%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU225+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n014.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 05:27:18 AM UTC 2025
% Result : Theorem 0.58s 0.75s
% Output : Proof 0.61s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SEU225+1 : TPTP v9.0.0. Released v3.3.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.32 % Computer : n014.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Fri Jun 20 06:22:54 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.58/0.75 % SZS status Theorem
% 0.58/0.75 % SZS output start Proof
% 0.58/0.75 tff(apply_type, type, (
% 0.58/0.75 apply: ( $i * $i ) > $i)).
% 0.58/0.75 tff(tptp_fun_B_11_type, type, (
% 0.58/0.75 tptp_fun_B_11: $i)).
% 0.58/0.75 tff(tptp_fun_C_10_type, type, (
% 0.58/0.75 tptp_fun_C_10: $i)).
% 0.58/0.75 tff(relation_dom_restriction_type, type, (
% 0.58/0.75 relation_dom_restriction: ( $i * $i ) > $i)).
% 0.58/0.75 tff(tptp_fun_A_12_type, type, (
% 0.58/0.75 tptp_fun_A_12: $i)).
% 0.58/0.75 tff(empty_set_type, type, (
% 0.58/0.75 empty_set: $i)).
% 0.58/0.75 tff(in_type, type, (
% 0.58/0.75 in: ( $i * $i ) > $o)).
% 0.58/0.75 tff(relation_dom_type, type, (
% 0.58/0.75 relation_dom: $i > $i)).
% 0.58/0.75 tff(ordered_pair_type, type, (
% 0.58/0.75 ordered_pair: ( $i * $i ) > $i)).
% 0.58/0.75 tff(function_type, type, (
% 0.58/0.75 function: $i > $o)).
% 0.58/0.75 tff(relation_type, type, (
% 0.58/0.75 relation: $i > $o)).
% 0.58/0.75 tff(set_intersection2_type, type, (
% 0.58/0.75 set_intersection2: ( $i * $i ) > $i)).
% 0.58/0.75 tff(tptp_fun_D_9_type, type, (
% 0.58/0.75 tptp_fun_D_9: ( $i * $i ) > $i)).
% 0.58/0.75 tff(1,plain,
% 0.58/0.75 ((empty_set = apply(C!10, B!11)) <=> (apply(C!10, B!11) = empty_set)),
% 0.58/0.75 inference(commutativity,[status(thm)],[])).
% 0.58/0.75 tff(2,plain,
% 0.58/0.75 ((~((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)) | (~(relation(C!10) & function(C!10))) | (~in(B!11, A!12)))) <=> (~((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)) | (~(relation(C!10) & function(C!10))) | (~in(B!11, A!12))))),
% 0.58/0.75 inference(rewrite,[status(thm)],[])).
% 0.58/0.75 tff(3,plain,
% 0.58/0.75 ((~![A: $i, B: $i, C: $i] : ((apply(relation_dom_restriction(C, A), B) = apply(C, B)) | (~(relation(C) & function(C))) | (~in(B, A)))) <=> (~![A: $i, B: $i, C: $i] : ((apply(relation_dom_restriction(C, A), B) = apply(C, B)) | (~(relation(C) & function(C))) | (~in(B, A))))),
% 0.58/0.75 inference(rewrite,[status(thm)],[])).
% 0.58/0.75 tff(4,plain,
% 0.58/0.75 ((~![A: $i, B: $i, C: $i] : ((relation(C) & function(C)) => (in(B, A) => (apply(relation_dom_restriction(C, A), B) = apply(C, B))))) <=> (~![A: $i, B: $i, C: $i] : ((apply(relation_dom_restriction(C, A), B) = apply(C, B)) | (~(relation(C) & function(C))) | (~in(B, A))))),
% 0.58/0.75 inference(rewrite,[status(thm)],[])).
% 0.58/0.75 tff(5,axiom,(~![A: $i, B: $i, C: $i] : ((relation(C) & function(C)) => (in(B, A) => (apply(relation_dom_restriction(C, A), B) = apply(C, B))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t72_funct_1')).
% 0.58/0.75 tff(6,plain,
% 0.58/0.75 (~![A: $i, B: $i, C: $i] : ((apply(relation_dom_restriction(C, A), B) = apply(C, B)) | (~(relation(C) & function(C))) | (~in(B, A)))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.58/0.75 tff(7,plain,
% 0.58/0.75 (~![A: $i, B: $i, C: $i] : ((apply(relation_dom_restriction(C, A), B) = apply(C, B)) | (~(relation(C) & function(C))) | (~in(B, A)))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[6, 3])).
% 0.58/0.75 tff(8,plain,
% 0.58/0.75 (~![A: $i, B: $i, C: $i] : ((apply(relation_dom_restriction(C, A), B) = apply(C, B)) | (~(relation(C) & function(C))) | (~in(B, A)))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.58/0.75 tff(9,plain,
% 0.58/0.75 (~![A: $i, B: $i, C: $i] : ((apply(relation_dom_restriction(C, A), B) = apply(C, B)) | (~(relation(C) & function(C))) | (~in(B, A)))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[8, 3])).
% 0.58/0.75 tff(10,plain,(
% 0.58/0.75 ~((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)) | (~(relation(C!10) & function(C!10))) | (~in(B!11, A!12)))),
% 0.58/0.75 inference(skolemize,[status(sab)],[9])).
% 0.58/0.75 tff(11,plain,
% 0.58/0.75 (~((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)) | (~(relation(C!10) & function(C!10))) | (~in(B!11, A!12)))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.58/0.75 tff(12,plain,
% 0.58/0.75 (relation(C!10) & function(C!10)),
% 0.58/0.75 inference(or_elim,[status(thm)],[11])).
% 0.58/0.75 tff(13,plain,
% 0.58/0.75 (function(C!10)),
% 0.58/0.75 inference(and_elim,[status(thm)],[12])).
% 0.58/0.75 tff(14,plain,
% 0.58/0.75 (relation(C!10)),
% 0.58/0.75 inference(and_elim,[status(thm)],[12])).
% 0.58/0.75 tff(15,plain,
% 0.58/0.75 (^[A: $i, B: $i] : refl(((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B)))))) <=> ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B)))))))),
% 0.58/0.75 inference(bind,[status(th)],[])).
% 0.58/0.75 tff(16,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B)))))) <=> ![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))),
% 0.58/0.75 inference(quant_intro,[status(thm)],[15])).
