%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP039-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:06:55 AM UTC 2025
% Result : Unsatisfiable 4.70s 4.89s
% Output : Proof 4.70s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GRP039-1 : TPTP v9.0.0. Released v1.0.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n003.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 15:30:24 EDT 2025
% 0.12/0.34 % CPUTime :
% 4.70/4.89 % SZS status Unsatisfiable
% 4.70/4.89 % SZS output start Proof
% 4.70/4.89 tff(product_type, type, (
% 4.70/4.89 product: ( $i * $i * $i ) > $o)).
% 4.70/4.89 tff(multiply_type, type, (
% 4.70/4.89 multiply: ( $i * $i ) > $i)).
% 4.70/4.89 tff(d_type, type, (
% 4.70/4.89 d: $i)).
% 4.70/4.89 tff(inverse_type, type, (
% 4.70/4.89 inverse: $i > $i)).
% 4.70/4.89 tff(identity_type, type, (
% 4.70/4.89 identity: $i)).
% 4.70/4.89 tff(c_type, type, (
% 4.70/4.89 c: $i)).
% 4.70/4.89 tff(a_type, type, (
% 4.70/4.89 a: $i)).
% 4.70/4.89 tff(subgroup_member_type, type, (
% 4.70/4.89 subgroup_member: $i > $o)).
% 4.70/4.89 tff(b_type, type, (
% 4.70/4.89 b: $i)).
% 4.70/4.89 tff(element_in_O2_type, type, (
% 4.70/4.89 element_in_O2: ( $i * $i ) > $i)).
% 4.70/4.89 tff(1,plain,
% 4.70/4.89 (^[X: $i] : refl(product(identity, X, X) <=> product(identity, X, X))),
% 4.70/4.89 inference(bind,[status(th)],[])).
% 4.70/4.89 tff(2,plain,
% 4.70/4.89 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 4.70/4.89 inference(quant_intro,[status(thm)],[1])).
% 4.70/4.89 tff(3,plain,
% 4.70/4.89 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(4,axiom,(![X: $i] : product(identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','left_identity')).
% 4.70/4.89 tff(5,plain,
% 4.70/4.89 (![X: $i] : product(identity, X, X)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[4, 3])).
% 4.70/4.89 tff(6,plain,(
% 4.70/4.89 ![X: $i] : product(identity, X, X)),
% 4.70/4.89 inference(skolemize,[status(sab)],[5])).
% 4.70/4.89 tff(7,plain,
% 4.70/4.89 (![X: $i] : product(identity, X, X)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[6, 2])).
% 4.70/4.89 tff(8,plain,
% 4.70/4.89 ((~![X: $i] : product(identity, X, X)) | product(identity, inverse(inverse(identity)), inverse(inverse(identity)))),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(9,plain,
% 4.70/4.89 (product(identity, inverse(inverse(identity)), inverse(inverse(identity)))),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[8, 7])).
% 4.70/4.89 tff(10,plain,
% 4.70/4.89 (^[X: $i] : refl(product(X, inverse(X), identity) <=> product(X, inverse(X), identity))),
% 4.70/4.89 inference(bind,[status(th)],[])).
% 4.70/4.89 tff(11,plain,
% 4.70/4.89 (![X: $i] : product(X, inverse(X), identity) <=> ![X: $i] : product(X, inverse(X), identity)),
% 4.70/4.89 inference(quant_intro,[status(thm)],[10])).
% 4.70/4.89 tff(12,plain,
% 4.70/4.89 (![X: $i] : product(X, inverse(X), identity) <=> ![X: $i] : product(X, inverse(X), identity)),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(13,axiom,(![X: $i] : product(X, inverse(X), identity)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','right_inverse')).
% 4.70/4.89 tff(14,plain,
% 4.70/4.89 (![X: $i] : product(X, inverse(X), identity)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[13, 12])).
% 4.70/4.89 tff(15,plain,(
% 4.70/4.89 ![X: $i] : product(X, inverse(X), identity)),
% 4.70/4.89 inference(skolemize,[status(sab)],[14])).
% 4.70/4.89 tff(16,plain,
% 4.70/4.89 (![X: $i] : product(X, inverse(X), identity)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[15, 11])).
% 4.70/4.89 tff(17,plain,
% 4.70/4.89 ((~![X: $i] : product(X, inverse(X), identity)) | product(inverse(identity), inverse(inverse(identity)), identity)),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(18,plain,
% 4.70/4.89 (product(inverse(identity), inverse(inverse(identity)), identity)),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[17, 16])).
% 4.70/4.89 tff(19,plain,
% 4.70/4.89 (^[X: $i] : refl(product(inverse(X), X, identity) <=> product(inverse(X), X, identity))),
% 4.70/4.89 inference(bind,[status(th)],[])).
% 4.70/4.89 tff(20,plain,
% 4.70/4.89 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 4.70/4.89 inference(quant_intro,[status(thm)],[19])).
% 4.70/4.89 tff(21,plain,
% 4.70/4.89 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(22,axiom,(![X: $i] : product(inverse(X), X, identity)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','left_inverse')).
% 4.70/4.89 tff(23,plain,
% 4.70/4.89 (![X: $i] : product(inverse(X), X, identity)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[22, 21])).
% 4.70/4.89 tff(24,plain,(
% 4.70/4.89 ![X: $i] : product(inverse(X), X, identity)),
% 4.70/4.89 inference(skolemize,[status(sab)],[23])).
% 4.70/4.89 tff(25,plain,
% 4.70/4.89 (![X: $i] : product(inverse(X), X, identity)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[24, 20])).
% 4.70/4.89 tff(26,plain,
% 4.70/4.89 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(identity), identity, identity)),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(27,plain,
% 4.70/4.89 (product(inverse(identity), identity, identity)),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[26, 25])).
% 4.70/4.89 tff(28,plain,
% 4.70/4.89 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))))),
% 4.70/4.89 inference(bind,[status(th)],[])).
% 4.70/4.89 tff(29,plain,
% 4.70/4.89 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 4.70/4.89 inference(quant_intro,[status(thm)],[28])).
% 4.70/4.89 tff(30,plain,
% 4.70/4.89 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(31,plain,
% 4.70/4.89 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) <=> ((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> (((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) | product(U, Z, W)))), rewrite((((~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W))) | product(U, Z, W)) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))))),
% 4.70/4.89 inference(bind,[status(th)],[])).
% 4.70/4.89 tff(32,plain,
% 4.70/4.89 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 4.70/4.89 inference(quant_intro,[status(thm)],[31])).
% 4.70/4.89 tff(33,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','associativity2')).
% 4.70/4.89 tff(34,plain,
% 4.70/4.89 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[33, 32])).
% 4.70/4.89 tff(35,plain,
% 4.70/4.89 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[34, 30])).
% 4.70/4.89 tff(36,plain,(
% 4.70/4.89 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 4.70/4.89 inference(skolemize,[status(sab)],[35])).
% 4.70/4.89 tff(37,plain,
% 4.70/4.89 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[36, 29])).
% 4.70/4.89 tff(38,plain,
% 4.70/4.89 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(inverse(identity), identity, identity)) | product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), inverse(inverse(identity)), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(identity), identity, identity)) | product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), inverse(inverse(identity)), identity)))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(39,plain,
% 4.70/4.89 ((product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), inverse(inverse(identity)), identity))) <=> ((~product(inverse(identity), identity, identity)) | product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), inverse(inverse(identity)), identity)))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(40,plain,
% 4.70/4.89 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), inverse(inverse(identity)), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(inverse(identity), identity, identity)) | product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), inverse(inverse(identity)), identity))))),
% 4.70/4.89 inference(monotonicity,[status(thm)],[39])).
% 4.70/4.89 tff(41,plain,
% 4.70/4.89 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), inverse(inverse(identity)), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(identity), identity, identity)) | product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), inverse(inverse(identity)), identity)))),
% 4.70/4.89 inference(transitivity,[status(thm)],[40, 38])).
% 4.70/4.89 tff(42,plain,
% 4.70/4.89 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), inverse(inverse(identity)), identity)))),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(43,plain,
% 4.70/4.89 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(identity), identity, identity)) | product(identity, inverse(inverse(identity)), identity) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(inverse(identity), inverse(inverse(identity)), identity))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[42, 41])).
% 4.70/4.89 tff(44,plain,
% 4.70/4.89 (product(identity, inverse(inverse(identity)), identity)),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[43, 37, 27, 18, 9])).
% 4.70/4.89 tff(45,plain,
% 4.70/4.89 (^[W: $i, Z: $i, Y: $i, X: $i] : refl(((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))))),
% 4.70/4.89 inference(bind,[status(th)],[])).
