (~![X: $i, Y: $i] : ((~(member(X, A) & member(Y, image2(F, A)))) | (apply(G, X, Y) <=> apply(F, X, Y)))) | surjective(G, A, image2(F, A)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[48, 44])).
% 0.21/0.42 tff(50,plain,
% 0.21/0.42 (~![F: $i, G: $i, A: $i, B: $i] : ((~subset(image2(F, A), B)) | (~maps(F, A, B)) | (~![X: $i, Y: $i] : ((~(member(X, A) & member(Y, image2(F, A)))) | (apply(G, X, Y) <=> apply(F, X, Y)))) | surjective(G, A, image2(F, A)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[49, 44])).
% 0.21/0.42 tff(51,plain,
% 0.21/0.42 (subset(image2(F!42, A!40), B!39) & maps(F!42, A!40, B!39) & ![X: $i, Y: $i] : ((~(member(X, A!40) & member(Y, image2(F!42, A!40)))) | (apply(G!41, X, Y) <=> apply(F!42, X, Y))) & (~surjective(G!41, A!40, image2(F!42, A!40)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[50, 43])).
% 0.21/0.42 tff(52,plain,
% 0.21/0.42 (~surjective(G!41, A!40, image2(F!42, A!40))),
% 0.21/0.42 inference(and_elim,[status(thm)],[51])).
% 0.21/0.42 tff(53,plain,
% 0.21/0.42 ((~(surjective(G!41, A!40, image2(F!42, A!40)) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))) | surjective(G!41, A!40, image2(F!42, A!40)) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(54,plain,
% 0.21/0.42 ((~(surjective(G!41, A!40, image2(F!42, A!40)) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))))) | (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[53, 52])).
% 0.21/0.42 tff(55,plain,
% 0.21/0.42 (~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[54, 42])).
% 0.21/0.42 tff(56,plain,
% 0.21/0.42 (((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(57,plain,
% 0.21/0.42 (member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[56, 55])).
% 0.21/0.42 tff(58,plain,
% 0.21/0.42 ((~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))) | (~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(59,plain,
% 0.21/0.42 ((~((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))))) | (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.21/0.42 tff(60,plain,
% 0.21/0.42 (~((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[59, 22])).
% 0.21/0.43 tff(61,plain,
% 0.21/0.43 (((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))) | member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(62,plain,
% 0.21/0.43 (member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[61, 60])).
% 0.21/0.43 tff(63,plain,
% 0.21/0.43 (((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))) | ![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(64,plain,
% 0.21/0.43 (![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[63, 55])).
% 0.21/0.43 tff(65,plain,
% 0.21/0.43 (((~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))) | ((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))) <=> ((~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(66,plain,
% 0.21/0.43 ((~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))) | ((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(67,plain,
% 0.21/0.43 ((~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.21/0.43 tff(68,plain,
% 0.21/0.43 (~apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[67, 64, 62])).
% 0.21/0.43 tff(69,plain,
% 0.21/0.43 (((~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))) | apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(70,plain,
% 0.21/0.43 (apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[69, 60])).
% 0.21/0.43 tff(71,plain,
% 0.21/0.43 ((~(apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))) | apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) | (~apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(72,plain,
% 0.21/0.43 (~(apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[71, 70, 68])).
% 0.21/0.43 tff(73,plain,
% 0.21/0.43 (^[X: $i, Y: $i] : refl(((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40)))) <=> ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40)))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(74,plain,
% 0.21/0.43 (![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40)))) <=> ![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[73])).
% 0.21/0.43 tff(75,plain,
% 0.21/0.43 (^[X: $i, Y: $i] : trans(monotonicity(trans(monotonicity(rewrite((member(X, A!40) & member(Y, image2(F!42, A!40))) <=> (~((~member(X, A!40)) | (~member(Y, image2(F!42, A!40)))))), ((~(member(X, A!40) & member(Y, image2(F!42, A!40)))) <=> (~(~((~member(X, A!40)) | (~member(Y, image2(F!42, A!40)))))))), rewrite((~(~((~member(X, A!40)) | (~member(Y, image2(F!42, A!40)))))) <=> ((~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))), ((~(member(X, A!40) & member(Y, image2(F!42, A!40)))) <=> ((~member(X, A!40)) | (~member(Y, image2(F!42, A!40)))))), (((~(member(X, A!40) & member(Y, image2(F!42, A!40)))) | (apply(G!41, X, Y) <=> apply(F!42, X, Y))) <=> (((~member(X, A!40)) | (~member(Y, image2(F!42, A!40)))) | (apply(G!41, X, Y) <=> apply(F!42, X, Y))))), rewrite((((~member(X, A!40)) | (~member(Y, image2(F!42, A!40)))) | (apply(G!41, X, Y) <=> apply(F!42, X, Y))) <=> ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))), (((~(member(X, A!40) & member(Y, image2(F!42, A!40)))) | (apply(G!41, X, Y) <=> apply(F!42, X, Y))) <=> ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(76,plain,
% 0.21/0.43 (![X: $i, Y: $i] : ((~(member(X, A!40) & member(Y, image2(F!42, A!40)))) | (apply(G!41, X, Y) <=> apply(F!42, X, Y))) <=> ![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[75])).
% 0.21/0.43 tff(77,plain,
% 0.21/0.43 (![X: $i, Y: $i] : ((~(member(X, A!40) & member(Y, image2(F!42, A!40)))) | (apply(G!41, X, Y) <=> apply(F!42, X, Y)))),
% 0.21/0.43 inference(and_elim,[status(thm)],[51])).
% 0.21/0.43 tff(78,plain,
% 0.21/0.43 (![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.21/0.43 tff(79,plain,
% 0.21/0.43 (![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[78, 74])).
% 0.21/0.43 tff(80,plain,
% 0.21/0.43 (((~![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))) | ((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))) <=> ((~![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))) | (~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(81,plain,
% 0.21/0.44 (((apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40)))) <=> ((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(82,plain,
% 0.21/0.44 (((~![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))) | ((apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))))) <=> ((~![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))) | ((~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[81])).
% 0.21/0.44 tff(83,plain,
% 0.21/0.44 (((~![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))) | ((apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))))) <=> ((~![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))) | (~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))))),
% 0.21/0.44 inference(transitivity,[status(thm)],[82, 80])).
% 0.21/0.44 tff(84,plain,
% 0.21/0.44 ((~![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))) | ((apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(85,plain,
% 0.21/0.44 ((~![X: $i, Y: $i] : ((apply(G!41, X, Y) <=> apply(F!42, X, Y)) | (~member(X, A!40)) | (~member(Y, image2(F!42, A!40))))) | (~member(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), image2(F!42, A!40))) | (~member(tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), A!40)) | (apply(G!41, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)) <=> apply(F!42, tptp_fun_X_23(tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41), A!40, F!42), tptp_fun_Y_20(image2(F!42, A!40), A!40, G!41)))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.21/0.44 tff(86,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[85, 79, 57, 62, 72])).
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44 % E exiting
%------------------------------------------------------------------------------