%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET795+4 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.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 05:26:22 AM UTC 2025
% Result : Theorem 0.20s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET795+4 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n002.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 05:44:46 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.20/0.39 % SZS status Theorem
% 0.20/0.39 % SZS output start Proof
% 0.20/0.39 tff(apply_type, type, (
% 0.20/0.39 apply: ( $i * $i * $i ) > $o)).
% 0.20/0.39 tff(tptp_fun_M_15_type, type, (
% 0.20/0.39 tptp_fun_M_15: ( $i * $i * $i * $i ) > $i)).
% 0.20/0.39 tff(tptp_fun_B_17_type, type, (
% 0.20/0.39 tptp_fun_B_17: $i)).
% 0.20/0.39 tff(unordered_pair_type, type, (
% 0.20/0.39 unordered_pair: ( $i * $i ) > $i)).
% 0.20/0.39 tff(tptp_fun_A_18_type, type, (
% 0.20/0.39 tptp_fun_A_18: $i)).
% 0.20/0.39 tff(tptp_fun_R_20_type, type, (
% 0.20/0.39 tptp_fun_R_20: $i)).
% 0.20/0.39 tff(tptp_fun_E_19_type, type, (
% 0.20/0.39 tptp_fun_E_19: $i)).
% 0.20/0.39 tff(member_type, type, (
% 0.20/0.39 member: ( $i * $i ) > $o)).
% 0.20/0.39 tff(upper_bound_type, type, (
% 0.20/0.39 upper_bound: ( $i * $i * $i ) > $o)).
% 0.20/0.39 tff(tptp_fun_X_11_type, type, (
% 0.20/0.39 tptp_fun_X_11: ( $i * $i * $i ) > $i)).
% 0.20/0.39 tff(order_type, type, (
% 0.20/0.39 order: ( $i * $i ) > $o)).
% 0.20/0.39 tff(tptp_fun_Z_6_type, type, (
% 0.20/0.39 tptp_fun_Z_6: ( $i * $i ) > $i)).
% 0.20/0.39 tff(tptp_fun_X_8_type, type, (
% 0.20/0.39 tptp_fun_X_8: ( $i * $i ) > $i)).
% 0.20/0.39 tff(tptp_fun_Y_7_type, type, (
% 0.20/0.39 tptp_fun_Y_7: ( $i * $i ) > $i)).
% 0.20/0.39 tff(tptp_fun_Y_4_type, type, (
% 0.20/0.39 tptp_fun_Y_4: ( $i * $i ) > $i)).
% 0.20/0.39 tff(tptp_fun_X_5_type, type, (
% 0.20/0.39 tptp_fun_X_5: ( $i * $i ) > $i)).
% 0.20/0.39 tff(tptp_fun_X_3_type, type, (
% 0.20/0.39 tptp_fun_X_3: ( $i * $i ) > $i)).
% 0.20/0.39 tff(least_upper_bound_type, type, (
% 0.20/0.39 least_upper_bound: ( $i * $i * $i * $i ) > $o)).
% 0.20/0.39 tff(1,plain,
% 0.20/0.39 (^[R: $i, E: $i, M: $i] : refl((~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(2,plain,
% 0.20/0.39 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.39 tff(3,plain,
% 0.20/0.39 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(4,plain,
% 0.20/0.39 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[3])).
% 0.20/0.39 tff(5,plain,
% 0.20/0.39 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.20/0.39 inference(transitivity,[status(thm)],[4, 2])).
% 0.20/0.39 tff(6,plain,
% 0.20/0.39 (^[R: $i, E: $i, M: $i] : trans(monotonicity(rewrite(((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) <=> ((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))), rewrite((upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))) <=> (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))), ((((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))) <=> (((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))), rewrite((((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))) <=> (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))), ((((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))) <=> (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(7,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[6])).
% 0.20/0.40 tff(8,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M))) <=> ![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(9,plain,
% 0.20/0.40 (^[R: $i, E: $i, M: $i] : rewrite((upper_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, X, M))) <=> (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(10,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, X, M))) <=> ![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[9])).