% 4.70/4.89 tff(46,plain,
% 4.70/4.89 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 4.70/4.89 inference(quant_intro,[status(thm)],[45])).
% 4.70/4.89 tff(47,plain,
% 4.70/4.89 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(48,plain,
% 4.70/4.89 (^[W: $i, Z: $i, Y: $i, X: $i] : rewrite((((~product(X, Y, Z)) | (~product(X, Y, W))) | (Z = W)) <=> ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z))))),
% 4.70/4.89 inference(bind,[status(th)],[])).
% 4.70/4.89 tff(49,plain,
% 4.70/4.89 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | (Z = W)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 4.70/4.89 inference(quant_intro,[status(thm)],[48])).
% 4.70/4.89 tff(50,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, Z)) | (~product(X, Y, W))) | (Z = W))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','total_function2')).
% 4.70/4.89 tff(51,plain,
% 4.70/4.89 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[50, 49])).
% 4.70/4.89 tff(52,plain,
% 4.70/4.89 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[51, 47])).
% 4.70/4.89 tff(53,plain,(
% 4.70/4.89 ![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 4.70/4.89 inference(skolemize,[status(sab)],[52])).
% 4.70/4.89 tff(54,plain,
% 4.70/4.89 (![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[53, 46])).
% 4.70/4.89 tff(55,plain,
% 4.70/4.89 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), identity)) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), identity)) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(56,plain,
% 4.70/4.89 (((identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(identity, inverse(inverse(identity)), identity))) <=> ((identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), identity)) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(57,plain,
% 4.70/4.89 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(identity, inverse(inverse(identity)), identity)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), identity)) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity))))))),
% 4.70/4.89 inference(monotonicity,[status(thm)],[56])).
% 4.70/4.89 tff(58,plain,
% 4.70/4.89 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(identity, inverse(inverse(identity)), identity)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), identity)) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))))),
% 4.70/4.89 inference(transitivity,[status(thm)],[57, 55])).
% 4.70/4.89 tff(59,plain,
% 4.70/4.89 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity)))) | (~product(identity, inverse(inverse(identity)), identity)))),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(60,plain,
% 4.70/4.89 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (identity = inverse(inverse(identity))) | (~product(identity, inverse(inverse(identity)), identity)) | (~product(identity, inverse(inverse(identity)), inverse(inverse(identity))))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[59, 58])).
% 4.70/4.89 tff(61,plain,
% 4.70/4.89 (identity = inverse(inverse(identity))),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[60, 54, 9, 44])).
% 4.70/4.89 tff(62,plain,
% 4.70/4.89 (inverse(inverse(identity)) = identity),
% 4.70/4.89 inference(symmetry,[status(thm)],[61])).
% 4.70/4.89 tff(63,plain,
% 4.70/4.89 (inverse(inverse(inverse(identity))) = inverse(identity)),
% 4.70/4.89 inference(monotonicity,[status(thm)],[62])).
% 4.70/4.89 tff(64,plain,
% 4.70/4.89 (inverse(identity) = inverse(inverse(inverse(identity)))),
% 4.70/4.89 inference(symmetry,[status(thm)],[63])).
% 4.70/4.89 tff(65,plain,
% 4.70/4.89 (^[X: $i] : refl(product(X, identity, X) <=> product(X, identity, X))),
% 4.70/4.89 inference(bind,[status(th)],[])).
% 4.70/4.89 tff(66,plain,
% 4.70/4.89 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 4.70/4.89 inference(quant_intro,[status(thm)],[65])).
% 4.70/4.89 tff(67,plain,
% 4.70/4.89 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(68,axiom,(![X: $i] : product(X, identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','right_identity')).
% 4.70/4.89 tff(69,plain,
% 4.70/4.89 (![X: $i] : product(X, identity, X)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[68, 67])).
% 4.70/4.89 tff(70,plain,(
% 4.70/4.89 ![X: $i] : product(X, identity, X)),
% 4.70/4.89 inference(skolemize,[status(sab)],[69])).
% 4.70/4.89 tff(71,plain,
% 4.70/4.89 (![X: $i] : product(X, identity, X)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[70, 66])).
% 4.70/4.89 tff(72,plain,
% 4.70/4.89 ((~![X: $i] : product(X, identity, X)) | product(inverse(identity), identity, inverse(identity))),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(73,plain,
% 4.70/4.89 (product(inverse(identity), identity, inverse(identity))),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[72, 71])).
% 4.70/4.89 tff(74,plain,
% 4.70/4.89 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))) | (inverse(identity) = identity))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))) | (inverse(identity) = identity))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(75,plain,
% 4.70/4.89 (((inverse(identity) = identity) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity)))) <=> ((~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))) | (inverse(identity) = identity))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(76,plain,
% 4.70/4.89 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((inverse(identity) = identity) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))) | (inverse(identity) = identity)))),
% 4.70/4.89 inference(monotonicity,[status(thm)],[75])).
% 4.70/4.89 tff(77,plain,
% 4.70/4.89 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((inverse(identity) = identity) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))) | (inverse(identity) = identity))),
% 4.70/4.89 inference(transitivity,[status(thm)],[76, 74])).
% 4.70/4.89 tff(78,plain,
% 4.70/4.89 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((inverse(identity) = identity) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))))),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(79,plain,
% 4.70/4.89 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(inverse(identity), identity, identity)) | (~product(inverse(identity), identity, inverse(identity))) | (inverse(identity) = identity)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[78, 77])).
% 4.70/4.89 tff(80,plain,
% 4.70/4.89 (inverse(identity) = identity),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[79, 54, 27, 73])).
% 4.70/4.89 tff(81,plain,
% 4.70/4.89 (identity = inverse(identity)),
% 4.70/4.89 inference(symmetry,[status(thm)],[80])).
% 4.70/4.89 tff(82,plain,
% 4.70/4.89 (identity = inverse(inverse(inverse(identity)))),
% 4.70/4.89 inference(transitivity,[status(thm)],[81, 64])).
% 4.70/4.89 tff(83,plain,
% 4.70/4.89 (product(identity, d, multiply(inverse(inverse(inverse(identity))), d)) <=> product(inverse(inverse(inverse(identity))), d, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.89 inference(monotonicity,[status(thm)],[82])).
% 4.70/4.89 tff(84,plain,
% 4.70/4.89 (product(inverse(inverse(inverse(identity))), d, multiply(inverse(inverse(inverse(identity))), d)) <=> product(identity, d, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.89 inference(symmetry,[status(thm)],[83])).
% 4.70/4.89 tff(85,plain,
% 4.70/4.89 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 4.70/4.89 inference(bind,[status(th)],[])).
% 4.70/4.89 tff(86,plain,
% 4.70/4.89 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 4.70/4.89 inference(quant_intro,[status(thm)],[85])).
% 4.70/4.89 tff(87,plain,
% 4.70/4.89 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(88,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','total_function1')).
% 4.70/4.89 tff(89,plain,
% 4.70/4.89 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[88, 87])).
% 4.70/4.89 tff(90,plain,(
% 4.70/4.89 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 4.70/4.89 inference(skolemize,[status(sab)],[89])).
% 4.70/4.89 tff(91,plain,
% 4.70/4.89 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[90, 86])).
% 4.70/4.89 tff(92,plain,
% 4.70/4.89 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(inverse(inverse(inverse(identity))), d, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(93,plain,
% 4.70/4.89 (product(inverse(inverse(inverse(identity))), d, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[92, 91])).
% 4.70/4.89 tff(94,plain,
% 4.70/4.89 (product(identity, d, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[93, 84])).
% 4.70/4.89 tff(95,plain,
% 4.70/4.89 ((~![X: $i] : product(identity, X, X)) | product(identity, d, d)),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(96,plain,
% 4.70/4.89 (product(identity, d, d)),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[95, 7])).
% 4.70/4.89 tff(97,plain,
% 4.70/4.89 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((~product(identity, d, d)) | (d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d))))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(identity, d, d)) | (d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d))))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(98,plain,
% 4.70/4.89 (((d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d))) | (~product(identity, d, d))) <=> ((~product(identity, d, d)) | (d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d))))),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(99,plain,
% 4.70/4.89 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d))) | (~product(identity, d, d)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((~product(identity, d, d)) | (d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d)))))),
% 4.70/4.89 inference(monotonicity,[status(thm)],[98])).
% 4.70/4.89 tff(100,plain,
% 4.70/4.89 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d))) | (~product(identity, d, d)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(identity, d, d)) | (d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d))))),
% 4.70/4.89 inference(transitivity,[status(thm)],[99, 97])).
% 4.70/4.89 tff(101,plain,
% 4.70/4.89 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d))) | (~product(identity, d, d)))),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(102,plain,
% 4.70/4.89 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(identity, d, d)) | (d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d)))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[101, 100])).