% 0.58/0.75 tff(17,plain,
% 0.58/0.75 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(A) & function(A)) <=> (~((~relation(A)) | (~function(A))))), ((~(relation(A) & function(A))) <=> (~(~((~relation(A)) | (~function(A))))))), rewrite((~(~((~relation(A)) | (~function(A))))) <=> ((~relation(A)) | (~function(A)))), ((~(relation(A) & function(A))) <=> ((~relation(A)) | (~function(A))))), rewrite((relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B))) <=> (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B)))))), (((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B)))) <=> (((~relation(A)) | (~function(A))) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B)))))))), rewrite((((~relation(A)) | (~function(A))) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B)))))) <=> ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))), (((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B)))) <=> ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))))),
% 0.58/0.75 inference(bind,[status(th)],[])).
% 0.58/0.75 tff(18,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B)))) <=> ![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))),
% 0.58/0.75 inference(quant_intro,[status(thm)],[17])).
% 0.58/0.75 tff(19,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B)))) <=> ![A: $i, B: $i] : ((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B))))),
% 0.58/0.75 inference(rewrite,[status(thm)],[])).
% 0.58/0.75 tff(20,plain,
% 0.58/0.75 (^[A: $i, B: $i] : rewrite(((relation(A) & function(A)) => (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B)))) <=> ((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B)))))),
% 0.58/0.75 inference(bind,[status(th)],[])).
% 0.58/0.75 tff(21,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((relation(A) & function(A)) => (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B)))) <=> ![A: $i, B: $i] : ((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B))))),
% 0.58/0.75 inference(quant_intro,[status(thm)],[20])).
% 0.58/0.75 tff(22,axiom,(![A: $i, B: $i] : ((relation(A) & function(A)) => (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','fc4_funct_1')).
% 0.58/0.75 tff(23,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B))))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.58/0.75 tff(24,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B))))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[23, 19])).
% 0.58/0.75 tff(25,plain,(
% 0.58/0.75 ![A: $i, B: $i] : ((~(relation(A) & function(A))) | (relation(relation_dom_restriction(A, B)) & function(relation_dom_restriction(A, B))))),
% 0.58/0.75 inference(skolemize,[status(sab)],[24])).
% 0.58/0.75 tff(26,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[25, 18])).
% 0.58/0.75 tff(27,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[26, 16])).
% 0.58/0.75 tff(28,plain,
% 0.58/0.75 (((~![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))) | ((~function(C!10)) | (~relation(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))))) <=> ((~![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))) | (~function(C!10)) | (~relation(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))))),
% 0.58/0.75 inference(rewrite,[status(thm)],[])).
% 0.58/0.75 tff(29,plain,
% 0.58/0.75 (((~relation(C!10)) | (~function(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12)))))) <=> ((~function(C!10)) | (~relation(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))))),
% 0.58/0.75 inference(rewrite,[status(thm)],[])).
% 0.58/0.75 tff(30,plain,
% 0.58/0.75 ((~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))) <=> (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12)))))),
% 0.58/0.75 inference(rewrite,[status(thm)],[])).
% 0.58/0.75 tff(31,plain,
% 0.58/0.75 (((~relation(C!10)) | (~function(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12)))))) <=> ((~relation(C!10)) | (~function(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))))),
% 0.58/0.75 inference(monotonicity,[status(thm)],[30])).
% 0.58/0.75 tff(32,plain,
% 0.58/0.75 (((~relation(C!10)) | (~function(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12)))))) <=> ((~function(C!10)) | (~relation(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))))),
% 0.58/0.75 inference(transitivity,[status(thm)],[31, 29])).
% 0.58/0.75 tff(33,plain,
% 0.58/0.75 (((~![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))) | ((~relation(C!10)) | (~function(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))))) <=> ((~![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))) | ((~function(C!10)) | (~relation(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12)))))))),
% 0.58/0.75 inference(monotonicity,[status(thm)],[32])).
% 0.58/0.75 tff(34,plain,
% 0.58/0.75 (((~![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))) | ((~relation(C!10)) | (~function(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))))) <=> ((~![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))) | (~function(C!10)) | (~relation(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))))),
% 0.58/0.75 inference(transitivity,[status(thm)],[33, 28])).
% 0.58/0.75 tff(35,plain,
% 0.58/0.75 ((~![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))) | ((~relation(C!10)) | (~function(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))))),
% 0.58/0.75 inference(quant_inst,[status(thm)],[])).
% 0.58/0.75 tff(36,plain,
% 0.58/0.75 ((~![A: $i, B: $i] : ((~relation(A)) | (~function(A)) | (~((~relation(relation_dom_restriction(A, B))) | (~function(relation_dom_restriction(A, B))))))) | (~function(C!10)) | (~relation(C!10)) | (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12)))))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.58/0.75 tff(37,plain,
% 0.58/0.75 (~((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))))),
% 0.58/0.75 inference(unit_resolution,[status(thm)],[36, 27, 14, 13])).
% 0.58/0.75 tff(38,plain,
% 0.58/0.75 (((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12)))) | function(relation_dom_restriction(C!10, A!12))),
% 0.58/0.75 inference(tautology,[status(thm)],[])).
% 0.58/0.75 tff(39,plain,
% 0.58/0.75 (function(relation_dom_restriction(C!10, A!12))),
% 0.58/0.75 inference(unit_resolution,[status(thm)],[38, 37])).
% 0.58/0.75 tff(40,plain,
% 0.58/0.75 (^[A: $i, B: $i] : refl(((~relation(A)) | relation(relation_dom_restriction(A, B))) <=> ((~relation(A)) | relation(relation_dom_restriction(A, B))))),
% 0.58/0.75 inference(bind,[status(th)],[])).
% 0.58/0.75 tff(41,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B))) <=> ![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))),
% 0.58/0.75 inference(quant_intro,[status(thm)],[40])).
% 0.58/0.75 tff(42,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B))) <=> ![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))),
% 0.58/0.75 inference(rewrite,[status(thm)],[])).
% 0.58/0.75 tff(43,plain,
% 0.58/0.75 (^[A: $i, B: $i] : rewrite((relation(A) => relation(relation_dom_restriction(A, B))) <=> ((~relation(A)) | relation(relation_dom_restriction(A, B))))),
% 0.58/0.75 inference(bind,[status(th)],[])).
% 0.58/0.75 tff(44,plain,
% 0.58/0.75 (![A: $i, B: $i] : (relation(A) => relation(relation_dom_restriction(A, B))) <=> ![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))),
% 0.58/0.75 inference(quant_intro,[status(thm)],[43])).
% 0.58/0.75 tff(45,axiom,(![A: $i, B: $i] : (relation(A) => relation(relation_dom_restriction(A, B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k7_relat_1')).
% 0.58/0.75 tff(46,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.58/0.75 tff(47,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.58/0.75 tff(48,plain,(
% 0.58/0.75 ![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))),
% 0.58/0.75 inference(skolemize,[status(sab)],[47])).