% 0.20/0.40 tff(11,axiom,(![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, X, M)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+3.ax','upper_bound')).
% 0.20/0.40 tff(12,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.20/0.40 tff(13,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.20/0.40 tff(14,plain,(
% 0.20/0.40 ![R: $i, E: $i, M: $i] : (((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[13])).
% 0.20/0.40 tff(15,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.20/0.40 tff(16,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.20/0.40 tff(17,plain,
% 0.20/0.40 ((~![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))) | (~((~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))) | (~(upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), unordered_pair(A!18, B!17), R!20), tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(18,plain,
% 0.20/0.40 (~((~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))) | (~(upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), unordered_pair(A!18, B!17), R!20), tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.20/0.40 tff(19,plain,
% 0.20/0.40 (((~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))) | (~(upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), unordered_pair(A!18, B!17), R!20), tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))))) | ((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(20,plain,
% 0.20/0.40 ((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.20/0.40 tff(21,assumption,(~member(B!17, unordered_pair(A!18, B!17))), introduced(assumption)).
% 0.20/0.40 tff(22,plain,
% 0.20/0.40 (^[X: $i, A: $i, B: $i] : refl((member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))) <=> (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(23,plain,
% 0.20/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))) <=> ![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[22])).
% 0.20/0.40 tff(24,plain,
% 0.20/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))) <=> ![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(25,plain,
% 0.20/0.40 (^[X: $i, A: $i, B: $i] : rewrite((member(X, unordered_pair(A, B)) <=> ((X = A) | (X = B))) <=> (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(26,plain,
% 0.20/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = A) | (X = B))) <=> ![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[25])).
% 0.20/0.40 tff(27,axiom,(![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = A) | (X = B)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','unordered_pair')).
% 0.20/0.40 tff(28,plain,
% 0.20/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.40 tff(29,plain,
% 0.20/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.20/0.40 tff(30,plain,(
% 0.20/0.40 ![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[29])).
% 0.20/0.40 tff(31,plain,
% 0.20/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[30, 23])).
% 0.20/0.40 tff(32,plain,
% 0.20/0.40 (((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(B!17, unordered_pair(A!18, B!17))) <=> ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(B!17, unordered_pair(A!18, B!17)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(33,plain,
% 0.20/0.40 ((member(B!17, unordered_pair(A!18, B!17)) <=> $true) <=> member(B!17, unordered_pair(A!18, B!17))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(34,plain,
% 0.20/0.40 (($true | (B!17 = A!18)) <=> $true),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(35,plain,
% 0.20/0.40 ((B!17 = B!17) <=> $true),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(36,plain,
% 0.20/0.40 (((B!17 = B!17) | (B!17 = A!18)) <=> ($true | (B!17 = A!18))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[35])).
% 0.20/0.40 tff(37,plain,
% 0.20/0.40 (((B!17 = B!17) | (B!17 = A!18)) <=> $true),
% 0.20/0.40 inference(transitivity,[status(thm)],[36, 34])).
% 0.20/0.40 tff(38,plain,
% 0.20/0.40 ((member(B!17, unordered_pair(A!18, B!17)) <=> ((B!17 = B!17) | (B!17 = A!18))) <=> (member(B!17, unordered_pair(A!18, B!17)) <=> $true)),
% 0.20/0.40 inference(monotonicity,[status(thm)],[37])).
% 0.20/0.40 tff(39,plain,
% 0.20/0.40 ((member(B!17, unordered_pair(A!18, B!17)) <=> ((B!17 = B!17) | (B!17 = A!18))) <=> member(B!17, unordered_pair(A!18, B!17))),
% 0.20/0.40 inference(transitivity,[status(thm)],[38, 33])).
% 0.20/0.40 tff(40,plain,
% 0.20/0.40 (((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | (member(B!17, unordered_pair(A!18, B!17)) <=> ((B!17 = B!17) | (B!17 = A!18)))) <=> ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(B!17, unordered_pair(A!18, B!17)))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[39])).
% 0.20/0.40 tff(41,plain,
% 0.20/0.40 (((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | (member(B!17, unordered_pair(A!18, B!17)) <=> ((B!17 = B!17) | (B!17 = A!18)))) <=> ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(B!17, unordered_pair(A!18, B!17)))),
% 0.20/0.40 inference(transitivity,[status(thm)],[40, 32])).