% 4.70/4.89 tff(103,plain,
% 4.70/4.89 ((d = multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, d, multiply(inverse(inverse(inverse(identity))), d)))),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[102, 54, 96])).
% 4.70/4.89 tff(104,plain,
% 4.70/4.89 (d = multiply(inverse(inverse(inverse(identity))), d)),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[103, 94])).
% 4.70/4.89 tff(105,plain,
% 4.70/4.89 (multiply(inverse(inverse(inverse(identity))), d) = d),
% 4.70/4.89 inference(symmetry,[status(thm)],[104])).
% 4.70/4.89 tff(106,plain,
% 4.70/4.89 (product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d)) <=> product(identity, d, d)),
% 4.70/4.89 inference(monotonicity,[status(thm)],[105, 105])).
% 4.70/4.89 tff(107,plain,
% 4.70/4.89 (product(identity, d, d) <=> product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.89 inference(symmetry,[status(thm)],[106])).
% 4.70/4.89 tff(108,plain,
% 4.70/4.89 (product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[96, 107])).
% 4.70/4.89 tff(109,plain,
% 4.70/4.89 (product(a, c, multiply(inverse(inverse(inverse(identity))), d)) <=> product(a, c, d)),
% 4.70/4.89 inference(monotonicity,[status(thm)],[105])).
% 4.70/4.89 tff(110,plain,
% 4.70/4.89 (product(a, c, d) <=> product(a, c, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.89 inference(symmetry,[status(thm)],[109])).
% 4.70/4.89 tff(111,plain,
% 4.70/4.89 (product(a, c, d) <=> product(a, c, d)),
% 4.70/4.89 inference(rewrite,[status(thm)],[])).
% 4.70/4.89 tff(112,axiom,(product(a, c, d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_times_c_is_d')).
% 4.70/4.89 tff(113,plain,
% 4.70/4.89 (product(a, c, d)),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[112, 111])).
% 4.70/4.89 tff(114,plain,
% 4.70/4.89 (product(a, c, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.89 inference(modus_ponens,[status(thm)],[113, 110])).
% 4.70/4.89 tff(115,plain,
% 4.70/4.89 ((~![X: $i] : product(identity, X, X)) | product(identity, c, c)),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(116,plain,
% 4.70/4.89 (product(identity, c, c)),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[115, 7])).
% 4.70/4.89 tff(117,plain,
% 4.70/4.89 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(a), a, identity)),
% 4.70/4.89 inference(quant_inst,[status(thm)],[])).
% 4.70/4.89 tff(118,plain,
% 4.70/4.89 (product(inverse(a), a, identity)),
% 4.70/4.89 inference(unit_resolution,[status(thm)],[117, 25])).
% 4.70/4.89 tff(119,plain,
% 4.70/4.89 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : refl((product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) <=> (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))))),
% 4.70/4.90 inference(bind,[status(th)],[])).
% 4.70/4.90 tff(120,plain,
% 4.70/4.90 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 4.70/4.90 inference(quant_intro,[status(thm)],[119])).
% 4.70/4.90 tff(121,plain,
% 4.70/4.90 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(122,plain,
% 4.70/4.90 (^[W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) <=> ((~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W)) <=> (((~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) | product(X, V, W)))), rewrite((((~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U))) | product(X, V, W)) <=> (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))), (((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W)) <=> (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))))),
% 4.70/4.90 inference(bind,[status(th)],[])).
% 4.70/4.90 tff(123,plain,
% 4.70/4.90 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W)) <=> ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 4.70/4.90 inference(quant_intro,[status(thm)],[122])).
% 4.70/4.90 tff(124,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','associativity1')).
% 4.70/4.90 tff(125,plain,
% 4.70/4.90 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[124, 123])).
% 4.70/4.90 tff(126,plain,
% 4.70/4.90 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[125, 121])).
% 4.70/4.90 tff(127,plain,(
% 4.70/4.90 ![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 4.70/4.90 inference(skolemize,[status(sab)],[126])).
% 4.70/4.90 tff(128,plain,
% 4.70/4.90 (![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[127, 120])).
% 4.70/4.90 tff(129,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(inverse(a), a, identity)) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(a), a, identity)) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(130,plain,
% 4.70/4.90 ((product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | (~product(inverse(a), a, identity))) <=> ((~product(inverse(a), a, identity)) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(131,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | (~product(inverse(a), a, identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(inverse(a), a, identity)) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)))),
% 4.70/4.90 inference(monotonicity,[status(thm)],[130])).
% 4.70/4.90 tff(132,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | (~product(inverse(a), a, identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(a), a, identity)) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c))),
% 4.70/4.90 inference(transitivity,[status(thm)],[131, 129])).
% 4.70/4.90 tff(133,plain,
% 4.70/4.90 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | (~product(inverse(a), a, identity)))),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(134,plain,
% 4.70/4.90 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(a), a, identity)) | (~product(identity, c, c)) | (~product(a, c, multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[133, 132])).
% 4.70/4.90 tff(135,plain,
% 4.70/4.90 (product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[134, 128, 118, 116, 114])).
% 4.70/4.90 tff(136,plain,
% 4.70/4.90 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(inverse(a)), inverse(a), identity)),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(137,plain,
% 4.70/4.90 (product(inverse(inverse(a)), inverse(a), identity)),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[136, 25])).
% 4.70/4.90 tff(138,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(inverse(inverse(a)), inverse(a), identity)) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(inverse(a)), inverse(a), identity)) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(139,plain,
% 4.70/4.90 ((product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(inverse(inverse(a)), inverse(a), identity))) <=> ((~product(inverse(inverse(a)), inverse(a), identity)) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(140,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(inverse(inverse(a)), inverse(a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(inverse(inverse(a)), inverse(a), identity)) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))))),
% 4.70/4.90 inference(monotonicity,[status(thm)],[139])).
% 4.70/4.90 tff(141,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(inverse(inverse(a)), inverse(a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(inverse(a)), inverse(a), identity)) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)))),
% 4.70/4.90 inference(transitivity,[status(thm)],[140, 138])).
% 4.70/4.90 tff(142,plain,
% 4.70/4.90 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(inverse(inverse(a)), inverse(a), identity)))),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(143,plain,
% 4.70/4.90 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(inverse(a)), inverse(a), identity)) | (~product(inverse(a), multiply(inverse(inverse(inverse(identity))), d), c)) | (~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[142, 141])).
% 4.70/4.90 tff(144,plain,
% 4.70/4.90 ((~product(identity, multiply(inverse(inverse(inverse(identity))), d), multiply(inverse(inverse(inverse(identity))), d))) | product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[143, 128, 137, 135])).
% 4.70/4.90 tff(145,plain,
% 4.70/4.90 (product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[144, 108])).
% 4.70/4.90 tff(146,plain,
% 4.70/4.90 (subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) <=> subgroup_member(d)),
% 4.70/4.90 inference(monotonicity,[status(thm)],[105])).
% 4.70/4.90 tff(147,plain,
% 4.70/4.90 (subgroup_member(d) <=> subgroup_member(multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.90 inference(symmetry,[status(thm)],[146])).
% 4.70/4.90 tff(148,plain,
% 4.70/4.90 ((~subgroup_member(d)) <=> (~subgroup_member(multiply(inverse(inverse(inverse(identity))), d)))),
% 4.70/4.90 inference(monotonicity,[status(thm)],[147])).
% 4.70/4.90 tff(149,plain,
% 4.70/4.90 ((~subgroup_member(d)) <=> (~subgroup_member(d))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(150,axiom,(~subgroup_member(d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_d_is_in_subgroup')).
% 4.70/4.90 tff(151,plain,
% 4.70/4.90 (~subgroup_member(d)),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[150, 149])).
% 4.70/4.90 tff(152,plain,
% 4.70/4.90 (~subgroup_member(multiply(inverse(inverse(inverse(identity))), d))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[151, 148])).
% 4.70/4.90 tff(153,plain,
% 4.70/4.90 ((~![X: $i] : product(X, identity, X)) | product(b, identity, b)),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(154,plain,
% 4.70/4.90 (product(b, identity, b)),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[153, 71])).
% 4.70/4.90 tff(155,plain,
% 4.70/4.90 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(c, a, multiply(c, a))),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(156,plain,
% 4.70/4.90 (product(c, a, multiply(c, a))),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[155, 91])).
% 4.70/4.90 tff(157,plain,
% 4.70/4.90 (product(b, inverse(a), c) <=> product(b, inverse(a), c)),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(158,axiom,(product(b, inverse(a), c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','b_times_a_inverse_is_c')).
% 4.70/4.90 tff(159,plain,
% 4.70/4.90 (product(b, inverse(a), c)),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[158, 157])).