% 0.58/0.75 tff(49,plain,
% 0.58/0.75 (![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[48, 41])).
% 0.58/0.75 tff(50,plain,
% 0.58/0.75 (((~![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))) | ((~relation(C!10)) | relation(relation_dom_restriction(C!10, A!12)))) <=> ((~![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))) | (~relation(C!10)) | relation(relation_dom_restriction(C!10, A!12)))),
% 0.58/0.75 inference(rewrite,[status(thm)],[])).
% 0.58/0.75 tff(51,plain,
% 0.58/0.75 ((~![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))) | ((~relation(C!10)) | relation(relation_dom_restriction(C!10, A!12)))),
% 0.58/0.75 inference(quant_inst,[status(thm)],[])).
% 0.58/0.75 tff(52,plain,
% 0.58/0.75 ((~![A: $i, B: $i] : ((~relation(A)) | relation(relation_dom_restriction(A, B)))) | (~relation(C!10)) | relation(relation_dom_restriction(C!10, A!12))),
% 0.58/0.75 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.58/0.75 tff(53,plain,
% 0.58/0.75 (relation(relation_dom_restriction(C!10, A!12))),
% 0.58/0.75 inference(unit_resolution,[status(thm)],[52, 49, 14])).
% 0.58/0.75 tff(54,plain,
% 0.58/0.75 (^[A: $i] : refl(((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))) <=> ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))))),
% 0.58/0.76 inference(bind,[status(th)],[])).
% 0.58/0.76 tff(55,plain,
% 0.58/0.76 (![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))) <=> ![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))),
% 0.58/0.76 inference(quant_intro,[status(thm)],[54])).
% 0.58/0.76 tff(56,plain,
% 0.58/0.76 (^[A: $i] : rewrite(((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))) <=> ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))))),
% 0.58/0.76 inference(bind,[status(th)],[])).
% 0.58/0.76 tff(57,plain,
% 0.58/0.76 (![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))) <=> ![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))),
% 0.58/0.76 inference(quant_intro,[status(thm)],[56])).
% 0.58/0.76 tff(58,plain,
% 0.58/0.76 (![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))) <=> ![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))),
% 0.58/0.76 inference(transitivity,[status(thm)],[57, 55])).
% 0.58/0.76 tff(59,plain,
% 0.58/0.76 (^[A: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(A) & function(A)) <=> (~((~relation(A)) | (~function(A))))), ((~(relation(A) & function(A))) <=> (~(~((~relation(A)) | (~function(A))))))), rewrite((~(~((~relation(A)) | (~function(A))))) <=> ((~relation(A)) | (~function(A)))), ((~(relation(A) & function(A))) <=> ((~relation(A)) | (~function(A))))), quant_intro(proof_bind(^[B: $i, C: $i] : rewrite((((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))) <=> (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))), (![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))) <=> ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))), (((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))) <=> (((~relation(A)) | (~function(A))) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))))), rewrite((((~relation(A)) | (~function(A))) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))) <=> ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))), (((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))) <=> ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))))),
% 0.58/0.76 inference(bind,[status(th)],[])).
% 0.58/0.76 tff(60,plain,
% 0.58/0.76 (![A: $i] : ((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))) <=> ![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))),
% 0.58/0.76 inference(quant_intro,[status(thm)],[59])).
% 0.58/0.76 tff(61,plain,
% 0.58/0.76 (![A: $i] : ((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))) <=> ![A: $i] : ((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))),
% 0.58/0.76 inference(rewrite,[status(thm)],[])).
% 0.58/0.76 tff(62,plain,
% 0.58/0.76 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i, C: $i] : rewrite(((in(B, relation_dom(A)) => ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & ((~in(B, relation_dom(A))) => ((C = apply(A, B)) <=> (C = empty_set)))) <=> (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))), (![B: $i, C: $i] : ((in(B, relation_dom(A)) => ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & ((~in(B, relation_dom(A))) => ((C = apply(A, B)) <=> (C = empty_set)))) <=> ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))), (((relation(A) & function(A)) => ![B: $i, C: $i] : ((in(B, relation_dom(A)) => ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & ((~in(B, relation_dom(A))) => ((C = apply(A, B)) <=> (C = empty_set))))) <=> ((relation(A) & function(A)) => ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))))), rewrite(((relation(A) & function(A)) => ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set))))) <=> ((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))), (((relation(A) & function(A)) => ![B: $i, C: $i] : ((in(B, relation_dom(A)) => ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & ((~in(B, relation_dom(A))) => ((C = apply(A, B)) <=> (C = empty_set))))) <=> ((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))),
% 0.58/0.76 inference(bind,[status(th)],[])).
% 0.58/0.76 tff(63,plain,
% 0.58/0.76 (![A: $i] : ((relation(A) & function(A)) => ![B: $i, C: $i] : ((in(B, relation_dom(A)) => ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & ((~in(B, relation_dom(A))) => ((C = apply(A, B)) <=> (C = empty_set))))) <=> ![A: $i] : ((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))),
% 0.58/0.76 inference(quant_intro,[status(thm)],[62])).
% 0.58/0.76 tff(64,axiom,(![A: $i] : ((relation(A) & function(A)) => ![B: $i, C: $i] : ((in(B, relation_dom(A)) => ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & ((~in(B, relation_dom(A))) => ((C = apply(A, B)) <=> (C = empty_set)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d4_funct_1')).
% 0.58/0.76 tff(65,plain,
% 0.58/0.76 (![A: $i] : ((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))),
% 0.58/0.76 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.58/0.76 tff(66,plain,
% 0.58/0.76 (![A: $i] : ((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))),
% 0.58/0.76 inference(modus_ponens,[status(thm)],[65, 61])).
% 0.58/0.76 tff(67,plain,(
% 0.58/0.76 ![A: $i] : ((~(relation(A) & function(A))) | ![B: $i, C: $i] : (((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A))) & (in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))),
% 0.58/0.76 inference(skolemize,[status(sab)],[66])).
% 0.58/0.76 tff(68,plain,
% 0.58/0.76 (![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))),
% 0.58/0.76 inference(modus_ponens,[status(thm)],[67, 60])).
% 0.58/0.76 tff(69,plain,
% 0.58/0.76 (![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))),
% 0.58/0.76 inference(modus_ponens,[status(thm)],[68, 58])).
% 0.58/0.76 tff(70,plain,
% 0.58/0.76 (((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | ((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set)))))))) <=> ((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set)))))))),
% 0.58/0.76 inference(rewrite,[status(thm)],[])).