% 0.20/0.40 tff(42,plain,
% 0.20/0.40 ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | (member(B!17, unordered_pair(A!18, B!17)) <=> ((B!17 = B!17) | (B!17 = A!18)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(43,plain,
% 0.20/0.40 ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(B!17, unordered_pair(A!18, B!17))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.20/0.40 tff(44,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[43, 31, 21])).
% 0.20/0.40 tff(45,plain,(member(B!17, unordered_pair(A!18, B!17))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(46,assumption,((~((~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, B!17)))) | (~(upper_bound(B!17, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)))))), introduced(assumption)).
% 0.20/0.40 tff(47,plain,
% 0.20/0.40 ((~![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))) | (~((~((~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, B!17)))) | (~(upper_bound(B!17, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)))))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(48,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[47, 16, 46])).
% 0.20/0.41 tff(49,plain,(~((~((~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, B!17)))) | (~(upper_bound(B!17, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17))))))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(50,plain,
% 0.20/0.41 (((~((~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, B!17)))) | (~(upper_bound(B!17, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)))))) | (upper_bound(B!17, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17))))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(51,plain,
% 0.20/0.41 (upper_bound(B!17, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.20/0.41 tff(52,plain,
% 0.20/0.41 ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | (member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17)) <=> ((tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(53,plain,
% 0.20/0.41 (member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17)) <=> ((tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[52, 31])).
% 0.20/0.41 tff(54,assumption,(~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17))), introduced(assumption)).
% 0.20/0.41 tff(55,plain,
% 0.20/0.41 (((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)) | member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(56,plain,
% 0.20/0.41 (member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[55, 54])).
% 0.20/0.41 tff(57,plain,
% 0.20/0.41 ((~(member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17)) <=> ((tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18)))) | (~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | ((tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(58,plain,
% 0.20/0.41 ((~(member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17)) <=> ((tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18)))) | ((tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[57, 56])).
% 0.20/0.41 tff(59,plain,
% 0.20/0.41 ((tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[58, 53])).
% 0.20/0.41 tff(60,plain,
% 0.20/0.41 (((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)) | (~apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 (~apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[60, 54])).
% 0.20/0.41 tff(62,assumption,(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17), introduced(assumption)).
% 0.20/0.41 tff(63,plain,
% 0.20/0.41 (apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17) <=> apply(R!20, B!17, B!17)),
% 0.20/0.41 inference(monotonicity,[status(thm)],[62])).
% 0.20/0.41 tff(64,plain,
% 0.20/0.41 (apply(R!20, B!17, B!17) <=> apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)),
% 0.20/0.41 inference(symmetry,[status(thm)],[63])).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 (^[R: $i, E: $i] : rewrite((~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[65])).
% 0.20/0.41 tff(67,plain,
% 0.20/0.41 (^[R: $i, E: $i] : refl((~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(68,plain,
% 0.20/0.41 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[67])).
% 0.20/0.41 tff(69,plain,
% 0.20/0.41 (^[R: $i, E: $i] : rewrite((~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(70,plain,
% 0.20/0.41 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[69])).
% 0.20/0.41 tff(71,plain,
% 0.20/0.41 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[70, 68])).
% 0.20/0.41 tff(72,plain,
% 0.20/0.41 (^[R: $i, E: $i] : trans(monotonicity(rewrite(((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) <=> ((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))), trans(monotonicity(rewrite((~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) <=> (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E))))), rewrite((~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))) <=> (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))), ((order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))) <=> (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))), rewrite((order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))) <=> (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))), ((order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))) <=> (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))), ((((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))) <=> (((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))), rewrite((((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) <=> (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))), ((((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))) <=> (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(73,plain,
% 0.20/0.42 (![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))) <=> ![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[72])).
% 0.20/0.42 tff(74,plain,
% 0.20/0.42 (^[R: $i, E: $i] : rewrite((((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | ((~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))))) <=> (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(75,plain,
% 0.20/0.42 (![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | ((~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[74])).
