%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET776+4 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:26:19 AM UTC 2025
% Result : Theorem 0.22s 0.42s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : SET776+4 : TPTP v9.0.0. Released v2.2.0.
% 0.08/0.13 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n001.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Jun 20 05:46:05 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.22/0.42 % SZS status Theorem
% 0.22/0.42 % SZS output start Proof
% 0.22/0.42 tff(apply_type, type, (
% 0.22/0.42 apply: ( $i * $i * $i ) > $o)).
% 0.22/0.42 tff(tptp_fun_Y2_23_type, type, (
% 0.22/0.42 tptp_fun_Y2_23: $i)).
% 0.22/0.42 tff(tptp_fun_X1_26_type, type, (
% 0.22/0.42 tptp_fun_X1_26: $i)).
% 0.22/0.42 tff(tptp_fun_P_21_type, type, (
% 0.22/0.42 tptp_fun_P_21: $i)).
% 0.22/0.42 tff(tptp_fun_Y1_24_type, type, (
% 0.22/0.42 tptp_fun_Y1_24: $i)).
% 0.22/0.42 tff(tptp_fun_R_20_type, type, (
% 0.22/0.42 tptp_fun_R_20: $i)).
% 0.22/0.42 tff(member_type, type, (
% 0.22/0.42 member: ( $i * $i ) > $o)).
% 0.22/0.42 tff(tptp_fun_E_22_type, type, (
% 0.22/0.42 tptp_fun_E_22: $i)).
% 0.22/0.42 tff(tptp_fun_X2_25_type, type, (
% 0.22/0.42 tptp_fun_X2_25: $i)).
% 0.22/0.42 tff(pre_order_type, type, (
% 0.22/0.42 pre_order: ( $i * $i ) > $o)).
% 0.22/0.42 tff(tptp_fun_Z_17_type, type, (
% 0.22/0.42 tptp_fun_Z_17: ( $i * $i ) > $i)).
% 0.22/0.42 tff(tptp_fun_X_19_type, type, (
% 0.22/0.42 tptp_fun_X_19: ( $i * $i ) > $i)).
% 0.22/0.42 tff(tptp_fun_Y_18_type, type, (
% 0.22/0.42 tptp_fun_Y_18: ( $i * $i ) > $i)).
% 0.22/0.42 tff(tptp_fun_X_16_type, type, (
% 0.22/0.42 tptp_fun_X_16: ( $i * $i ) > $i)).
% 0.22/0.42 tff(1,plain,
% 0.22/0.42 (((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))) <=> (pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(2,plain,
% 0.22/0.42 ((~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))) <=> (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(3,plain,
% 0.22/0.42 ((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) <=> (pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(4,plain,
% 0.22/0.42 (((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))) <=> ((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))))),
% 0.22/0.42 inference(monotonicity,[status(thm)],[3, 2])).
% 0.22/0.42 tff(5,plain,
% 0.22/0.42 (((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))) <=> (pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))))),
% 0.22/0.42 inference(transitivity,[status(thm)],[4, 1])).
% 0.22/0.42 tff(6,plain,
% 0.22/0.42 ((~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))) <=> (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E))))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(7,plain,
% 0.22/0.42 ((~![E: $i, P: $i, R: $i] : ((pre_order(P, E) & ![A: $i, B: $i] : ((member(A, E) & member(B, E)) => (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A))))) => ![X1: $i, X2: $i, Y1: $i, Y2: $i] : ((((member(X1, E) & member(X2, E)) & member(Y1, E)) & member(Y2, E)) => (((apply(R, X1, Y1) & apply(R, X2, Y2)) & apply(P, X1, X2)) => apply(P, Y1, Y2))))) <=> (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E))))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(8,axiom,(~![E: $i, P: $i, R: $i] : ((pre_order(P, E) & ![A: $i, B: $i] : ((member(A, E) & member(B, E)) => (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A))))) => ![X1: $i, X2: $i, Y1: $i, Y2: $i] : ((((member(X1, E) & member(X2, E)) & member(Y1, E)) & member(Y2, E)) => (((apply(R, X1, Y1) & apply(R, X2, Y2)) & apply(P, X1, X2)) => apply(P, Y1, Y2))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thIII12')).
