%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL029+4 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 12:33:19 PM UTC 2026
% Result : Theorem 129.46s 16.88s
% Output : Proof 132.91s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : REL029+4 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.06 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.15/0.41 % Computer : n010.cluster.edu
% 0.15/0.41 % Model : x86_64 x86_64
% 0.15/0.41 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.41 % Memory : 8046.5625MB
% 0.15/0.41 % OS : Linux 6.8.0-71-generic
% 0.15/0.41 % CPULimit : 300
% 0.15/0.41 % WCLimit : 300
% 0.15/0.41 % DateTime : Sun Sep 27 22:54:31 UTC 2026
% 0.15/0.42 % CPUTime :
% 0.15/0.42 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 129.46/16.88 Command-line arguments: --no-flatten-goal
% 129.46/16.88
% 129.46/16.88 % SZS status Theorem
% 129.46/16.88
% 131.64/17.17 % SZS output start Proof
% 131.64/17.17 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 131.64/17.17 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 131.64/17.17 Axiom 3 (goals): join(x1, one) = one.
% 131.64/17.17 Axiom 4 (goals_1): join(x0, one) = one.
% 131.64/17.17 Axiom 5 (composition_identity): composition(X, one) = X.
% 131.64/17.17 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 131.64/17.17 Axiom 7 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 131.64/17.17 Axiom 8 (def_top): top = join(X, complement(X)).
% 131.64/17.17 Axiom 9 (def_zero): zero = meet(X, complement(X)).
% 131.64/17.17 Axiom 10 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 131.64/17.17 Axiom 11 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 131.64/17.17 Axiom 12 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 131.64/17.17 Axiom 13 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 131.64/17.17 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 131.64/17.18 Axiom 15 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 131.64/17.18
% 131.64/17.18 Lemma 16: complement(top) = zero.
% 131.64/17.18 Proof:
% 131.64/17.18 complement(top)
% 131.64/17.18 = { by axiom 8 (def_top) }
% 131.64/17.18 complement(join(complement(X), complement(complement(X))))
% 131.64/17.18 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 131.64/17.18 meet(X, complement(X))
% 131.64/17.18 = { by axiom 9 (def_zero) R->L }
% 131.64/17.18 zero
% 131.64/17.18
% 131.64/17.18 Lemma 17: composition(converse(one), X) = X.
% 131.64/17.18 Proof:
% 131.64/17.18 composition(converse(one), X)
% 131.64/17.18 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.18 composition(converse(one), converse(converse(X)))
% 131.64/17.18 = { by axiom 7 (converse_multiplicativity) R->L }
% 131.64/17.18 converse(composition(converse(X), one))
% 131.64/17.18 = { by axiom 5 (composition_identity) }
% 131.64/17.18 converse(converse(X))
% 131.64/17.18 = { by axiom 1 (converse_idempotence) }
% 131.64/17.18 X
% 131.64/17.18
% 131.64/17.18 Lemma 18: converse(one) = one.
% 131.64/17.18 Proof:
% 131.64/17.18 converse(one)
% 131.64/17.18 = { by axiom 5 (composition_identity) R->L }
% 131.64/17.18 composition(converse(one), one)
% 131.64/17.18 = { by lemma 17 }
% 131.64/17.18 one
% 131.64/17.18
% 131.64/17.18 Lemma 19: composition(one, X) = X.
% 131.64/17.18 Proof:
% 131.64/17.18 composition(one, X)
% 131.64/17.18 = { by lemma 18 R->L }
% 131.64/17.18 composition(converse(one), X)
% 131.64/17.18 = { by lemma 17 }
% 131.64/17.18 X
% 131.64/17.18
% 131.64/17.18 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 131.64/17.18 Proof:
% 131.64/17.18 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 131.64/17.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.18 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 131.64/17.18 = { by axiom 14 (converse_cancellativity) }
% 131.64/17.18 complement(X)
% 131.64/17.18
% 131.64/17.18 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 131.64/17.18 Proof:
% 131.64/17.18 join(complement(X), complement(X))
% 131.64/17.18 = { by lemma 17 R->L }
% 131.64/17.18 join(complement(X), composition(converse(one), complement(X)))
% 131.64/17.18 = { by lemma 19 R->L }
% 131.64/17.18 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 131.64/17.18 = { by lemma 20 }
% 131.64/17.18 complement(X)
% 131.64/17.18
% 131.64/17.18 Lemma 22: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 131.64/17.18 Proof:
% 131.64/17.18 join(meet(X, Y), complement(join(complement(X), Y)))
% 131.64/17.18 = { by axiom 12 (maddux4_definiton_of_meet) }
% 131.64/17.18 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 131.64/17.18 = { by axiom 15 (maddux3_a_kind_of_de_Morgan) R->L }
% 131.64/17.18 X
% 131.64/17.18
% 131.64/17.18 Lemma 23: join(zero, complement(complement(X))) = X.