% 0.20/0.42 tff(76,plain,
% 0.20/0.42 (![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) <=> ![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(77,plain,
% 0.20/0.42 (^[R: $i, E: $i] : rewrite((order(R, E) <=> ((![X: $i] : (member(X, E) => apply(R, X, X)) & ![X: $i, Y: $i] : ((member(X, E) & member(Y, E)) => ((apply(R, X, Y) & apply(R, Y, X)) => (X = Y)))) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z))))) <=> (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(78,plain,
% 0.20/0.42 (![R: $i, E: $i] : (order(R, E) <=> ((![X: $i] : (member(X, E) => apply(R, X, X)) & ![X: $i, Y: $i] : ((member(X, E) & member(Y, E)) => ((apply(R, X, Y) & apply(R, Y, X)) => (X = Y)))) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z))))) <=> ![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[77])).
% 0.20/0.42 tff(79,axiom,(![R: $i, E: $i] : (order(R, E) <=> ((![X: $i] : (member(X, E) => apply(R, X, X)) & ![X: $i, Y: $i] : ((member(X, E) & member(Y, E)) => ((apply(R, X, Y) & apply(R, Y, X)) => (X = Y)))) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z)))))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+3.ax','order')).
% 0.20/0.42 tff(80,plain,
% 0.20/0.42 (![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.20/0.42 tff(81,plain,
% 0.20/0.42 (![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[80, 76])).
% 0.20/0.42 tff(82,plain,(
% 0.20/0.42 ![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | ((~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))))),
% 0.20/0.42 inference(skolemize,[status(sab)],[81])).
% 0.20/0.42 tff(83,plain,
% 0.20/0.42 (![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[82, 75])).
% 0.20/0.42 tff(84,plain,
% 0.20/0.42 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[83, 73])).
% 0.20/0.42 tff(85,plain,
% 0.20/0.42 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[84, 71])).
% 0.20/0.42 tff(86,plain,
% 0.20/0.42 (![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[85, 66])).
% 0.20/0.42 tff(87,plain,
% 0.20/0.42 (((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))) <=> ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(88,plain,
% 0.20/0.43 ((~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~member(tptp_fun_Y_4(E!19, R!20), E!19)))) | (~(apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_Z_6(E!19, R!20), E!19)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E!19)) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, X)) | (~member(X, E!19)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))) <=> (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(89,plain,
% 0.20/0.43 (((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~member(tptp_fun_Y_4(E!19, R!20), E!19)))) | (~(apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_Z_6(E!19, R!20), E!19)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E!19)) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, X)) | (~member(X, E!19)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))) <=> ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[88])).
% 0.20/0.43 tff(90,plain,
% 0.20/0.43 (((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~member(tptp_fun_Y_4(E!19, R!20), E!19)))) | (~(apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_Z_6(E!19, R!20), E!19)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E!19)) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, X)) | (~member(X, E!19)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))) <=> ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))))),
% 0.20/0.44 inference(transitivity,[status(thm)],[89, 87])).
% 0.20/0.44 tff(91,plain,
% 0.20/0.44 ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~member(tptp_fun_Y_4(E!19, R!20), E!19)))) | (~(apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_Z_6(E!19, R!20), E!19)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E!19)) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, X)) | (~member(X, E!19)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(92,plain,
% 0.20/0.44 ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.20/0.44 tff(93,plain,
% 0.20/0.44 (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[92, 86])).
% 0.20/0.44 tff(94,plain,
% 0.20/0.44 (((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))) | ((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(95,plain,
% 0.20/0.45 ((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[94, 93])).
% 0.20/0.45 tff(96,plain,
% 0.20/0.45 ((~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | least_upper_bound(B, unordered_pair(A, B), R, E))) <=> (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | least_upper_bound(B, unordered_pair(A, B), R, E)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(97,plain,
% 0.20/0.45 ((~![R: $i, E: $i, A: $i, B: $i] : ((((order(R, E) & member(A, E)) & member(B, E)) & apply(R, A, B)) => least_upper_bound(B, unordered_pair(A, B), R, E))) <=> (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | least_upper_bound(B, unordered_pair(A, B), R, E)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(98,axiom,(~![R: $i, E: $i, A: $i, B: $i] : ((((order(R, E) & member(A, E)) & member(B, E)) & apply(R, A, B)) => least_upper_bound(B, unordered_pair(A, B), R, E))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thIV7')).