% 0.22/0.42 tff(9,plain,
% 0.22/0.42 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.22/0.42 tff(10,plain,
% 0.22/0.42 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[9, 6])).
% 0.22/0.42 tff(11,plain,
% 0.22/0.42 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[10, 6])).
% 0.22/0.42 tff(12,plain,
% 0.22/0.42 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[11, 6])).
% 0.22/0.42 tff(13,plain,
% 0.22/0.42 (pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[12, 5])).
% 0.22/0.43 tff(14,plain,
% 0.22/0.43 (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))),
% 0.22/0.43 inference(and_elim,[status(thm)],[13])).
% 0.22/0.43 tff(15,plain,
% 0.22/0.43 (member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)),
% 0.22/0.43 inference(or_elim,[status(thm)],[14])).
% 0.22/0.43 tff(16,plain,
% 0.22/0.43 (member(Y1!24, E!22)),
% 0.22/0.43 inference(and_elim,[status(thm)],[15])).
% 0.22/0.43 tff(17,plain,
% 0.22/0.43 (member(X1!26, E!22)),
% 0.22/0.43 inference(and_elim,[status(thm)],[15])).
% 0.22/0.43 tff(18,plain,
% 0.22/0.43 (^[A: $i, B: $i] : refl(((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22))) <=> ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(19,plain,
% 0.22/0.43 (![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22))) <=> ![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[18])).
% 0.22/0.43 tff(20,plain,
% 0.22/0.43 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((member(A, E!22) & member(B, E!22)) <=> (~((~member(A, E!22)) | (~member(B, E!22))))), ((~(member(A, E!22) & member(B, E!22))) <=> (~(~((~member(A, E!22)) | (~member(B, E!22))))))), rewrite((~(~((~member(A, E!22)) | (~member(B, E!22))))) <=> ((~member(A, E!22)) | (~member(B, E!22)))), ((~(member(A, E!22) & member(B, E!22))) <=> ((~member(A, E!22)) | (~member(B, E!22))))), trans(trans(monotonicity(rewrite((apply(P!21, A, B) & apply(P!21, B, A)) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))), ((apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))) <=> (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))), rewrite((apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))) <=> (~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B)))), ((apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))) <=> (~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))))), rewrite((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) <=> (~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B)))), ((apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))) <=> (~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))))), (((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) <=> (((~member(A, E!22)) | (~member(B, E!22))) | (~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B)))))), rewrite((((~member(A, E!22)) | (~member(B, E!22))) | (~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B)))) <=> ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))), (((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) <=> ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))))),
% 0.22/0.43 inference(bind,[status(th)],[])).
% 0.22/0.43 tff(21,plain,
% 0.22/0.43 (![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) <=> ![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))),
% 0.22/0.43 inference(quant_intro,[status(thm)],[20])).
% 0.22/0.43 tff(22,plain,
% 0.22/0.43 (![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))),
% 0.22/0.43 inference(and_elim,[status(thm)],[13])).
% 0.22/0.43 tff(23,plain,
% 0.22/0.43 (![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.22/0.44 tff(24,plain,
% 0.22/0.44 (![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[23, 19])).
% 0.22/0.44 tff(25,plain,
% 0.22/0.44 (((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~member(Y1!24, E!22)) | (~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)))) <=> ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | (~member(Y1!24, E!22)) | (~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(26,plain,
% 0.22/0.44 (((~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)) | (~member(Y1!24, E!22))) <=> ((~member(Y1!24, E!22)) | (~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(27,plain,
% 0.22/0.44 ((~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) <=> (~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24)))),
% 0.22/0.44 inference(rewrite,[status(thm)],[])).
% 0.22/0.44 tff(28,plain,
% 0.22/0.44 (((~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)) | (~member(Y1!24, E!22))) <=> ((~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)) | (~member(Y1!24, E!22)))),
% 0.22/0.44 inference(monotonicity,[status(thm)],[27])).
% 0.22/0.44 tff(29,plain,
% 0.22/0.44 (((~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)) | (~member(Y1!24, E!22))) <=> ((~member(Y1!24, E!22)) | (~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)))),
% 0.22/0.44 inference(transitivity,[status(thm)],[28, 26])).