% 4.70/4.90 tff(160,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(161,plain,
% 4.70/4.90 ((product(b, identity, multiply(c, a)) | (~product(c, a, multiply(c, a))) | (~product(inverse(a), a, identity)) | (~product(b, inverse(a), c))) <=> ((~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(162,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(b, identity, multiply(c, a)) | (~product(c, a, multiply(c, a))) | (~product(inverse(a), a, identity)) | (~product(b, inverse(a), c)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a))))),
% 4.70/4.90 inference(monotonicity,[status(thm)],[161])).
% 4.70/4.90 tff(163,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(b, identity, multiply(c, a)) | (~product(c, a, multiply(c, a))) | (~product(inverse(a), a, identity)) | (~product(b, inverse(a), c)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a)))),
% 4.70/4.90 inference(transitivity,[status(thm)],[162, 160])).
% 4.70/4.90 tff(164,plain,
% 4.70/4.90 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(b, identity, multiply(c, a)) | (~product(c, a, multiply(c, a))) | (~product(inverse(a), a, identity)) | (~product(b, inverse(a), c)))),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(165,plain,
% 4.70/4.90 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[164, 163])).
% 4.70/4.90 tff(166,plain,
% 4.70/4.90 (product(b, identity, multiply(c, a))),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[165, 128, 159, 118, 156])).
% 4.70/4.90 tff(167,plain,
% 4.70/4.90 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((~product(b, identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, b)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(b, identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, b)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(168,plain,
% 4.70/4.90 (((b = multiply(c, a)) | (~product(b, identity, multiply(c, a))) | (~product(b, identity, b))) <=> ((~product(b, identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, b)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(169,plain,
% 4.70/4.90 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((b = multiply(c, a)) | (~product(b, identity, multiply(c, a))) | (~product(b, identity, b)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((~product(b, identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, b))))),
% 4.70/4.90 inference(monotonicity,[status(thm)],[168])).
% 4.70/4.90 tff(170,plain,
% 4.70/4.90 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((b = multiply(c, a)) | (~product(b, identity, multiply(c, a))) | (~product(b, identity, b)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(b, identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, b)))),
% 4.70/4.90 inference(transitivity,[status(thm)],[169, 167])).
% 4.70/4.90 tff(171,plain,
% 4.70/4.90 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | ((b = multiply(c, a)) | (~product(b, identity, multiply(c, a))) | (~product(b, identity, b)))),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(172,plain,
% 4.70/4.90 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((Z = W) | (~product(X, Y, W)) | (~product(X, Y, Z)))) | (~product(b, identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, b))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[171, 170])).
% 4.70/4.90 tff(173,plain,
% 4.70/4.90 (b = multiply(c, a)),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[172, 54, 166, 154])).
% 4.70/4.90 tff(174,plain,
% 4.70/4.90 (multiply(c, a) = b),
% 4.70/4.90 inference(symmetry,[status(thm)],[173])).
% 4.70/4.90 tff(175,plain,
% 4.70/4.90 (product(inverse(b), multiply(c, a), identity) <=> product(inverse(b), b, identity)),
% 4.70/4.90 inference(monotonicity,[status(thm)],[174])).
% 4.70/4.90 tff(176,plain,
% 4.70/4.90 (product(inverse(b), b, identity) <=> product(inverse(b), multiply(c, a), identity)),
% 4.70/4.90 inference(symmetry,[status(thm)],[175])).
% 4.70/4.90 tff(177,plain,
% 4.70/4.90 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(b), b, identity)),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(178,plain,
% 4.70/4.90 (product(inverse(b), b, identity)),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[177, 25])).
% 4.70/4.90 tff(179,plain,
% 4.70/4.90 (product(inverse(b), multiply(c, a), identity)),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[178, 176])).
% 4.70/4.90 tff(180,plain,
% 4.70/4.90 (product(multiply(c, a), inverse(a), c) <=> product(b, inverse(a), c)),
% 4.70/4.90 inference(monotonicity,[status(thm)],[174])).
% 4.70/4.90 tff(181,plain,
% 4.70/4.90 (product(b, inverse(a), c) <=> product(multiply(c, a), inverse(a), c)),
% 4.70/4.90 inference(symmetry,[status(thm)],[180])).
% 4.70/4.90 tff(182,plain,
% 4.70/4.90 (product(multiply(c, a), inverse(a), c)),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[159, 181])).
% 4.70/4.90 tff(183,plain,
% 4.70/4.90 ((~![X: $i] : product(identity, X, X)) | product(identity, inverse(a), inverse(a))),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(184,plain,
% 4.70/4.90 (product(identity, inverse(a), inverse(a))),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[183, 7])).
% 4.70/4.90 tff(185,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(identity, inverse(a), inverse(a))) | (~product(inverse(b), multiply(c, a), identity)) | (~product(multiply(c, a), inverse(a), c)) | product(inverse(b), c, inverse(a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(b), multiply(c, a), identity)) | (~product(multiply(c, a), inverse(a), c)) | product(inverse(b), c, inverse(a)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(186,plain,
% 4.70/4.90 ((product(inverse(b), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(multiply(c, a), inverse(a), c)) | (~product(inverse(b), multiply(c, a), identity))) <=> ((~product(identity, inverse(a), inverse(a))) | (~product(inverse(b), multiply(c, a), identity)) | (~product(multiply(c, a), inverse(a), c)) | product(inverse(b), c, inverse(a)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(187,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(b), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(multiply(c, a), inverse(a), c)) | (~product(inverse(b), multiply(c, a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(identity, inverse(a), inverse(a))) | (~product(inverse(b), multiply(c, a), identity)) | (~product(multiply(c, a), inverse(a), c)) | product(inverse(b), c, inverse(a))))),
% 4.70/4.90 inference(monotonicity,[status(thm)],[186])).
% 4.70/4.90 tff(188,plain,
% 4.70/4.90 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(b), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(multiply(c, a), inverse(a), c)) | (~product(inverse(b), multiply(c, a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(b), multiply(c, a), identity)) | (~product(multiply(c, a), inverse(a), c)) | product(inverse(b), c, inverse(a)))),
% 4.70/4.90 inference(transitivity,[status(thm)],[187, 185])).
% 4.70/4.90 tff(189,plain,
% 4.70/4.90 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(b), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(multiply(c, a), inverse(a), c)) | (~product(inverse(b), multiply(c, a), identity)))),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(190,plain,
% 4.70/4.90 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(identity, inverse(a), inverse(a))) | (~product(inverse(b), multiply(c, a), identity)) | (~product(multiply(c, a), inverse(a), c)) | product(inverse(b), c, inverse(a))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[189, 188])).
% 4.70/4.90 tff(191,plain,
% 4.70/4.90 (product(inverse(b), c, inverse(a))),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[190, 128, 184, 182, 179])).
% 4.70/4.90 tff(192,plain,
% 4.70/4.90 (subgroup_member(b) <=> subgroup_member(b)),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(193,axiom,(subgroup_member(b)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','b_is_in_subgroup')).
% 4.70/4.90 tff(194,plain,
% 4.70/4.90 (subgroup_member(b)),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[193, 192])).
% 4.70/4.90 tff(195,plain,
% 4.70/4.90 (^[X: $i] : refl(((~subgroup_member(X)) | subgroup_member(inverse(X))) <=> ((~subgroup_member(X)) | subgroup_member(inverse(X))))),
% 4.70/4.90 inference(bind,[status(th)],[])).
% 4.70/4.90 tff(196,plain,
% 4.70/4.90 (![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X))) <=> ![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 4.70/4.90 inference(quant_intro,[status(thm)],[195])).
% 4.70/4.90 tff(197,plain,
% 4.70/4.90 (![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X))) <=> ![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(198,axiom,(![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-1.ax','closure_of_inverse')).
% 4.70/4.90 tff(199,plain,
% 4.70/4.90 (![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[198, 197])).
% 4.70/4.90 tff(200,plain,(
% 4.70/4.90 ![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 4.70/4.90 inference(skolemize,[status(sab)],[199])).
% 4.70/4.90 tff(201,plain,
% 4.70/4.90 (![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[200, 196])).
% 4.70/4.90 tff(202,plain,
% 4.70/4.90 (((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(b)) | subgroup_member(inverse(b)))) <=> ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(b)) | subgroup_member(inverse(b)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(203,plain,
% 4.70/4.90 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(b)) | subgroup_member(inverse(b)))),
% 4.70/4.90 inference(quant_inst,[status(thm)],[])).
% 4.70/4.90 tff(204,plain,
% 4.70/4.90 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(b)) | subgroup_member(inverse(b))),
% 4.70/4.90 inference(modus_ponens,[status(thm)],[203, 202])).