% 0.20/0.45 tff(99,plain,
% 0.20/0.45 (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | least_upper_bound(B, unordered_pair(A, B), R, E))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.20/0.45 tff(100,plain,
% 0.20/0.45 (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | least_upper_bound(B, unordered_pair(A, B), R, E))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[99, 96])).
% 0.20/0.45 tff(101,plain,
% 0.20/0.45 (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | least_upper_bound(B, unordered_pair(A, B), R, E))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[100, 96])).
% 0.20/0.45 tff(102,plain,
% 0.20/0.45 (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | least_upper_bound(B, unordered_pair(A, B), R, E))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[101, 96])).
% 0.20/0.45 tff(103,plain,(
% 0.20/0.45 ~((~(order(R!20, E!19) & member(A!18, E!19) & member(B!17, E!19) & apply(R!20, A!18, B!17))) | least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19))),
% 0.20/0.45 inference(skolemize,[status(sab)],[102])).
% 0.20/0.45 tff(104,plain,
% 0.20/0.45 (order(R!20, E!19) & member(A!18, E!19) & member(B!17, E!19) & apply(R!20, A!18, B!17)),
% 0.20/0.45 inference(or_elim,[status(thm)],[103])).
% 0.20/0.45 tff(105,plain,
% 0.20/0.45 (order(R!20, E!19)),
% 0.20/0.45 inference(and_elim,[status(thm)],[104])).
% 0.20/0.45 tff(106,plain,
% 0.20/0.45 ((~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))) | (~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(107,plain,
% 0.20/0.45 ((~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[106, 105])).
% 0.20/0.45 tff(108,plain,
% 0.20/0.45 (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[107, 95])).
% 0.20/0.45 tff(109,plain,
% 0.20/0.45 (((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))) | ![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(110,plain,
% 0.20/0.45 (![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[109, 108])).
% 0.20/0.45 tff(111,plain,
% 0.20/0.45 (member(B!17, E!19)),
% 0.20/0.45 inference(and_elim,[status(thm)],[104])).
% 0.20/0.45 tff(112,plain,
% 0.20/0.45 (((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | ((~member(B!17, E!19)) | apply(R!20, B!17, B!17))) <=> ((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~member(B!17, E!19)) | apply(R!20, B!17, B!17))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(113,plain,
% 0.20/0.45 ((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | ((~member(B!17, E!19)) | apply(R!20, B!17, B!17))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(114,plain,
% 0.20/0.45 ((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~member(B!17, E!19)) | apply(R!20, B!17, B!17)),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.20/0.45 tff(115,plain,
% 0.20/0.45 (apply(R!20, B!17, B!17)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[114, 111, 110])).
% 0.20/0.45 tff(116,plain,
% 0.20/0.45 (apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[115, 64])).
% 0.20/0.45 tff(117,assumption,(~apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)), introduced(assumption)).
% 0.20/0.45 tff(118,plain,
% 0.20/0.45 ($false),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[117, 116])).
% 0.20/0.45 tff(119,plain,((~(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17)) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.45 tff(120,plain,
% 0.20/0.45 (~(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[119, 61])).
% 0.20/0.45 tff(121,plain,
% 0.20/0.45 ((~((tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18))) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18)),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(122,plain,
% 0.20/0.45 (tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20) = A!18),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[121, 120, 59])).
% 0.20/0.45 tff(123,plain,
% 0.20/0.45 (apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17) <=> apply(R!20, A!18, B!17)),
% 0.20/0.45 inference(monotonicity,[status(thm)],[122])).
% 0.20/0.45 tff(124,plain,
% 0.20/0.45 (apply(R!20, A!18, B!17) <=> apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)),
% 0.20/0.45 inference(symmetry,[status(thm)],[123])).