% 0.22/0.44 tff(30,plain,
% 0.22/0.44 (((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)) | (~member(Y1!24, E!22)))) <=> ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~member(Y1!24, E!22)) | (~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22))))),
% 0.22/0.44 inference(monotonicity,[status(thm)],[29])).
% 0.22/0.44 tff(31,plain,
% 0.22/0.44 (((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)) | (~member(Y1!24, E!22)))) <=> ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | (~member(Y1!24, E!22)) | (~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)))),
% 0.22/0.44 inference(transitivity,[status(thm)],[30, 25])).
% 0.22/0.44 tff(32,plain,
% 0.22/0.44 ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22)) | (~member(Y1!24, E!22)))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(33,plain,
% 0.22/0.44 ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | (~member(Y1!24, E!22)) | (~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))) | (~member(X1!26, E!22))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.22/0.44 tff(34,plain,
% 0.22/0.44 (~(((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[33, 24, 17, 16])).
% 0.22/0.44 tff(35,plain,
% 0.22/0.44 (apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25)),
% 0.22/0.44 inference(or_elim,[status(thm)],[14])).
% 0.22/0.44 tff(36,plain,
% 0.22/0.44 (apply(R!20, X1!26, Y1!24)),
% 0.22/0.44 inference(and_elim,[status(thm)],[35])).
% 0.22/0.44 tff(37,plain,
% 0.22/0.44 ((((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24)) | (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26)))) | (~apply(R!20, X1!26, Y1!24))),
% 0.22/0.44 inference(tautology,[status(thm)],[])).
% 0.22/0.44 tff(38,plain,
% 0.22/0.44 ((((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) <=> apply(R!20, X1!26, Y1!24)) | (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[37, 36])).
% 0.22/0.44 tff(39,plain,
% 0.22/0.44 (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26)))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[38, 34])).
% 0.22/0.44 tff(40,plain,
% 0.22/0.44 (((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) | apply(P!21, Y1!24, X1!26)),
% 0.22/0.44 inference(tautology,[status(thm)],[])).
% 0.22/0.44 tff(41,plain,
% 0.22/0.44 (apply(P!21, Y1!24, X1!26)),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[40, 39])).
% 0.22/0.44 tff(42,plain,
% 0.22/0.44 (^[R: $i, E: $i] : rewrite((~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))))))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(43,plain,
% 0.22/0.44 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[42])).
% 0.22/0.44 tff(44,plain,
% 0.22/0.44 (^[R: $i, E: $i] : refl((~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(45,plain,
% 0.22/0.44 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[44])).
% 0.22/0.44 tff(46,plain,
% 0.22/0.44 (^[R: $i, E: $i] : rewrite((~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))))),
% 0.22/0.44 inference(bind,[status(th)],[])).
% 0.22/0.44 tff(47,plain,
% 0.22/0.44 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.22/0.44 inference(quant_intro,[status(thm)],[46])).
% 0.22/0.44 tff(48,plain,
% 0.22/0.44 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.22/0.44 inference(transitivity,[status(thm)],[47, 45])).
% 0.22/0.44 tff(49,plain,
% 0.22/0.44 (^[R: $i, E: $i] : trans(monotonicity(rewrite(((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) <=> ((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))), trans(monotonicity(rewrite((~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))) <=> (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))), ((pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))) <=> (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))), rewrite((pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))) <=> (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))), ((pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))) <=> (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))), ((((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))) <=> (((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))), rewrite((((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) <=> (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))), ((((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))) <=> (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(50,plain,
% 0.22/0.45 (![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))) <=> ![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[49])).
% 0.22/0.45 tff(51,plain,
% 0.22/0.45 (^[R: $i, E: $i] : rewrite((((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) & (pre_order(R, E) | ((~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))))) <=> (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(52,plain,
% 0.22/0.45 (![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) & (pre_order(R, E) | ((~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[51])).
% 0.22/0.45 tff(53,plain,
% 0.22/0.45 (![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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))))))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(54,plain,
% 0.22/0.45 (^[R: $i, E: $i] : rewrite((pre_order(R, E) <=> (![X: $i] : (member(X, E) => apply(R, X, X)) & ![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))))) <=> (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(55,plain,
% 0.22/0.45 (![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : (member(X, E) => apply(R, X, X)) & ![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] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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))))))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[54])).