% 131.64/17.18 Proof:
% 131.64/17.18 join(zero, complement(complement(X)))
% 131.64/17.18 = { by axiom 9 (def_zero) }
% 131.64/17.18 join(meet(X, complement(X)), complement(complement(X)))
% 131.64/17.18 = { by lemma 21 R->L }
% 131.64/17.18 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 131.64/17.18 = { by lemma 22 }
% 131.64/17.18 X
% 131.64/17.18
% 131.64/17.18 Lemma 24: meet(Y, X) = meet(X, Y).
% 131.64/17.18 Proof:
% 131.64/17.18 meet(Y, X)
% 131.64/17.18 = { by axiom 12 (maddux4_definiton_of_meet) }
% 131.64/17.18 complement(join(complement(Y), complement(X)))
% 131.64/17.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.18 complement(join(complement(X), complement(Y)))
% 131.64/17.18 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 131.64/17.18 meet(X, Y)
% 131.64/17.18
% 131.64/17.18 Lemma 25: complement(join(zero, complement(X))) = meet(X, top).
% 131.64/17.18 Proof:
% 131.64/17.18 complement(join(zero, complement(X)))
% 131.64/17.18 = { by lemma 16 R->L }
% 131.64/17.18 complement(join(complement(top), complement(X)))
% 131.64/17.18 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 131.64/17.18 meet(top, X)
% 131.64/17.18 = { by lemma 24 R->L }
% 131.64/17.18 meet(X, top)
% 131.64/17.18
% 131.64/17.18 Lemma 26: join(X, join(Y, complement(X))) = join(Y, top).
% 131.64/17.18 Proof:
% 131.64/17.18 join(X, join(Y, complement(X)))
% 131.64/17.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.18 join(X, join(complement(X), Y))
% 131.64/17.18 = { by axiom 10 (maddux2_join_associativity) }
% 131.64/17.18 join(join(X, complement(X)), Y)
% 131.64/17.18 = { by axiom 8 (def_top) R->L }
% 131.64/17.18 join(top, Y)
% 131.64/17.18 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.18 join(Y, top)
% 131.64/17.18
% 131.64/17.18 Lemma 27: join(top, complement(X)) = top.
% 131.64/17.18 Proof:
% 131.64/17.18 join(top, complement(X))
% 131.64/17.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.18 join(complement(X), top)
% 131.64/17.18 = { by lemma 26 R->L }
% 131.64/17.18 join(X, join(complement(X), complement(X)))
% 131.64/17.18 = { by lemma 21 }
% 131.64/17.18 join(X, complement(X))
% 131.64/17.18 = { by axiom 8 (def_top) R->L }
% 131.64/17.18 top
% 131.64/17.18
% 131.64/17.18 Lemma 28: join(X, join(complement(X), Y)) = join(Z, top).
% 131.64/17.18 Proof:
% 131.64/17.18 join(X, join(complement(X), Y))
% 131.64/17.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.18 join(X, join(Y, complement(X)))
% 131.64/17.18 = { by lemma 26 }
% 131.64/17.18 join(Y, top)
% 131.64/17.18 = { by lemma 27 R->L }
% 131.64/17.18 join(Y, join(top, complement(Y)))
% 131.64/17.18 = { by lemma 26 }
% 131.64/17.18 join(top, top)
% 131.64/17.18 = { by lemma 26 R->L }
% 131.64/17.18 join(Z, join(top, complement(Z)))
% 131.64/17.18 = { by lemma 27 }
% 131.64/17.18 join(Z, top)
% 131.64/17.18
% 131.64/17.18 Lemma 29: join(X, top) = top.
% 131.64/17.18 Proof:
% 131.64/17.18 join(X, top)
% 131.64/17.18 = { by lemma 28 R->L }
% 131.64/17.18 join(Y, join(complement(Y), composition(converse(Z), complement(composition(Z, Y)))))
% 131.64/17.18 = { by lemma 20 }
% 131.64/17.18 join(Y, complement(Y))
% 131.64/17.18 = { by axiom 8 (def_top) R->L }
% 131.64/17.18 top
% 131.64/17.18
% 131.64/17.18 Lemma 30: join(X, converse(top)) = top.