% 0.20/0.45 tff(125,plain,
% 0.20/0.45 (apply(R!20, A!18, B!17)),
% 0.20/0.45 inference(and_elim,[status(thm)],[104])).
% 0.20/0.45 tff(126,plain,
% 0.20/0.45 (apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.20/0.45 tff(127,plain,
% 0.20/0.45 ($false),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[61, 126])).
% 0.20/0.45 tff(128,plain,((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.45 tff(129,plain,
% 0.20/0.45 ((~(upper_bound(B!17, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17))))) | upper_bound(B!17, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17)))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(130,plain,
% 0.20/0.45 ((~(upper_bound(B!17, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_X_11(B!17, unordered_pair(A!18, B!17), R!20), B!17))))) | upper_bound(B!17, R!20, unordered_pair(A!18, B!17))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[129, 128])).
% 0.20/0.45 tff(131,plain,
% 0.20/0.45 (upper_bound(B!17, R!20, unordered_pair(A!18, B!17))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[130, 51])).
% 0.20/0.45 tff(132,plain,
% 0.20/0.45 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(133,plain,
% 0.20/0.45 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[132])).
% 0.20/0.45 tff(134,plain,
% 0.20/0.45 (^[A: $i, X: $i, R: $i, E: $i] : refl((~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(135,plain,
% 0.20/0.45 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[134])).
% 0.20/0.45 tff(136,plain,
% 0.20/0.45 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(137,plain,
% 0.20/0.45 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[136])).
% 0.20/0.45 tff(138,plain,
% 0.20/0.45 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.20/0.45 inference(transitivity,[status(thm)],[137, 135])).
% 0.20/0.45 tff(139,plain,
% 0.20/0.45 (^[A: $i, X: $i, R: $i, E: $i] : trans(monotonicity(rewrite(((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) <=> ((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))), trans(monotonicity(rewrite((~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))) <=> (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))), ((least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))) <=> (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))), rewrite((least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))) <=> (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))), ((least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))) <=> (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))), ((((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))) <=> (((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))), rewrite((((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) <=> (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))), ((((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))) <=> (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))))),
% 0.20/0.46 inference(bind,[status(th)],[])).
% 0.20/0.46 tff(140,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.20/0.46 inference(quant_intro,[status(thm)],[139])).
% 0.20/0.46 tff(141,plain,
% 0.20/0.46 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | ((~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))))) <=> (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))))),
% 0.20/0.46 inference(bind,[status(th)],[])).
% 0.20/0.46 tff(142,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | ((~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))))),
% 0.20/0.46 inference(quant_intro,[status(thm)],[141])).
% 0.20/0.46 tff(143,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) <=> ![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(144,plain,
% 0.20/0.46 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((least_upper_bound(A, X, R, E) <=> ((member(A, X) & upper_bound(A, R, X)) & ![M: $i] : ((member(M, E) & upper_bound(M, R, X)) => apply(R, A, M)))) <=> (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))))),
% 0.20/0.46 inference(bind,[status(th)],[])).
% 0.20/0.46 tff(145,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> ((member(A, X) & upper_bound(A, R, X)) & ![M: $i] : ((member(M, E) & upper_bound(M, R, X)) => apply(R, A, M)))) <=> ![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M))))),
% 0.20/0.46 inference(quant_intro,[status(thm)],[144])).
% 0.20/0.46 tff(146,axiom,(![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> ((member(A, X) & upper_bound(A, R, X)) & ![M: $i] : ((member(M, E) & upper_bound(M, R, X)) => apply(R, A, M))))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+3.ax','least_upper_bound')).
% 0.20/0.46 tff(147,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[146, 145])).
% 0.20/0.46 tff(148,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[147, 143])).
% 0.20/0.46 tff(149,plain,(
% 0.20/0.46 ![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | ((~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))))),
% 0.20/0.46 inference(skolemize,[status(sab)],[148])).
% 0.20/0.46 tff(150,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[149, 142])).
% 0.20/0.46 tff(151,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[150, 140])).
% 0.20/0.46 tff(152,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[151, 138])).
% 0.20/0.46 tff(153,plain,
% 0.20/0.46 (![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[152, 133])).