% 0.22/0.45 tff(56,axiom,(![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : (member(X, E) => apply(R, X, X)) & ![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+2.ax','pre_order')).
% 0.22/0.45 tff(57,plain,
% 0.22/0.45 (![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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))))))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.22/0.45 tff(58,plain,
% 0.22/0.45 (![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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))))))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[57, 53])).
% 0.22/0.45 tff(59,plain,(
% 0.22/0.45 ![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) & (pre_order(R, E) | ((~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))))),
% 0.22/0.46 inference(skolemize,[status(sab)],[58])).
% 0.22/0.46 tff(60,plain,
% 0.22/0.46 (![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![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)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[59, 52])).
% 0.22/0.46 tff(61,plain,
% 0.22/0.46 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[60, 50])).
% 0.22/0.46 tff(62,plain,
% 0.22/0.46 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[61, 48])).
% 0.22/0.46 tff(63,plain,
% 0.22/0.46 (![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[62, 43])).
% 0.22/0.46 tff(64,plain,
% 0.22/0.46 (((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~((~member(tptp_fun_Z_17(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y))))))))))) <=> ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~((~member(tptp_fun_Z_17(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))))))))))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(65,plain,
% 0.22/0.46 ((~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(P!21, X, Z) | (~member(Y, E!22)) | (~member(Z, E!22)) | (~member(X, E!22)) | (~apply(P!21, X, Y)) | (~apply(P!21, Y, Z)))))))))) <=> (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~((~member(tptp_fun_Z_17(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y))))))))))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(66,plain,
% 0.22/0.46 (((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(P!21, X, Z) | (~member(Y, E!22)) | (~member(Z, E!22)) | (~member(X, E!22)) | (~apply(P!21, X, Y)) | (~apply(P!21, Y, Z))))))))))) <=> ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~((~member(tptp_fun_Z_17(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))))))))))),
% 0.22/0.47 inference(monotonicity,[status(thm)],[65])).
% 0.22/0.47 tff(67,plain,
% 0.22/0.47 (((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(P!21, X, Z) | (~member(Y, E!22)) | (~member(Z, E!22)) | (~member(X, E!22)) | (~apply(P!21, X, Y)) | (~apply(P!21, Y, Z))))))))))) <=> ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~((~member(tptp_fun_Z_17(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))))))))))),
% 0.22/0.47 inference(transitivity,[status(thm)],[66, 64])).
% 0.22/0.47 tff(68,plain,
% 0.22/0.47 ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(P!21, X, Z) | (~member(Y, E!22)) | (~member(Z, E!22)) | (~member(X, E!22)) | (~apply(P!21, X, Y)) | (~apply(P!21, Y, Z))))))))))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(69,plain,
% 0.22/0.47 ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~((~member(tptp_fun_Z_17(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y))))))))))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.22/0.48 tff(70,plain,
% 0.22/0.48 (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~((~member(tptp_fun_Z_17(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))))))))),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[69, 63])).
% 0.22/0.48 tff(71,plain,
% 0.22/0.48 (((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~((~member(tptp_fun_Z_17(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y))))))))) | ((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))))))),
% 0.22/0.48 inference(tautology,[status(thm)],[])).
% 0.22/0.48 tff(72,plain,
% 0.22/0.48 ((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y))))))),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[71, 70])).
% 0.22/0.48 tff(73,plain,
% 0.22/0.48 (pre_order(P!21, E!22)),
% 0.22/0.48 inference(and_elim,[status(thm)],[13])).
% 0.22/0.48 tff(74,plain,
% 0.22/0.48 ((~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))))))) | (~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y))))))),
% 0.22/0.48 inference(tautology,[status(thm)],[])).
% 0.22/0.48 tff(75,plain,
% 0.22/0.48 ((~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))))))) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y))))))),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[74, 73])).
% 0.22/0.48 tff(76,plain,
% 0.22/0.48 (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))))),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[75, 72])).
% 0.22/0.48 tff(77,plain,
% 0.22/0.48 (((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y))))) | ![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))),
% 0.22/0.49 inference(tautology,[status(thm)],[])).
% 0.22/0.49 tff(78,plain,
% 0.22/0.49 (![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))),
% 0.22/0.49 inference(unit_resolution,[status(thm)],[77, 76])).