% 0.58/0.76 tff(71,plain,
% 0.58/0.76 ((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | ((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set)))))))),
% 0.58/0.76 inference(quant_inst,[status(thm)],[])).
% 0.58/0.76 tff(72,plain,
% 0.58/0.76 ((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))),
% 0.58/0.76 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.58/0.76 tff(73,plain,
% 0.58/0.76 (![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))),
% 0.58/0.76 inference(unit_resolution,[status(thm)],[72, 69, 53, 39])).
% 0.58/0.76 tff(74,plain,
% 0.58/0.76 (((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12))))))) <=> ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12)))))))),
% 0.58/0.76 inference(rewrite,[status(thm)],[])).
% 0.58/0.76 tff(75,plain,
% 0.58/0.76 ((~((~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(relation_dom_restriction(C!10, A!12), B!11)) <=> in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12))))) | (~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(relation_dom_restriction(C!10, A!12), B!11)) <=> (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set)))))) <=> (~((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12))))))),
% 0.58/0.76 inference(rewrite,[status(thm)],[])).
% 0.58/0.76 tff(76,plain,
% 0.58/0.76 (((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))) | (~((~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(relation_dom_restriction(C!10, A!12), B!11)) <=> in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12))))) | (~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(relation_dom_restriction(C!10, A!12), B!11)) <=> (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))))))) <=> ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12)))))))),
% 0.58/0.76 inference(monotonicity,[status(thm)],[75])).
% 0.58/0.76 tff(77,plain,
% 0.58/0.76 (((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))) | (~((~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(relation_dom_restriction(C!10, A!12), B!11)) <=> in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12))))) | (~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(relation_dom_restriction(C!10, A!12), B!11)) <=> (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))))))) <=> ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12)))))))),
% 0.58/0.76 inference(transitivity,[status(thm)],[76, 74])).
% 0.58/0.76 tff(78,plain,
% 0.58/0.76 ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))) | (~((~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(relation_dom_restriction(C!10, A!12), B!11)) <=> in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12))))) | (~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(relation_dom_restriction(C!10, A!12), B!11)) <=> (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))))))),
% 0.58/0.76 inference(quant_inst,[status(thm)],[])).
% 0.58/0.76 tff(79,plain,
% 0.58/0.76 ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> in(ordered_pair(B, C), relation_dom_restriction(C!10, A!12))))) | (~(in(B, relation_dom(relation_dom_restriction(C!10, A!12))) | ((C = apply(relation_dom_restriction(C!10, A!12), B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12))))))),
% 0.58/0.76 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.58/0.76 tff(80,plain,
% 0.58/0.76 (~((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12)))))),
% 0.58/0.76 inference(unit_resolution,[status(thm)],[79, 73])).
% 0.58/0.76 tff(81,plain,
% 0.58/0.76 (((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | in(ordered_pair(B!11, apply(relation_dom_restriction(C!10, A!12), B!11)), relation_dom_restriction(C!10, A!12))))) | (in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))),
% 0.58/0.76 inference(tautology,[status(thm)],[])).
% 0.58/0.76 tff(82,plain,
% 0.58/0.76 (in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set)),
% 0.58/0.76 inference(unit_resolution,[status(thm)],[81, 80])).
% 0.58/0.76 tff(83,plain,
% 0.58/0.76 (^[A: $i, B: $i, C: $i, D: $i] : trans(monotonicity(trans(monotonicity(rewrite((~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))) <=> (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))), (((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))), rewrite(((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))), (((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))), (((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))))), rewrite(((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))), (((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))))),
% 0.58/0.76 inference(bind,[status(th)],[])).
% 0.58/0.76 tff(84,plain,
% 0.58/0.76 (![A: $i, B: $i, C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))),
% 0.58/0.76 inference(quant_intro,[status(thm)],[83])).
% 0.58/0.76 tff(85,plain,
% 0.58/0.76 (^[A: $i, B: $i, C: $i, D: $i] : refl(((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))))),
% 0.58/0.76 inference(bind,[status(th)],[])).
% 0.58/0.76 tff(86,plain,
% 0.58/0.76 (![A: $i, B: $i, C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.58/0.76 inference(quant_intro,[status(thm)],[85])).
% 0.58/0.76 tff(87,plain,
% 0.58/0.76 (![A: $i, B: $i] : ![C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.58/0.76 inference(pull_quant,[status(thm)],[])).
% 0.58/0.76 tff(88,plain,
% 0.58/0.76 (^[A: $i, B: $i] : trans(monotonicity(trans(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(trans(monotonicity(trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant((~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))) <=> ?[D: $i] : (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))), (((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) <=> ((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | ?[D: $i] : (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))), pull_quant(((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | ?[D: $i] : (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) <=> ?[D: $i] : ((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))), (((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) <=> ?[D: $i] : ((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))), ((~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))) <=> (~?[D: $i] : ((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))), pull_quant((~?[D: $i] : ((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))) <=> ![D: $i] : (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))), ((~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))) <=> ![D: $i] : (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))), (((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))) <=> ((~(B = relation_dom_restriction(C, A))) | ![D: $i] : (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))))), pull_quant(((~(B = relation_dom_restriction(C, A))) | ![D: $i] : (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))) <=> ![D: $i] : ((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))), (((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))) <=> ![D: $i] : ((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))))), ((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) <=> (~![D: $i] : ((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))))), pull_quant((~![D: $i] : ((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) <=> ?[D: $i] : (~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))))), ((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) <=> ?[D: $i] : (~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))))), (((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))) <=> (?[D: $i] : (~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))), pull_quant((?[D: $i] : (~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))) <=> ?[D: $i] : ((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))), (((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))) <=> ?[D: $i] : ((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))), ((~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))) <=> (~?[D: $i] : ((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))), pull_quant((~?[D: $i] : ((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))) <=> ![D: $i] : (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))), ((~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))) <=> ![D: $i] : (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))), (((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ((~relation(C)) | (~function(C)) | ![D: $i] : (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))), pull_quant(((~relation(C)) | (~function(C)) | ![D: $i] : (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ![D: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))), (((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ![D: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))))), (![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ![C: $i] : ![D: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))), pull_quant(![C: $i] : ![D: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ![C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))), (![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ![C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))), (((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ((~relation(B)) | (~function(B)) | ![C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))))), pull_quant(((~relation(B)) | (~function(B)) | ![C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))), (((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))))),
% 0.61/0.77 inference(bind,[status(th)],[])).
% 0.61/0.77 tff(89,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![A: $i, B: $i] : ![C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.61/0.77 inference(quant_intro,[status(thm)],[88])).
% 0.61/0.77 tff(90,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.61/0.77 inference(transitivity,[status(thm)],[89, 87])).