% 4.70/4.90 tff(205,plain,
% 4.70/4.90 (subgroup_member(inverse(b))),
% 4.70/4.90 inference(unit_resolution,[status(thm)],[204, 201, 194])).
% 4.70/4.90 tff(206,plain,
% 4.70/4.90 (product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a) <=> product(d, element_in_O2(d, a), a)),
% 4.70/4.90 inference(monotonicity,[status(thm)],[105])).
% 4.70/4.90 tff(207,plain,
% 4.70/4.90 (product(d, element_in_O2(d, a), a) <=> product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)),
% 4.70/4.90 inference(symmetry,[status(thm)],[206])).
% 4.70/4.90 tff(208,assumption,(~subgroup_member(c)), introduced(assumption)).
% 4.70/4.90 tff(209,plain,
% 4.70/4.90 (^[B: $i, A: $i, C: $i] : refl((subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))))),
% 4.70/4.90 inference(bind,[status(th)],[])).
% 4.70/4.90 tff(210,plain,
% 4.70/4.90 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 4.70/4.90 inference(quant_intro,[status(thm)],[209])).
% 4.70/4.90 tff(211,plain,
% 4.70/4.90 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 4.70/4.90 inference(rewrite,[status(thm)],[])).
% 4.70/4.90 tff(212,plain,
% 4.70/4.90 (^[B: $i, A: $i, C: $i] : trans(monotonicity(rewrite((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) <=> ((~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))), (((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) | subgroup_member(C)) <=> (((~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) | subgroup_member(C)))), rewrite((((~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) | subgroup_member(C)) <=> (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))), (((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) | subgroup_member(C)) <=> (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))))),
% 4.70/4.91 inference(bind,[status(th)],[])).
% 4.70/4.91 tff(213,plain,
% 4.70/4.91 (![B: $i, A: $i, C: $i] : ((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) | subgroup_member(C)) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 4.70/4.91 inference(quant_intro,[status(thm)],[212])).
% 4.70/4.91 tff(214,axiom,(![B: $i, A: $i, C: $i] : ((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) | subgroup_member(C))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-1.ax','closure_of_product')).
% 4.70/4.91 tff(215,plain,
% 4.70/4.91 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[214, 213])).
% 4.70/4.91 tff(216,plain,
% 4.70/4.91 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[215, 211])).
% 4.70/4.91 tff(217,plain,(
% 4.70/4.91 ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 4.70/4.91 inference(skolemize,[status(sab)],[216])).
% 4.70/4.91 tff(218,plain,
% 4.70/4.91 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[217, 210])).
% 4.70/4.91 tff(219,plain,
% 4.70/4.91 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(220,plain,
% 4.70/4.91 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(221,plain,
% 4.70/4.91 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(inverse(a))) | (~subgroup_member(b))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[220, 219])).
% 4.70/4.91 tff(222,plain,
% 4.70/4.91 (subgroup_member(c) | (~subgroup_member(inverse(a)))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[221, 218, 194, 159])).
% 4.70/4.91 tff(223,plain,
% 4.70/4.91 (~subgroup_member(inverse(a))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[222, 208])).
% 4.70/4.91 tff(224,plain,
% 4.70/4.91 (((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(a)) | subgroup_member(inverse(a)))) <=> ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(a)) | subgroup_member(inverse(a)))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(225,plain,
% 4.70/4.91 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(a)) | subgroup_member(inverse(a)))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(226,plain,
% 4.70/4.91 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(a)) | subgroup_member(inverse(a))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[225, 224])).
% 4.70/4.91 tff(227,plain,
% 4.70/4.91 ((~subgroup_member(a)) | subgroup_member(inverse(a))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[226, 201])).
% 4.70/4.91 tff(228,plain,
% 4.70/4.91 (~subgroup_member(a)),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[227, 223])).
% 4.70/4.91 tff(229,plain,
% 4.70/4.91 (^[B: $i, A: $i] : refl((subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)) <=> (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)))),
% 4.70/4.91 inference(bind,[status(th)],[])).
% 4.70/4.91 tff(230,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 4.70/4.91 inference(quant_intro,[status(thm)],[229])).
% 4.70/4.91 tff(231,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(232,plain,
% 4.70/4.91 (^[B: $i, A: $i] : trans(monotonicity(rewrite((product(A, element_in_O2(A, B), B) | subgroup_member(B)) <=> (subgroup_member(B) | product(A, element_in_O2(A, B), B))), (((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> ((subgroup_member(B) | product(A, element_in_O2(A, B), B)) | subgroup_member(A)))), rewrite(((subgroup_member(B) | product(A, element_in_O2(A, B), B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))), (((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))))),
% 4.70/4.91 inference(bind,[status(th)],[])).
% 4.70/4.91 tff(233,plain,
% 4.70/4.91 (![B: $i, A: $i] : ((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 4.70/4.91 inference(quant_intro,[status(thm)],[232])).
% 4.70/4.91 tff(234,axiom,(![B: $i, A: $i] : ((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','property_of_O2')).
% 4.70/4.91 tff(235,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[234, 233])).
% 4.70/4.91 tff(236,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[235, 231])).
% 4.70/4.91 tff(237,plain,(
% 4.70/4.91 ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 4.70/4.91 inference(skolemize,[status(sab)],[236])).
% 4.70/4.91 tff(238,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[237, 230])).
% 4.70/4.91 tff(239,plain,
% 4.70/4.91 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(d) | subgroup_member(a) | product(d, element_in_O2(d, a), a))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(d) | subgroup_member(a) | product(d, element_in_O2(d, a), a))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(240,plain,
% 4.70/4.91 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(d) | subgroup_member(a) | product(d, element_in_O2(d, a), a))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(241,plain,
% 4.70/4.91 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(d) | subgroup_member(a) | product(d, element_in_O2(d, a), a)),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[240, 239])).
% 4.70/4.91 tff(242,plain,
% 4.70/4.91 (subgroup_member(a) | product(d, element_in_O2(d, a), a)),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[241, 238, 151])).
% 4.70/4.91 tff(243,plain,
% 4.70/4.91 (product(d, element_in_O2(d, a), a)),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[242, 228])).
% 4.70/4.91 tff(244,plain,
% 4.70/4.91 (product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[243, 207])).
% 4.70/4.91 tff(245,plain,
% 4.70/4.91 (product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a) <=> product(d, inverse(c), a)),
% 4.70/4.91 inference(monotonicity,[status(thm)],[105])).
% 4.70/4.91 tff(246,plain,
% 4.70/4.91 (product(d, inverse(c), a) <=> product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)),
% 4.70/4.91 inference(symmetry,[status(thm)],[245])).
% 4.70/4.91 tff(247,plain,
% 4.70/4.91 ((~![X: $i] : product(X, inverse(X), identity)) | product(c, inverse(c), identity)),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(248,plain,
% 4.70/4.91 (product(c, inverse(c), identity)),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[247, 16])).
% 4.70/4.91 tff(249,plain,
% 4.70/4.91 ((~![X: $i] : product(X, identity, X)) | product(a, identity, a)),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(250,plain,
% 4.70/4.91 (product(a, identity, a)),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[249, 71])).
% 4.70/4.91 tff(251,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(a, c, d)) | (~product(a, identity, a)) | product(d, inverse(c), a) | (~product(c, inverse(c), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(a, c, d)) | (~product(a, identity, a)) | product(d, inverse(c), a) | (~product(c, inverse(c), identity)))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(252,plain,
% 4.70/4.91 ((product(d, inverse(c), a) | (~product(c, inverse(c), identity)) | (~product(a, c, d)) | (~product(a, identity, a))) <=> ((~product(a, c, d)) | (~product(a, identity, a)) | product(d, inverse(c), a) | (~product(c, inverse(c), identity)))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(253,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(d, inverse(c), a) | (~product(c, inverse(c), identity)) | (~product(a, c, d)) | (~product(a, identity, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(a, c, d)) | (~product(a, identity, a)) | product(d, inverse(c), a) | (~product(c, inverse(c), identity))))),
% 4.70/4.91 inference(monotonicity,[status(thm)],[252])).
% 4.70/4.91 tff(254,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(d, inverse(c), a) | (~product(c, inverse(c), identity)) | (~product(a, c, d)) | (~product(a, identity, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(a, c, d)) | (~product(a, identity, a)) | product(d, inverse(c), a) | (~product(c, inverse(c), identity)))),
% 4.70/4.91 inference(transitivity,[status(thm)],[253, 251])).
% 4.70/4.91 tff(255,plain,
% 4.70/4.91 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(d, inverse(c), a) | (~product(c, inverse(c), identity)) | (~product(a, c, d)) | (~product(a, identity, a)))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(256,plain,
% 4.70/4.91 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(a, c, d)) | (~product(a, identity, a)) | product(d, inverse(c), a) | (~product(c, inverse(c), identity))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[255, 254])).