% 0.22/0.49 tff(79,plain,
% 0.22/0.49 (member(Y2!23, E!22)),
% 0.22/0.49 inference(and_elim,[status(thm)],[15])).
% 0.22/0.49 tff(80,plain,
% 0.22/0.49 (~apply(P!21, Y1!24, Y2!23)),
% 0.22/0.49 inference(or_elim,[status(thm)],[14])).
% 0.22/0.49 tff(81,plain,
% 0.22/0.49 (((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | (apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, Y2!23)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, Y2!23)))),
% 0.22/0.49 inference(rewrite,[status(thm)],[])).
% 0.22/0.49 tff(82,plain,
% 0.22/0.49 (((~apply(P!21, X1!26, Y2!23)) | (~member(X1!26, E!22)) | (~member(Y2!23, E!22)) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~apply(P!21, Y1!24, X1!26))) <=> (apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, Y2!23)))),
% 0.22/0.49 inference(rewrite,[status(thm)],[])).
% 0.22/0.49 tff(83,plain,
% 0.22/0.49 (((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | ((~apply(P!21, X1!26, Y2!23)) | (~member(X1!26, E!22)) | (~member(Y2!23, E!22)) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~apply(P!21, Y1!24, X1!26)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | (apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, Y2!23))))),
% 0.22/0.49 inference(monotonicity,[status(thm)],[82])).
% 0.22/0.49 tff(84,plain,
% 0.22/0.49 (((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | ((~apply(P!21, X1!26, Y2!23)) | (~member(X1!26, E!22)) | (~member(Y2!23, E!22)) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~apply(P!21, Y1!24, X1!26)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, Y2!23)))),
% 0.22/0.49 inference(transitivity,[status(thm)],[83, 81])).
% 0.22/0.49 tff(85,plain,
% 0.22/0.49 ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | ((~apply(P!21, X1!26, Y2!23)) | (~member(X1!26, E!22)) | (~member(Y2!23, E!22)) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~apply(P!21, Y1!24, X1!26)))),
% 0.22/0.49 inference(quant_inst,[status(thm)],[])).
% 0.22/0.49 tff(86,plain,
% 0.22/0.49 ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, Y2!23))),
% 0.22/0.49 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.22/0.49 tff(87,plain,
% 0.22/0.49 (~apply(P!21, X1!26, Y2!23)),
% 0.22/0.49 inference(unit_resolution,[status(thm)],[86, 80, 17, 16, 79, 78, 41])).
% 0.22/0.49 tff(88,plain,
% 0.22/0.49 (member(X2!25, E!22)),
% 0.22/0.49 inference(and_elim,[status(thm)],[15])).
% 0.22/0.49 tff(89,plain,
% 0.22/0.49 (((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~member(Y2!23, E!22)) | (~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)))) <=> ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | (~member(Y2!23, E!22)) | (~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)))),
% 0.22/0.49 inference(rewrite,[status(thm)],[])).
% 0.22/0.49 tff(90,plain,
% 0.22/0.49 (((~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22))) <=> ((~member(Y2!23, E!22)) | (~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)))),
% 0.22/0.49 inference(rewrite,[status(thm)],[])).
% 0.22/0.49 tff(91,plain,
% 0.22/0.49 ((~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) <=> (~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23)))),
% 0.22/0.49 inference(rewrite,[status(thm)],[])).
% 0.22/0.49 tff(92,plain,
% 0.22/0.49 (((~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22))) <=> ((~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)))),
% 0.22/0.49 inference(monotonicity,[status(thm)],[91])).
% 0.22/0.49 tff(93,plain,
% 0.22/0.49 (((~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22))) <=> ((~member(Y2!23, E!22)) | (~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)))),
% 0.22/0.49 inference(transitivity,[status(thm)],[92, 90])).
% 0.22/0.49 tff(94,plain,
% 0.22/0.49 (((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)))) <=> ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~member(Y2!23, E!22)) | (~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22))))),
% 0.22/0.49 inference(monotonicity,[status(thm)],[93])).
% 0.22/0.49 tff(95,plain,
% 0.22/0.49 (((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)))) <=> ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | (~member(Y2!23, E!22)) | (~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)))),
% 0.22/0.49 inference(transitivity,[status(thm)],[94, 89])).