% 0.61/0.77 tff(91,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.61/0.77 inference(transitivity,[status(thm)],[90, 86])).
% 0.61/0.77 tff(92,plain,
% 0.61/0.77 (^[A: $i, B: $i] : rewrite(((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))))),
% 0.61/0.77 inference(bind,[status(th)],[])).
% 0.61/0.77 tff(93,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.61/0.77 inference(quant_intro,[status(thm)],[92])).
% 0.61/0.77 tff(94,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.61/0.77 inference(transitivity,[status(thm)],[93, 91])).
% 0.61/0.77 tff(95,plain,
% 0.61/0.77 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(B) & function(B)) <=> (~((~relation(B)) | (~function(B))))), ((~(relation(B) & function(B))) <=> (~(~((~relation(B)) | (~function(B))))))), rewrite((~(~((~relation(B)) | (~function(B))))) <=> ((~relation(B)) | (~function(B)))), ((~(relation(B) & function(B))) <=> ((~relation(B)) | (~function(B))))), quant_intro(proof_bind(^[C: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(C) & function(C)) <=> (~((~relation(C)) | (~function(C))))), ((~(relation(C) & function(C))) <=> (~(~((~relation(C)) | (~function(C))))))), rewrite((~(~((~relation(C)) | (~function(C))))) <=> ((~relation(C)) | (~function(C)))), ((~(relation(C) & function(C))) <=> ((~relation(C)) | (~function(C))))), trans(monotonicity(rewrite(((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) <=> ((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))), rewrite(((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))) <=> ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))), ((((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))) <=> (((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))), rewrite((((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))) <=> (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))), ((((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))) <=> (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))), (((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))) <=> (((~relation(C)) | (~function(C))) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))), rewrite((((~relation(C)) | (~function(C))) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))), (((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))) <=> ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))))), (![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))) <=> ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))), (((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))) <=> (((~relation(B)) | (~function(B))) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))))), rewrite((((~relation(B)) | (~function(B))) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))), (((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))))),
% 0.61/0.77 inference(bind,[status(th)],[])).
% 0.61/0.77 tff(96,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.61/0.77 inference(quant_intro,[status(thm)],[95])).
% 0.61/0.77 tff(97,plain,
% 0.61/0.77 (^[A: $i, B: $i] : rewrite(((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | ((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))),
% 0.61/0.77 inference(bind,[status(th)],[])).
% 0.61/0.77 tff(98,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | ((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))),
% 0.61/0.77 inference(quant_intro,[status(thm)],[97])).
% 0.61/0.77 tff(99,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))),
% 0.61/0.77 inference(rewrite,[status(thm)],[])).
% 0.61/0.77 tff(100,plain,
% 0.61/0.77 (^[A: $i, B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(rewrite(((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : (in(D, relation_dom(B)) => (apply(B, D) = apply(C, D))))) <=> ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))), (((relation(C) & function(C)) => ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : (in(D, relation_dom(B)) => (apply(B, D) = apply(C, D)))))) <=> ((relation(C) & function(C)) => ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))), rewrite(((relation(C) & function(C)) => ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))) <=> ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))), (((relation(C) & function(C)) => ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : (in(D, relation_dom(B)) => (apply(B, D) = apply(C, D)))))) <=> ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))))), (![C: $i] : ((relation(C) & function(C)) => ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : (in(D, relation_dom(B)) => (apply(B, D) = apply(C, D)))))) <=> ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))), (((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : (in(D, relation_dom(B)) => (apply(B, D) = apply(C, D))))))) <=> ((relation(B) & function(B)) => ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))))), rewrite(((relation(B) & function(B)) => ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))))) <=> ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))), (((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : (in(D, relation_dom(B)) => (apply(B, D) = apply(C, D))))))) <=> ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))))),
% 0.61/0.77 inference(bind,[status(th)],[])).
% 0.61/0.77 tff(101,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : (in(D, relation_dom(B)) => (apply(B, D) = apply(C, D))))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))),
% 0.61/0.77 inference(quant_intro,[status(thm)],[100])).
% 0.61/0.77 tff(102,axiom,(![A: $i, B: $i] : ((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : (in(D, relation_dom(B)) => (apply(B, D) = apply(C, D)))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t68_funct_1')).
% 0.61/0.77 tff(103,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))),
% 0.61/0.77 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.61/0.77 tff(104,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | ((B = relation_dom_restriction(C, A)) <=> ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))),
% 0.61/0.77 inference(modus_ponens,[status(thm)],[103, 99])).
% 0.61/0.77 tff(105,plain,(
% 0.61/0.77 ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | ((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))),
% 0.61/0.77 inference(skolemize,[status(sab)],[104])).
% 0.61/0.77 tff(106,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (((~(B = relation_dom_restriction(C, A))) | ((relation_dom(B) = set_intersection2(relation_dom(C), A)) & ![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D))))) & ((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))),
% 0.61/0.77 inference(modus_ponens,[status(thm)],[105, 98])).
% 0.61/0.77 tff(107,plain,
% 0.61/0.77 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~![D: $i] : ((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.61/0.77 inference(modus_ponens,[status(thm)],[106, 96])).
% 0.61/0.77 tff(108,plain,
% 0.61/0.77 (![A: $i, B: $i, C: $i, D: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((B = relation_dom_restriction(C, A)) | (~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B)))))))))))),
% 0.61/0.77 inference(modus_ponens,[status(thm)],[107, 94])).
% 0.61/0.77 tff(109,plain,
% 0.61/0.77 (![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))),
% 0.61/0.77 inference(modus_ponens,[status(thm)],[108, 84])).
% 0.61/0.77 tff(110,plain,
% 0.61/0.77 (((~![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) | ((~function(C!10)) | (~relation(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) | (~function(C!10)) | (~relation(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))),
% 0.61/0.77 inference(rewrite,[status(thm)],[])).
% 0.61/0.78 tff(111,plain,
% 0.61/0.78 (((~relation(C!10)) | (~function(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))))))) <=> ((~function(C!10)) | (~relation(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))),
% 0.61/0.78 inference(rewrite,[status(thm)],[])).
% 0.61/0.78 tff(112,plain,
% 0.61/0.78 ((~((~((~(relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12)) | (~((~in(tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)), relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12))) = apply(C!10, tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)))))))))) <=> (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))))))),
% 0.61/0.78 inference(rewrite,[status(thm)],[])).
% 0.61/0.78 tff(113,plain,
% 0.61/0.78 (((~relation(C!10)) | (~function(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~((~(relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12)) | (~((~in(tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)), relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12))) = apply(C!10, tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12))))))))))) <=> ((~relation(C!10)) | (~function(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))),
% 0.61/0.78 inference(monotonicity,[status(thm)],[112])).