% 0.20/0.46 tff(154,plain,
% 0.20/0.46 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, B!17, M) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, B!17, M) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17))))))))))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(155,plain,
% 0.20/0.46 ((~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~(apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | (~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : (apply(R!20, B!17, M) | (~member(M, E!19)) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17))))))))))) <=> (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, B!17, M) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(156,plain,
% 0.20/0.46 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~(apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | (~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : (apply(R!20, B!17, M) | (~member(M, E!19)) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, B!17, M) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17))))))))))))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[155])).
% 0.20/0.46 tff(157,plain,
% 0.20/0.46 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~(apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | (~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : (apply(R!20, B!17, M) | (~member(M, E!19)) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, B!17, M) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17))))))))))))),
% 0.20/0.47 inference(transitivity,[status(thm)],[156, 154])).
% 0.20/0.47 tff(158,plain,
% 0.20/0.47 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~(apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | (~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : (apply(R!20, B!17, M) | (~member(M, E!19)) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))),
% 0.20/0.47 inference(quant_inst,[status(thm)],[])).
% 0.20/0.47 tff(159,plain,
% 0.20/0.47 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, B!17, M) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[158, 157])).
% 0.20/0.47 tff(160,plain,
% 0.20/0.47 (~((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, B!17, M) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17))))))))))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[159, 153])).
% 0.20/0.47 tff(161,plain,
% 0.20/0.47 (((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | (~((~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(B!17, unordered_pair(A!18, B!17))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, B!17, M) | (~upper_bound(M, R!20, unordered_pair(A!18, B!17)))))))))) | (least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(162,plain,
% 0.20/0.47 (least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[161, 160])).
% 0.20/0.47 tff(163,plain,
% 0.20/0.47 (~least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19)),
% 0.20/0.47 inference(or_elim,[status(thm)],[103])).
% 0.20/0.47 tff(164,plain,
% 0.20/0.47 ((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(165,plain,
% 0.20/0.47 ((~(least_upper_bound(B!17, unordered_pair(A!18, B!17), R!20, E!19) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))))) | (~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[164, 163])).
% 0.20/0.47 tff(166,plain,
% 0.20/0.47 ((~upper_bound(B!17, R!20, unordered_pair(A!18, B!17))) | (~member(B!17, unordered_pair(A!18, B!17))) | (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[165, 162])).
% 0.20/0.47 tff(167,plain,
% 0.20/0.47 (~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[166, 131, 45])).
% 0.20/0.47 tff(168,plain,
% 0.20/0.47 (((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))) | upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(169,plain,
% 0.20/0.47 (upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[168, 167])).
% 0.20/0.47 tff(170,plain,
% 0.20/0.47 ((~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))) | (~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(171,plain,
% 0.20/0.47 ((~((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[170, 169])).
% 0.20/0.47 tff(172,plain,
% 0.20/0.47 (![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[171, 20])).
% 0.20/0.47 tff(173,plain,
% 0.20/0.47 (((~upper_bound(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17), E!19)) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))) | (~apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(174,plain,
% 0.20/0.47 (~apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[173, 167])).
% 0.20/0.47 tff(175,plain,
% 0.20/0.47 (((~![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))) | ((~member(B!17, unordered_pair(A!18, B!17))) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))) <=> ((~![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))) | (~member(B!17, unordered_pair(A!18, B!17))) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(176,plain,
% 0.20/0.47 ((~![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))) | ((~member(B!17, unordered_pair(A!18, B!17))) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))),
% 0.20/0.47 inference(quant_inst,[status(thm)],[])).
% 0.20/0.47 tff(177,plain,
% 0.20/0.47 ((~![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, X, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17)))) | (~member(B!17, unordered_pair(A!18, B!17))) | apply(R!20, B!17, tptp_fun_M_15(E!19, R!20, unordered_pair(A!18, B!17), B!17))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[176, 175])).
% 0.20/0.47 tff(178,plain,
% 0.20/0.47 ($false),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[177, 45, 174, 172])).
% 0.20/0.47 % SZS output end Proof
% 0.20/0.47 % E exiting
%------------------------------------------------------------------------------