% 4.70/4.91 tff(257,plain,
% 4.70/4.91 (product(d, inverse(c), a)),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[256, 37, 113, 250, 248])).
% 4.70/4.91 tff(258,plain,
% 4.70/4.91 (product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[257, 246])).
% 4.70/4.91 tff(259,plain,
% 4.70/4.91 (product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity) <=> product(inverse(d), d, identity)),
% 4.70/4.91 inference(monotonicity,[status(thm)],[105])).
% 4.70/4.91 tff(260,plain,
% 4.70/4.91 (product(inverse(d), d, identity) <=> product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)),
% 4.70/4.91 inference(symmetry,[status(thm)],[259])).
% 4.70/4.91 tff(261,plain,
% 4.70/4.91 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(d), d, identity)),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(262,plain,
% 4.70/4.91 (product(inverse(d), d, identity)),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[261, 25])).
% 4.70/4.91 tff(263,plain,
% 4.70/4.91 (product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[262, 260])).
% 4.70/4.91 tff(264,plain,
% 4.70/4.91 ((~![X: $i] : product(identity, X, X)) | product(identity, inverse(c), inverse(c))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(265,plain,
% 4.70/4.91 (product(identity, inverse(c), inverse(c))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[264, 7])).
% 4.70/4.91 tff(266,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c))))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(267,plain,
% 4.70/4.91 ((product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c))) | (~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity))) <=> ((~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c))))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(268,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c))) | (~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c)))))),
% 4.70/4.91 inference(monotonicity,[status(thm)],[267])).
% 4.70/4.91 tff(269,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c))) | (~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c))))),
% 4.70/4.91 inference(transitivity,[status(thm)],[268, 266])).
% 4.70/4.91 tff(270,plain,
% 4.70/4.91 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c))) | (~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(271,plain,
% 4.70/4.91 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(inverse(d), a, inverse(c)) | (~product(identity, inverse(c), inverse(c)))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[270, 269])).
% 4.70/4.91 tff(272,plain,
% 4.70/4.91 ((~product(multiply(inverse(inverse(inverse(identity))), d), inverse(c), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(inverse(d), a, inverse(c))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[271, 128, 265])).
% 4.70/4.91 tff(273,plain,
% 4.70/4.91 (product(inverse(d), a, inverse(c))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[272, 263, 258])).
% 4.70/4.91 tff(274,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(inverse(d), a, inverse(c))) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(d), a, inverse(c))) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(275,plain,
% 4.70/4.91 ((product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | (~product(inverse(d), a, inverse(c)))) <=> ((~product(inverse(d), a, inverse(c))) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(276,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | (~product(inverse(d), a, inverse(c))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(inverse(d), a, inverse(c))) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a))))),
% 4.70/4.91 inference(monotonicity,[status(thm)],[275])).
% 4.70/4.91 tff(277,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | (~product(inverse(d), a, inverse(c))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(d), a, inverse(c))) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)))),
% 4.70/4.91 inference(transitivity,[status(thm)],[276, 274])).
% 4.70/4.91 tff(278,plain,
% 4.70/4.91 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a)) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | (~product(inverse(d), a, inverse(c))))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(279,plain,
% 4.70/4.91 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(inverse(d), a, inverse(c))) | (~product(inverse(d), multiply(inverse(inverse(inverse(identity))), d), identity)) | product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[278, 277])).
% 4.70/4.91 tff(280,plain,
% 4.70/4.91 (product(identity, element_in_O2(d, a), inverse(c)) | (~product(multiply(inverse(inverse(inverse(identity))), d), element_in_O2(d, a), a))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[279, 37, 263, 273])).
% 4.70/4.91 tff(281,plain,
% 4.70/4.91 (product(identity, element_in_O2(d, a), inverse(c))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[280, 244])).
% 4.70/4.91 tff(282,plain,
% 4.70/4.91 (^[B: $i, A: $i] : refl((subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) <=> (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B)))),
% 4.70/4.91 inference(bind,[status(th)],[])).
% 4.70/4.91 tff(283,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))),
% 4.70/4.91 inference(quant_intro,[status(thm)],[282])).
% 4.70/4.91 tff(284,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(285,plain,
% 4.70/4.91 (^[B: $i, A: $i] : trans(monotonicity(rewrite((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) <=> (subgroup_member(element_in_O2(A, B)) | subgroup_member(B))), (((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> ((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)))), rewrite(((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))), (((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))))),
% 4.70/4.91 inference(bind,[status(th)],[])).
% 4.70/4.91 tff(286,plain,
% 4.70/4.91 (![B: $i, A: $i] : ((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))),
% 4.70/4.91 inference(quant_intro,[status(thm)],[285])).
% 4.70/4.91 tff(287,axiom,(![B: $i, A: $i] : ((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','an_element_in_O2')).
% 4.70/4.91 tff(288,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[287, 286])).
% 4.70/4.91 tff(289,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[288, 284])).
% 4.70/4.91 tff(290,plain,(
% 4.70/4.91 ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))),
% 4.70/4.91 inference(skolemize,[status(sab)],[289])).
% 4.70/4.91 tff(291,plain,
% 4.70/4.91 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[290, 283])).
% 4.70/4.91 tff(292,plain,
% 4.70/4.91 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(d, a)))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(d, a)))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(293,plain,
% 4.70/4.91 ((subgroup_member(d) | subgroup_member(element_in_O2(d, a)) | subgroup_member(a)) <=> (subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(d, a)))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(294,plain,
% 4.70/4.91 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(d) | subgroup_member(element_in_O2(d, a)) | subgroup_member(a))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(d, a))))),
% 4.70/4.91 inference(monotonicity,[status(thm)],[293])).
% 4.70/4.91 tff(295,plain,
% 4.70/4.91 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(d) | subgroup_member(element_in_O2(d, a)) | subgroup_member(a))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(d, a)))),
% 4.70/4.91 inference(transitivity,[status(thm)],[294, 292])).
% 4.70/4.91 tff(296,plain,
% 4.70/4.91 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(d) | subgroup_member(element_in_O2(d, a)) | subgroup_member(a))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(297,plain,
% 4.70/4.91 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(d, a))),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[296, 295])).
% 4.70/4.91 tff(298,plain,
% 4.70/4.91 (subgroup_member(a) | subgroup_member(element_in_O2(d, a))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[297, 291, 151])).
% 4.70/4.91 tff(299,plain,
% 4.70/4.91 (subgroup_member(element_in_O2(d, a))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[298, 228])).
% 4.70/4.91 tff(300,plain,
% 4.70/4.91 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(a) | subgroup_member(c) | product(a, element_in_O2(a, c), c))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(a) | subgroup_member(c) | product(a, element_in_O2(a, c), c))),
% 4.70/4.91 inference(rewrite,[status(thm)],[])).
% 4.70/4.91 tff(301,plain,
% 4.70/4.91 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(a) | subgroup_member(c) | product(a, element_in_O2(a, c), c))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(302,plain,
% 4.70/4.91 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(a) | subgroup_member(c) | product(a, element_in_O2(a, c), c)),
% 4.70/4.91 inference(modus_ponens,[status(thm)],[301, 300])).
% 4.70/4.91 tff(303,plain,
% 4.70/4.91 (subgroup_member(a) | subgroup_member(c) | product(a, element_in_O2(a, c), c)),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[302, 238])).
% 4.70/4.91 tff(304,plain,
% 4.70/4.91 (product(a, element_in_O2(a, c), c)),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[303, 228, 208])).
% 4.70/4.91 tff(305,plain,
% 4.70/4.91 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(c, c, multiply(c, c))),
% 4.70/4.91 inference(quant_inst,[status(thm)],[])).
% 4.70/4.91 tff(306,plain,
% 4.70/4.91 (product(c, c, multiply(c, c))),
% 4.70/4.91 inference(unit_resolution,[status(thm)],[305, 91])).
% 4.70/4.91 tff(307,plain,
% 4.70/4.91 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c))) | (~product(a, element_in_O2(a, c), c)) | product(multiply(c, a), element_in_O2(a, c), multiply(c, c)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c))) | (~product(a, element_in_O2(a, c), c)) | product(multiply(c, a), element_in_O2(a, c), multiply(c, c)))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(308,plain,
% 4.70/4.92 ((product(multiply(c, a), element_in_O2(a, c), multiply(c, c)) | (~product(a, element_in_O2(a, c), c)) | (~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c)))) <=> ((~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c))) | (~product(a, element_in_O2(a, c), c)) | product(multiply(c, a), element_in_O2(a, c), multiply(c, c)))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(309,plain,
% 4.70/4.92 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(multiply(c, a), element_in_O2(a, c), multiply(c, c)) | (~product(a, element_in_O2(a, c), c)) | (~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c))) | (~product(a, element_in_O2(a, c), c)) | product(multiply(c, a), element_in_O2(a, c), multiply(c, c))))),
% 4.70/4.92 inference(monotonicity,[status(thm)],[308])).