% 0.61/0.78 tff(114,plain,
% 0.61/0.78 (((~relation(C!10)) | (~function(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~((~(relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12)) | (~((~in(tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)), relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12))) = apply(C!10, tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12))))))))))) <=> ((~function(C!10)) | (~relation(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))),
% 0.61/0.78 inference(transitivity,[status(thm)],[113, 111])).
% 0.61/0.78 tff(115,plain,
% 0.61/0.78 (((~![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) | ((~relation(C!10)) | (~function(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~((~(relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12)) | (~((~in(tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)), relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12))) = apply(C!10, tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)))))))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) | ((~function(C!10)) | (~relation(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))))))))),
% 0.61/0.78 inference(monotonicity,[status(thm)],[114])).
% 0.61/0.78 tff(116,plain,
% 0.61/0.78 (((~![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) | ((~relation(C!10)) | (~function(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~((~(relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12)) | (~((~in(tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)), relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12))) = apply(C!10, tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)))))))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) | (~function(C!10)) | (~relation(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))),
% 0.61/0.78 inference(transitivity,[status(thm)],[115, 110])).
% 0.61/0.78 tff(117,plain,
% 0.61/0.78 ((~![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) | ((~relation(C!10)) | (~function(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~((~(relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (relation_dom_restriction(C!10, A!12) = relation_dom_restriction(C!10, A!12)) | (~((~in(tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)), relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12))) = apply(C!10, tptp_fun_D_9(C!10, relation_dom_restriction(C!10, A!12)))))))))))),
% 0.61/0.78 inference(quant_inst,[status(thm)],[])).
% 0.61/0.78 tff(118,plain,
% 0.61/0.78 ((~![A: $i, B: $i, C: $i, D: $i] : ((~relation(C)) | (~function(C)) | (~relation(B)) | (~function(B)) | (~((~((~(B = relation_dom_restriction(C, A))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (~((~in(D, relation_dom(B))) | (apply(B, D) = apply(C, D)))))))) | (~((~(relation_dom(B) = set_intersection2(relation_dom(C), A))) | (B = relation_dom_restriction(C, A)) | (~((~in(tptp_fun_D_9(C, B), relation_dom(B))) | (apply(B, tptp_fun_D_9(C, B)) = apply(C, tptp_fun_D_9(C, B))))))))))) | (~function(C!10)) | (~relation(C!10)) | (~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))))))),
% 0.61/0.78 inference(modus_ponens,[status(thm)],[117, 116])).
% 0.61/0.78 tff(119,plain,
% 0.61/0.78 ((~relation(relation_dom_restriction(C!10, A!12))) | (~function(relation_dom_restriction(C!10, A!12))) | (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))))))),
% 0.61/0.78 inference(unit_resolution,[status(thm)],[118, 109, 14, 13])).
% 0.61/0.78 tff(120,plain,
% 0.61/0.78 (~((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))))),
% 0.61/0.78 inference(unit_resolution,[status(thm)],[119, 39, 53])).
% 0.61/0.78 tff(121,plain,
% 0.61/0.78 (((~(relation_dom(relation_dom_restriction(C!10, A!12)) = set_intersection2(relation_dom(C!10), A!12))) | (~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))))) | ((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))),
% 0.61/0.78 inference(tautology,[status(thm)],[])).
% 0.61/0.78 tff(122,plain,
% 0.61/0.78 ((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))),
% 0.61/0.78 inference(unit_resolution,[status(thm)],[121, 120])).
% 0.61/0.78 tff(123,plain,
% 0.61/0.78 (~(apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))),
% 0.61/0.78 inference(or_elim,[status(thm)],[11])).
% 0.61/0.78 tff(124,plain,
% 0.61/0.78 ((~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))) | (~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))),
% 0.61/0.78 inference(tautology,[status(thm)],[])).
% 0.61/0.78 tff(125,plain,
% 0.61/0.78 ((~((~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)))) | (~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))),
% 0.61/0.78 inference(unit_resolution,[status(thm)],[124, 123])).
% 0.61/0.78 tff(126,plain,
% 0.61/0.78 (~in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))),
% 0.61/0.78 inference(unit_resolution,[status(thm)],[125, 122])).
% 0.61/0.78 tff(127,plain,
% 0.61/0.78 ((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set)),
% 0.61/0.78 inference(tautology,[status(thm)],[])).
% 0.61/0.78 tff(128,plain,
% 0.61/0.78 ((~(in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set))) | (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set)),
% 0.61/0.78 inference(unit_resolution,[status(thm)],[127, 126])).
% 0.61/0.78 tff(129,plain,
% 0.61/0.78 (apply(relation_dom_restriction(C!10, A!12), B!11) = empty_set),
% 0.61/0.78 inference(unit_resolution,[status(thm)],[128, 82])).
% 0.61/0.78 tff(130,plain,
% 0.61/0.78 ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)) <=> (empty_set = apply(C!10, B!11))),
% 0.61/0.78 inference(monotonicity,[status(thm)],[129])).
% 0.61/0.78 tff(131,plain,
% 0.61/0.78 ((apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)) <=> (apply(C!10, B!11) = empty_set)),
% 0.61/0.78 inference(transitivity,[status(thm)],[130, 1])).
% 0.61/0.78 tff(132,plain,
% 0.61/0.78 ((apply(C!10, B!11) = empty_set) <=> (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11))),
% 0.61/0.78 inference(symmetry,[status(thm)],[131])).
% 0.61/0.78 tff(133,plain,
% 0.61/0.78 (((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | (![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set)))))) | (~function(C!10)) | (~relation(C!10)))) <=> ((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set)))))) | (~function(C!10)) | (~relation(C!10)))),
% 0.61/0.78 inference(rewrite,[status(thm)],[])).
% 0.61/0.78 tff(134,plain,
% 0.61/0.78 (((~relation(C!10)) | (~function(C!10)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))) <=> (![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set)))))) | (~function(C!10)) | (~relation(C!10)))),
% 0.61/0.78 inference(rewrite,[status(thm)],[])).
% 0.61/0.78 tff(135,plain,
% 0.61/0.78 (((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | ((~relation(C!10)) | (~function(C!10)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set)))))))) <=> ((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | (![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set)))))) | (~function(C!10)) | (~relation(C!10))))),
% 0.61/0.78 inference(monotonicity,[status(thm)],[134])).
% 0.61/0.78 tff(136,plain,
% 0.61/0.78 (((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | ((~relation(C!10)) | (~function(C!10)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set)))))))) <=> ((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set)))))) | (~function(C!10)) | (~relation(C!10)))),
% 0.61/0.78 inference(transitivity,[status(thm)],[135, 133])).