% 0.22/0.49 tff(96,plain,
% 0.22/0.49 ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | ((~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)))),
% 0.22/0.49 inference(quant_inst,[status(thm)],[])).
% 0.22/0.49 tff(97,plain,
% 0.22/0.49 ((~![A: $i, B: $i] : ((~(((~apply(P!21, A, B)) | (~apply(P!21, B, A))) <=> apply(R!20, A, B))) | (~member(A, E!22)) | (~member(B, E!22)))) | (~member(Y2!23, E!22)) | (~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))) | (~member(X2!25, E!22))),
% 0.22/0.49 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.22/0.49 tff(98,plain,
% 0.22/0.49 (~(((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23))),
% 0.22/0.50 inference(unit_resolution,[status(thm)],[97, 24, 88, 79])).
% 0.22/0.50 tff(99,plain,
% 0.22/0.50 (apply(R!20, X2!25, Y2!23)),
% 0.22/0.50 inference(and_elim,[status(thm)],[35])).
% 0.22/0.50 tff(100,plain,
% 0.22/0.50 ((((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23)) | (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25)))) | (~apply(R!20, X2!25, Y2!23))),
% 0.22/0.50 inference(tautology,[status(thm)],[])).
% 0.22/0.50 tff(101,plain,
% 0.22/0.50 ((((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) <=> apply(R!20, X2!25, Y2!23)) | (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))))),
% 0.22/0.50 inference(unit_resolution,[status(thm)],[100, 99])).
% 0.22/0.50 tff(102,plain,
% 0.22/0.50 (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25)))),
% 0.22/0.50 inference(unit_resolution,[status(thm)],[101, 98])).
% 0.22/0.50 tff(103,plain,
% 0.22/0.50 (((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) | apply(P!21, X2!25, Y2!23)),
% 0.22/0.50 inference(tautology,[status(thm)],[])).
% 0.22/0.50 tff(104,plain,
% 0.22/0.50 (apply(P!21, X2!25, Y2!23)),
% 0.22/0.50 inference(unit_resolution,[status(thm)],[103, 102])).
% 0.22/0.50 tff(105,plain,
% 0.22/0.50 (apply(P!21, X1!26, X2!25)),
% 0.22/0.50 inference(and_elim,[status(thm)],[35])).
% 0.22/0.50 tff(106,plain,
% 0.22/0.50 (((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | ((~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23))),
% 0.22/0.50 inference(rewrite,[status(thm)],[])).
% 0.22/0.50 tff(107,plain,
% 0.22/0.50 (((~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | apply(P!21, X1!26, Y2!23) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, X2!25))) <=> ((~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23))),
% 0.22/0.50 inference(rewrite,[status(thm)],[])).
% 0.22/0.50 tff(108,plain,
% 0.22/0.50 (((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | ((~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | apply(P!21, X1!26, Y2!23) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, X2!25)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | ((~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23)))),
% 0.22/0.50 inference(monotonicity,[status(thm)],[107])).
% 0.22/0.50 tff(109,plain,
% 0.22/0.50 (((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | ((~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | apply(P!21, X1!26, Y2!23) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, X2!25)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23))),
% 0.22/0.50 inference(transitivity,[status(thm)],[108, 106])).
% 0.22/0.50 tff(110,plain,
% 0.22/0.50 ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | ((~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | apply(P!21, X1!26, Y2!23) | (~member(X1!26, E!22)) | (~apply(P!21, X1!26, X2!25)))),
% 0.22/0.51 inference(quant_inst,[status(thm)],[])).
% 0.22/0.51 tff(111,plain,
% 0.22/0.51 ((~![X: $i, Y: $i, Z: $i] : ((~apply(P!21, Y, Z)) | (~member(Y, E!22)) | (~member(Z, E!22)) | apply(P!21, X, Z) | (~member(X, E!22)) | (~apply(P!21, X, Y)))) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~member(X2!25, E!22)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23)),
% 0.22/0.51 inference(modus_ponens,[status(thm)],[110, 109])).
% 0.22/0.51 tff(112,plain,
% 0.22/0.51 ($false),
% 0.22/0.51 inference(unit_resolution,[status(thm)],[111, 105, 17, 88, 79, 78, 104, 87])).
% 0.22/0.51 % SZS output end Proof
% 0.22/0.51 % E exiting
%------------------------------------------------------------------------------