% 4.70/4.92 tff(310,plain,
% 4.70/4.92 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(multiply(c, a), element_in_O2(a, c), multiply(c, c)) | (~product(a, element_in_O2(a, c), c)) | (~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c))) | (~product(a, element_in_O2(a, c), c)) | product(multiply(c, a), element_in_O2(a, c), multiply(c, c)))),
% 4.70/4.92 inference(transitivity,[status(thm)],[309, 307])).
% 4.70/4.92 tff(311,plain,
% 4.70/4.92 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(multiply(c, a), element_in_O2(a, c), multiply(c, c)) | (~product(a, element_in_O2(a, c), c)) | (~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c))))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(312,plain,
% 4.70/4.92 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(c, a, multiply(c, a))) | (~product(c, c, multiply(c, c))) | (~product(a, element_in_O2(a, c), c)) | product(multiply(c, a), element_in_O2(a, c), multiply(c, c))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[311, 310])).
% 4.70/4.92 tff(313,plain,
% 4.70/4.92 ((~product(a, element_in_O2(a, c), c)) | product(multiply(c, a), element_in_O2(a, c), multiply(c, c))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[312, 37, 306, 156])).
% 4.70/4.92 tff(314,plain,
% 4.70/4.92 (product(multiply(c, a), element_in_O2(a, c), multiply(c, c))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[313, 304])).
% 4.70/4.92 tff(315,plain,
% 4.70/4.92 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(c) | subgroup_member(a) | subgroup_member(element_in_O2(a, c)))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | subgroup_member(c) | subgroup_member(a) | subgroup_member(element_in_O2(a, c)))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(316,plain,
% 4.70/4.92 ((subgroup_member(a) | subgroup_member(element_in_O2(a, c)) | subgroup_member(c)) <=> (subgroup_member(c) | subgroup_member(a) | subgroup_member(element_in_O2(a, c)))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(317,plain,
% 4.70/4.92 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(a) | subgroup_member(element_in_O2(a, c)) | subgroup_member(c))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(c) | subgroup_member(a) | subgroup_member(element_in_O2(a, c))))),
% 4.70/4.92 inference(monotonicity,[status(thm)],[316])).
% 4.70/4.92 tff(318,plain,
% 4.70/4.92 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(a) | subgroup_member(element_in_O2(a, c)) | subgroup_member(c))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | subgroup_member(c) | subgroup_member(a) | subgroup_member(element_in_O2(a, c)))),
% 4.70/4.92 inference(transitivity,[status(thm)],[317, 315])).
% 4.70/4.92 tff(319,plain,
% 4.70/4.92 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | (subgroup_member(a) | subgroup_member(element_in_O2(a, c)) | subgroup_member(c))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(320,plain,
% 4.70/4.92 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(element_in_O2(A, B)) | subgroup_member(B))) | subgroup_member(c) | subgroup_member(a) | subgroup_member(element_in_O2(a, c))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[319, 318])).
% 4.70/4.92 tff(321,plain,
% 4.70/4.92 (subgroup_member(c) | subgroup_member(a) | subgroup_member(element_in_O2(a, c))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[320, 291])).
% 4.70/4.92 tff(322,plain,
% 4.70/4.92 (subgroup_member(element_in_O2(a, c))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[321, 228, 208])).
% 4.70/4.92 tff(323,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(b)) | subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(inverse(b))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(b)) | subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(inverse(b))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(324,plain,
% 4.70/4.92 ((subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(b)) | (~subgroup_member(inverse(b)))) <=> ((~subgroup_member(b)) | subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(inverse(b))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(325,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(b)) | (~subgroup_member(inverse(b))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(b)) | subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(inverse(b)))))),
% 4.70/4.92 inference(monotonicity,[status(thm)],[324])).
% 4.70/4.92 tff(326,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(b)) | (~subgroup_member(inverse(b))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(b)) | subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(inverse(b))))),
% 4.70/4.92 inference(transitivity,[status(thm)],[325, 323])).
% 4.70/4.92 tff(327,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(b)) | (~subgroup_member(inverse(b))))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(328,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(b)) | subgroup_member(identity) | (~product(inverse(b), b, identity)) | (~subgroup_member(inverse(b)))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[327, 326])).
% 4.70/4.92 tff(329,plain,
% 4.70/4.92 (subgroup_member(identity)),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[328, 218, 194, 205, 178])).
% 4.70/4.92 tff(330,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(b)) | (~subgroup_member(identity)) | subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(b)) | (~subgroup_member(identity)) | subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(331,plain,
% 4.70/4.92 ((subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a))) | (~subgroup_member(identity)) | (~subgroup_member(b))) <=> ((~subgroup_member(b)) | (~subgroup_member(identity)) | subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(332,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a))) | (~subgroup_member(identity)) | (~subgroup_member(b)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(b)) | (~subgroup_member(identity)) | subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a)))))),
% 4.70/4.92 inference(monotonicity,[status(thm)],[331])).
% 4.70/4.92 tff(333,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a))) | (~subgroup_member(identity)) | (~subgroup_member(b)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(b)) | (~subgroup_member(identity)) | subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a))))),
% 4.70/4.92 inference(transitivity,[status(thm)],[332, 330])).
% 4.70/4.92 tff(334,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a))) | (~subgroup_member(identity)) | (~subgroup_member(b)))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(335,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(b)) | (~subgroup_member(identity)) | subgroup_member(multiply(c, a)) | (~product(b, identity, multiply(c, a)))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[334, 333])).
% 4.70/4.92 tff(336,plain,
% 4.70/4.92 (subgroup_member(multiply(c, a))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[335, 218, 194, 329, 166])).
% 4.70/4.92 tff(337,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(element_in_O2(a, c))) | subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(multiply(c, a))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(element_in_O2(a, c))) | subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(multiply(c, a))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(338,plain,
% 4.70/4.92 ((subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(element_in_O2(a, c))) | (~subgroup_member(multiply(c, a)))) <=> ((~subgroup_member(element_in_O2(a, c))) | subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(multiply(c, a))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(339,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(element_in_O2(a, c))) | (~subgroup_member(multiply(c, a))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(element_in_O2(a, c))) | subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(multiply(c, a)))))),
% 4.70/4.92 inference(monotonicity,[status(thm)],[338])).
% 4.70/4.92 tff(340,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(element_in_O2(a, c))) | (~subgroup_member(multiply(c, a))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(element_in_O2(a, c))) | subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(multiply(c, a))))),
% 4.70/4.92 inference(transitivity,[status(thm)],[339, 337])).
% 4.70/4.92 tff(341,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(element_in_O2(a, c))) | (~subgroup_member(multiply(c, a))))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(342,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(element_in_O2(a, c))) | subgroup_member(multiply(c, c)) | (~product(multiply(c, a), element_in_O2(a, c), multiply(c, c))) | (~subgroup_member(multiply(c, a)))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[341, 340])).
% 4.70/4.92 tff(343,plain,
% 4.70/4.92 (subgroup_member(multiply(c, c))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[342, 218, 336, 322, 314])).
% 4.70/4.92 tff(344,plain,
% 4.70/4.92 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(c), c, identity)),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(345,plain,
% 4.70/4.92 (product(inverse(c), c, identity)),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[344, 25])).
% 4.70/4.92 tff(346,plain,
% 4.70/4.92 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(c), multiply(c, c), c) | (~product(inverse(c), c, identity)) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | product(inverse(c), multiply(c, c), c) | (~product(inverse(c), c, identity)) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(347,plain,
% 4.70/4.92 ((product(inverse(c), multiply(c, c), c) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c))) | (~product(inverse(c), c, identity))) <=> (product(inverse(c), multiply(c, c), c) | (~product(inverse(c), c, identity)) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(348,plain,
% 4.70/4.92 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(c), multiply(c, c), c) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c))) | (~product(inverse(c), c, identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(c), multiply(c, c), c) | (~product(inverse(c), c, identity)) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c)))))),
% 4.70/4.92 inference(monotonicity,[status(thm)],[347])).
% 4.70/4.92 tff(349,plain,
% 4.70/4.92 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(c), multiply(c, c), c) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c))) | (~product(inverse(c), c, identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | product(inverse(c), multiply(c, c), c) | (~product(inverse(c), c, identity)) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c))))),
% 4.70/4.92 inference(transitivity,[status(thm)],[348, 346])).