% 0.61/0.78 tff(137,plain,
% 0.61/0.78 ((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | ((~relation(C!10)) | (~function(C!10)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set)))))))),
% 0.61/0.78 inference(quant_inst,[status(thm)],[])).
% 0.61/0.78 tff(138,plain,
% 0.61/0.78 ((~![A: $i] : ((~relation(A)) | (~function(A)) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(A))) | ((C = apply(A, B)) <=> in(ordered_pair(B, C), A)))) | (~(in(B, relation_dom(A)) | ((C = apply(A, B)) <=> (C = empty_set)))))))) | ![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set)))))) | (~function(C!10)) | (~relation(C!10))),
% 0.61/0.78 inference(modus_ponens,[status(thm)],[137, 136])).
% 0.61/0.78 tff(139,plain,
% 0.61/0.78 (![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))),
% 0.61/0.78 inference(unit_resolution,[status(thm)],[138, 69, 14, 13])).
% 0.61/0.78 tff(140,plain,
% 0.61/0.78 (((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | (~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10))))))) <=> ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | (~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10)))))))),
% 0.61/0.78 inference(rewrite,[status(thm)],[])).
% 0.61/0.78 tff(141,plain,
% 0.61/0.78 ((~((~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10)))) | (~(in(B!11, relation_dom(C!10)) | ((empty_set = apply(C!10, B!11)) <=> (empty_set = empty_set)))))) <=> (~((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | (~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10))))))),
% 0.61/0.78 inference(rewrite,[status(thm)],[])).
% 0.61/0.78 tff(142,plain,
% 0.61/0.78 (((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))) | (~((~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10)))) | (~(in(B!11, relation_dom(C!10)) | ((empty_set = apply(C!10, B!11)) <=> (empty_set = empty_set))))))) <=> ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | (~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10)))))))),
% 0.61/0.78 inference(monotonicity,[status(thm)],[141])).
% 0.61/0.78 tff(143,plain,
% 0.61/0.78 (((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))) | (~((~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10)))) | (~(in(B!11, relation_dom(C!10)) | ((empty_set = apply(C!10, B!11)) <=> (empty_set = empty_set))))))) <=> ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | (~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10)))))))),
% 0.61/0.78 inference(transitivity,[status(thm)],[142, 140])).
% 0.61/0.78 tff(144,plain,
% 0.61/0.78 ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))) | (~((~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10)))) | (~(in(B!11, relation_dom(C!10)) | ((empty_set = apply(C!10, B!11)) <=> (empty_set = empty_set))))))),
% 0.61/0.78 inference(quant_inst,[status(thm)],[])).
% 0.61/0.78 tff(145,plain,
% 0.61/0.78 ((~![B: $i, C: $i] : (~((~((~in(B, relation_dom(C!10))) | ((C = apply(C!10, B)) <=> in(ordered_pair(B, C), C!10)))) | (~(in(B, relation_dom(C!10)) | ((C = apply(C!10, B)) <=> (C = empty_set))))))) | (~((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | (~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10))))))),
% 0.61/0.78 inference(modus_ponens,[status(thm)],[144, 143])).
% 0.61/0.78 tff(146,plain,
% 0.61/0.78 (~((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | (~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10)))))),
% 0.61/0.78 inference(unit_resolution,[status(thm)],[145, 139])).
% 0.61/0.78 tff(147,plain,
% 0.61/0.78 (((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | (~((~in(B!11, relation_dom(C!10))) | ((empty_set = apply(C!10, B!11)) <=> in(ordered_pair(B!11, empty_set), C!10))))) | (in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))),
% 0.61/0.79 inference(tautology,[status(thm)],[])).
% 0.61/0.79 tff(148,plain,
% 0.61/0.79 (in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11))),
% 0.61/0.79 inference(unit_resolution,[status(thm)],[147, 146])).
% 0.61/0.79 tff(149,plain,
% 0.61/0.79 (^[A: $i, B: $i, C: $i] : refl(((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A)))))) <=> ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A)))))))),
% 0.61/0.79 inference(bind,[status(th)],[])).
% 0.61/0.79 tff(150,plain,
% 0.61/0.79 (![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A)))))) <=> ![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))),
% 0.61/0.79 inference(quant_intro,[status(thm)],[149])).
% 0.61/0.79 tff(151,plain,
% 0.61/0.79 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(C) & function(C)) <=> (~((~relation(C)) | (~function(C))))), ((~(relation(C) & function(C))) <=> (~(~((~relation(C)) | (~function(C))))))), rewrite((~(~((~relation(C)) | (~function(C))))) <=> ((~relation(C)) | (~function(C)))), ((~(relation(C) & function(C))) <=> ((~relation(C)) | (~function(C))))), trans(trans(monotonicity(rewrite((in(B, relation_dom(C)) & in(B, A)) <=> (~((~in(B, A)) | (~in(B, relation_dom(C)))))), ((in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A))) <=> (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (~((~in(B, A)) | (~in(B, relation_dom(C)))))))), rewrite((in(B, relation_dom(relation_dom_restriction(C, A))) <=> (~((~in(B, A)) | (~in(B, relation_dom(C)))))) <=> (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A)))))), ((in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A))) <=> (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))), rewrite((~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))) <=> (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A)))))), ((in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A))) <=> (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))), (((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A)))) <=> (((~relation(C)) | (~function(C))) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A)))))))), rewrite((((~relation(C)) | (~function(C))) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A)))))) <=> ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))), (((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A)))) <=> ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))))),
% 0.61/0.79 inference(bind,[status(th)],[])).
% 0.61/0.79 tff(152,plain,
% 0.61/0.79 (![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A)))) <=> ![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))),
% 0.61/0.79 inference(quant_intro,[status(thm)],[151])).
% 0.61/0.79 tff(153,plain,
% 0.61/0.79 (![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A)))) <=> ![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A))))),
% 0.61/0.79 inference(rewrite,[status(thm)],[])).
% 0.61/0.79 tff(154,plain,
% 0.61/0.79 (^[A: $i, B: $i, C: $i] : rewrite(((relation(C) & function(C)) => (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A)))) <=> ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A)))))),
% 0.61/0.79 inference(bind,[status(th)],[])).
% 0.61/0.79 tff(155,plain,
% 0.61/0.79 (![A: $i, B: $i, C: $i] : ((relation(C) & function(C)) => (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A)))) <=> ![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A))))),
% 0.61/0.79 inference(quant_intro,[status(thm)],[154])).
% 0.61/0.79 tff(156,axiom,(![A: $i, B: $i, C: $i] : ((relation(C) & function(C)) => (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l82_funct_1')).