% 4.70/4.92 tff(350,plain,
% 4.70/4.92 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(c), multiply(c, c), c) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c))) | (~product(inverse(c), c, identity)))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(351,plain,
% 4.70/4.92 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | product(inverse(c), multiply(c, c), c) | (~product(inverse(c), c, identity)) | (~product(identity, c, c)) | (~product(c, c, multiply(c, c)))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[350, 349])).
% 4.70/4.92 tff(352,plain,
% 4.70/4.92 (product(inverse(c), multiply(c, c), c)),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[351, 128, 345, 116, 306])).
% 4.70/4.92 tff(353,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~subgroup_member(inverse(c))) | (~subgroup_member(multiply(c, c))) | (~product(inverse(c), multiply(c, c), c)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(c) | (~subgroup_member(inverse(c))) | (~subgroup_member(multiply(c, c))) | (~product(inverse(c), multiply(c, c), c)))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(354,plain,
% 4.70/4.92 ((subgroup_member(c) | (~product(inverse(c), multiply(c, c), c)) | (~subgroup_member(multiply(c, c))) | (~subgroup_member(inverse(c)))) <=> (subgroup_member(c) | (~subgroup_member(inverse(c))) | (~subgroup_member(multiply(c, c))) | (~product(inverse(c), multiply(c, c), c)))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(355,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(inverse(c), multiply(c, c), c)) | (~subgroup_member(multiply(c, c))) | (~subgroup_member(inverse(c))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~subgroup_member(inverse(c))) | (~subgroup_member(multiply(c, c))) | (~product(inverse(c), multiply(c, c), c))))),
% 4.70/4.92 inference(monotonicity,[status(thm)],[354])).
% 4.70/4.92 tff(356,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(inverse(c), multiply(c, c), c)) | (~subgroup_member(multiply(c, c))) | (~subgroup_member(inverse(c))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(c) | (~subgroup_member(inverse(c))) | (~subgroup_member(multiply(c, c))) | (~product(inverse(c), multiply(c, c), c)))),
% 4.70/4.92 inference(transitivity,[status(thm)],[355, 353])).
% 4.70/4.92 tff(357,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(inverse(c), multiply(c, c), c)) | (~subgroup_member(multiply(c, c))) | (~subgroup_member(inverse(c))))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(358,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(c) | (~subgroup_member(inverse(c))) | (~subgroup_member(multiply(c, c))) | (~product(inverse(c), multiply(c, c), c))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[357, 356])).
% 4.70/4.92 tff(359,plain,
% 4.70/4.92 (subgroup_member(c) | (~subgroup_member(inverse(c))) | (~subgroup_member(multiply(c, c)))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[358, 218, 352])).
% 4.70/4.92 tff(360,plain,
% 4.70/4.92 (~subgroup_member(inverse(c))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[359, 343, 208])).
% 4.70/4.92 tff(361,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(identity)) | subgroup_member(inverse(c)) | (~subgroup_member(element_in_O2(d, a))) | (~product(identity, element_in_O2(d, a), inverse(c))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(identity)) | subgroup_member(inverse(c)) | (~subgroup_member(element_in_O2(d, a))) | (~product(identity, element_in_O2(d, a), inverse(c))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(362,plain,
% 4.70/4.92 ((subgroup_member(inverse(c)) | (~product(identity, element_in_O2(d, a), inverse(c))) | (~subgroup_member(element_in_O2(d, a))) | (~subgroup_member(identity))) <=> ((~subgroup_member(identity)) | subgroup_member(inverse(c)) | (~subgroup_member(element_in_O2(d, a))) | (~product(identity, element_in_O2(d, a), inverse(c))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(363,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(c)) | (~product(identity, element_in_O2(d, a), inverse(c))) | (~subgroup_member(element_in_O2(d, a))) | (~subgroup_member(identity)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(identity)) | subgroup_member(inverse(c)) | (~subgroup_member(element_in_O2(d, a))) | (~product(identity, element_in_O2(d, a), inverse(c)))))),
% 4.70/4.92 inference(monotonicity,[status(thm)],[362])).
% 4.70/4.92 tff(364,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(c)) | (~product(identity, element_in_O2(d, a), inverse(c))) | (~subgroup_member(element_in_O2(d, a))) | (~subgroup_member(identity)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(identity)) | subgroup_member(inverse(c)) | (~subgroup_member(element_in_O2(d, a))) | (~product(identity, element_in_O2(d, a), inverse(c))))),
% 4.70/4.92 inference(transitivity,[status(thm)],[363, 361])).
% 4.70/4.92 tff(365,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(c)) | (~product(identity, element_in_O2(d, a), inverse(c))) | (~subgroup_member(element_in_O2(d, a))) | (~subgroup_member(identity)))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(366,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(identity)) | subgroup_member(inverse(c)) | (~subgroup_member(element_in_O2(d, a))) | (~product(identity, element_in_O2(d, a), inverse(c)))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[365, 364])).
% 4.70/4.92 tff(367,plain,
% 4.70/4.92 ($false),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[366, 218, 329, 360, 299, 281])).
% 4.70/4.92 tff(368,plain,(subgroup_member(c)), inference(lemma,lemma(discharge,[]))).
% 4.70/4.92 tff(369,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(370,plain,
% 4.70/4.92 ((subgroup_member(inverse(a)) | (~product(inverse(b), c, inverse(a))) | (~subgroup_member(c)) | (~subgroup_member(inverse(b)))) <=> (subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(371,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(inverse(b), c, inverse(a))) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a)))))),
% 4.70/4.92 inference(monotonicity,[status(thm)],[370])).
% 4.70/4.92 tff(372,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(inverse(b), c, inverse(a))) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a))))),
% 4.70/4.92 inference(transitivity,[status(thm)],[371, 369])).
% 4.70/4.92 tff(373,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(inverse(b), c, inverse(a))) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(374,plain,
% 4.70/4.92 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a)))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[373, 372])).
% 4.70/4.92 tff(375,plain,
% 4.70/4.92 (subgroup_member(inverse(a))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[374, 218, 368, 205, 191])).
% 4.70/4.92 tff(376,plain,
% 4.70/4.92 (((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))) <=> ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))),
% 4.70/4.92 inference(rewrite,[status(thm)],[])).
% 4.70/4.92 tff(377,plain,
% 4.70/4.92 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))),
% 4.70/4.92 inference(quant_inst,[status(thm)],[])).
% 4.70/4.92 tff(378,plain,
% 4.70/4.92 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a)))),
% 4.70/4.92 inference(modus_ponens,[status(thm)],[377, 376])).
% 4.70/4.92 tff(379,plain,
% 4.70/4.92 ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a)))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[378, 201])).
% 4.70/4.92 tff(380,plain,
% 4.70/4.92 (subgroup_member(inverse(inverse(a)))),
% 4.70/4.92 inference(unit_resolution,[status(thm)],[379, 375])).
% 4.70/4.92 tff(381,plain,
% 4.70/4.92 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a)))) | subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a)))) | subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))))),
% 4.70/4.93 inference(rewrite,[status(thm)],[])).
% 4.70/4.93 tff(382,plain,
% 4.70/4.93 ((subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))) | (~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a))))) <=> ((~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a)))) | subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))))),
% 4.70/4.93 inference(rewrite,[status(thm)],[])).
% 4.70/4.93 tff(383,plain,
% 4.70/4.93 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))) | (~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a)))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a)))) | subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)))))),
% 4.70/4.93 inference(monotonicity,[status(thm)],[382])).
% 4.70/4.93 tff(384,plain,
% 4.70/4.93 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))) | (~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a)))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a)))) | subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))))),
% 4.70/4.93 inference(transitivity,[status(thm)],[383, 381])).
% 4.70/4.93 tff(385,plain,
% 4.70/4.93 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d))) | (~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a)))))),
% 4.70/4.93 inference(quant_inst,[status(thm)],[])).
% 4.70/4.93 tff(386,plain,
% 4.70/4.93 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~subgroup_member(c)) | (~subgroup_member(inverse(inverse(a)))) | subgroup_member(multiply(inverse(inverse(inverse(identity))), d)) | (~product(inverse(inverse(a)), c, multiply(inverse(inverse(inverse(identity))), d)))),
% 4.70/4.93 inference(modus_ponens,[status(thm)],[385, 384])).
% 4.70/4.93 tff(387,plain,
% 4.70/4.93 ($false),
% 4.70/4.93 inference(unit_resolution,[status(thm)],[386, 218, 368, 380, 152, 145])).
% 4.70/4.93 % SZS output end Proof
% 4.70/4.93 % E exiting
%------------------------------------------------------------------------------