% 0.61/0.79 tff(157,plain,
% 0.61/0.79 (![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A))))),
% 0.61/0.79 inference(modus_ponens,[status(thm)],[156, 155])).
% 0.61/0.79 tff(158,plain,
% 0.61/0.79 (![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A))))),
% 0.61/0.79 inference(modus_ponens,[status(thm)],[157, 153])).
% 0.61/0.79 tff(159,plain,(
% 0.61/0.79 ![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_dom_restriction(C, A))) <=> (in(B, relation_dom(C)) & in(B, A))))),
% 0.61/0.79 inference(skolemize,[status(sab)],[158])).
% 0.61/0.79 tff(160,plain,
% 0.61/0.79 (![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))),
% 0.61/0.79 inference(modus_ponens,[status(thm)],[159, 152])).
% 0.61/0.79 tff(161,plain,
% 0.61/0.79 (![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))),
% 0.61/0.79 inference(modus_ponens,[status(thm)],[160, 150])).
% 0.61/0.79 tff(162,plain,
% 0.61/0.79 (((~![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))) | ((~function(C!10)) | (~relation(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))))) <=> ((~![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))) | (~function(C!10)) | (~relation(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))))),
% 0.61/0.79 inference(rewrite,[status(thm)],[])).
% 0.61/0.79 tff(163,plain,
% 0.61/0.79 (((~relation(C!10)) | (~function(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))))) <=> ((~function(C!10)) | (~relation(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))))),
% 0.61/0.79 inference(rewrite,[status(thm)],[])).
% 0.61/0.79 tff(164,plain,
% 0.61/0.79 ((~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))) <=> (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))))),
% 0.61/0.79 inference(rewrite,[status(thm)],[])).
% 0.61/0.79 tff(165,plain,
% 0.61/0.79 (((~relation(C!10)) | (~function(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))))) <=> ((~relation(C!10)) | (~function(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))))),
% 0.61/0.79 inference(monotonicity,[status(thm)],[164])).
% 0.61/0.79 tff(166,plain,
% 0.61/0.79 (((~relation(C!10)) | (~function(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))))) <=> ((~function(C!10)) | (~relation(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))))),
% 0.61/0.79 inference(transitivity,[status(thm)],[165, 163])).
% 0.61/0.79 tff(167,plain,
% 0.61/0.79 (((~![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))) | ((~relation(C!10)) | (~function(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))))) <=> ((~![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))) | ((~function(C!10)) | (~relation(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))))))),
% 0.61/0.79 inference(monotonicity,[status(thm)],[166])).
% 0.61/0.79 tff(168,plain,
% 0.61/0.79 (((~![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))) | ((~relation(C!10)) | (~function(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))))) <=> ((~![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))) | (~function(C!10)) | (~relation(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))))),
% 0.61/0.79 inference(transitivity,[status(thm)],[167, 162])).
% 0.61/0.79 tff(169,plain,
% 0.61/0.79 ((~![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))) | ((~relation(C!10)) | (~function(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))))),
% 0.61/0.79 inference(quant_inst,[status(thm)],[])).
% 0.61/0.79 tff(170,plain,
% 0.61/0.79 ((~![A: $i, B: $i, C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(B, A)) | (~in(B, relation_dom(C)))) <=> in(B, relation_dom(relation_dom_restriction(C, A))))))) | (~function(C!10)) | (~relation(C!10)) | (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))))),
% 0.61/0.79 inference(modus_ponens,[status(thm)],[169, 168])).
% 0.61/0.79 tff(171,plain,
% 0.61/0.79 (~(((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12))))),
% 0.61/0.79 inference(unit_resolution,[status(thm)],[170, 161, 14, 13])).
% 0.61/0.79 tff(172,plain,
% 0.61/0.79 ((((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) | in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))),
% 0.61/0.79 inference(tautology,[status(thm)],[])).
% 0.61/0.79 tff(173,plain,
% 0.61/0.79 ((((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))) <=> in(B!11, relation_dom(relation_dom_restriction(C!10, A!12)))) | ((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10))))),
% 0.61/0.79 inference(unit_resolution,[status(thm)],[172, 126])).
% 0.61/0.79 tff(174,plain,
% 0.61/0.79 ((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))),
% 0.61/0.79 inference(unit_resolution,[status(thm)],[173, 171])).
% 0.61/0.79 tff(175,plain,
% 0.61/0.79 (in(B!11, A!12)),
% 0.61/0.79 inference(or_elim,[status(thm)],[11])).
% 0.61/0.79 tff(176,plain,
% 0.61/0.79 ((~((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10))))) | (~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10)))),
% 0.61/0.79 inference(tautology,[status(thm)],[])).
% 0.61/0.79 tff(177,plain,
% 0.61/0.79 ((~((~in(B!11, A!12)) | (~in(B!11, relation_dom(C!10))))) | (~in(B!11, relation_dom(C!10)))),
% 0.61/0.79 inference(unit_resolution,[status(thm)],[176, 175])).
% 0.61/0.79 tff(178,plain,
% 0.61/0.79 (~in(B!11, relation_dom(C!10))),
% 0.61/0.79 inference(unit_resolution,[status(thm)],[177, 174])).
% 0.61/0.79 tff(179,plain,
% 0.61/0.79 ((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11))),
% 0.61/0.79 inference(tautology,[status(thm)],[])).
% 0.61/0.79 tff(180,plain,
% 0.61/0.79 ((~(in(B!11, relation_dom(C!10)) | (empty_set = apply(C!10, B!11)))) | (empty_set = apply(C!10, B!11))),
% 0.61/0.79 inference(unit_resolution,[status(thm)],[179, 178])).
% 0.61/0.79 tff(181,plain,
% 0.61/0.79 (empty_set = apply(C!10, B!11)),
% 0.61/0.79 inference(unit_resolution,[status(thm)],[180, 148])).
% 0.61/0.79 tff(182,plain,
% 0.61/0.79 (apply(C!10, B!11) = empty_set),
% 0.61/0.79 inference(symmetry,[status(thm)],[181])).
% 0.61/0.79 tff(183,plain,
% 0.61/0.79 (apply(relation_dom_restriction(C!10, A!12), B!11) = apply(C!10, B!11)),
% 0.61/0.79 inference(modus_ponens,[status(thm)],[182, 132])).
% 0.61/0.79 tff(184,plain,
% 0.61/0.79 ($false),
% 0.61/0.79 inference(unit_resolution,[status(thm)],[123, 183])).
% 0.61/0.79 % SZS output end Proof
% 0.61/0.79 % E exiting
%------------------------------------------------------------------------------