% 131.64/17.18 Proof:
% 131.64/17.18 join(X, converse(top))
% 131.64/17.18 = { by axiom 8 (def_top) }
% 131.64/17.18 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 131.64/17.18 = { by axiom 6 (converse_additivity) }
% 131.64/17.18 join(X, join(converse(converse(complement(X))), converse(complement(converse(complement(X))))))
% 131.64/17.18 = { by axiom 1 (converse_idempotence) }
% 131.64/17.18 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 131.64/17.19 = { by lemma 28 }
% 131.64/17.19 join(Y, top)
% 131.64/17.19 = { by lemma 29 }
% 131.64/17.19 top
% 131.64/17.19
% 131.64/17.19 Lemma 31: join(zero, meet(X, top)) = X.
% 131.64/17.19 Proof:
% 131.64/17.19 join(zero, meet(X, top))
% 131.64/17.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.19 join(meet(X, top), zero)
% 131.64/17.19 = { by lemma 16 R->L }
% 131.64/17.19 join(meet(X, top), complement(top))
% 131.64/17.19 = { by lemma 29 R->L }
% 131.64/17.19 join(meet(X, join(converse(Y), top)), complement(top))
% 131.64/17.19 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.19 join(meet(X, join(converse(Y), converse(converse(top)))), complement(top))
% 131.64/17.19 = { by axiom 6 (converse_additivity) R->L }
% 131.64/17.19 join(meet(X, converse(join(Y, converse(top)))), complement(top))
% 131.64/17.19 = { by lemma 30 }
% 131.64/17.19 join(meet(X, converse(top)), complement(top))
% 131.64/17.19 = { by lemma 30 R->L }
% 131.64/17.19 join(meet(X, converse(top)), complement(join(complement(X), converse(top))))
% 131.64/17.19 = { by lemma 22 }
% 131.64/17.19 X
% 131.64/17.19
% 131.64/17.19 Lemma 32: join(zero, complement(X)) = complement(X).
% 131.64/17.19 Proof:
% 131.64/17.19 join(zero, complement(X))
% 131.64/17.19 = { by lemma 23 R->L }
% 131.64/17.19 join(zero, complement(join(zero, complement(complement(X)))))
% 131.64/17.19 = { by lemma 25 }
% 131.64/17.19 join(zero, meet(complement(X), top))
% 131.64/17.19 = { by lemma 31 }
% 131.64/17.19 complement(X)
% 131.64/17.19
% 131.64/17.19 Lemma 33: complement(complement(X)) = X.
% 131.64/17.19 Proof:
% 131.64/17.19 complement(complement(X))
% 131.64/17.19 = { by lemma 32 R->L }
% 131.64/17.19 join(zero, complement(complement(X)))
% 131.64/17.19 = { by lemma 23 }
% 131.64/17.19 X
% 131.64/17.19
% 131.64/17.19 Lemma 34: join(zero, meet(X, X)) = X.
% 131.64/17.19 Proof:
% 131.64/17.19 join(zero, meet(X, X))
% 131.64/17.19 = { by axiom 12 (maddux4_definiton_of_meet) }
% 131.64/17.19 join(zero, complement(join(complement(X), complement(X))))
% 131.64/17.19 = { by axiom 9 (def_zero) }
% 131.64/17.19 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 131.64/17.19 = { by lemma 22 }
% 131.64/17.19 X
% 131.64/17.19
% 131.64/17.19 Lemma 35: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 131.64/17.19 Proof:
% 131.64/17.19 join(zero, join(X, complement(complement(Y))))
% 131.64/17.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.19 join(zero, join(complement(complement(Y)), X))
% 131.64/17.19 = { by lemma 21 R->L }
% 131.64/17.19 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 131.64/17.19 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 131.64/17.19 join(zero, join(meet(Y, Y), X))
% 131.64/17.19 = { by axiom 10 (maddux2_join_associativity) }
% 131.64/17.19 join(join(zero, meet(Y, Y)), X)
% 131.64/17.19 = { by lemma 34 }
% 131.64/17.19 join(Y, X)
% 131.64/17.19 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.19 join(X, Y)
% 131.64/17.19
% 131.64/17.19 Lemma 36: join(X, X) = X.
% 131.64/17.19 Proof:
% 131.64/17.19 join(X, X)
% 131.64/17.19 = { by lemma 33 R->L }
% 131.64/17.19 join(X, complement(complement(X)))
% 131.64/17.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.19 join(complement(complement(X)), X)
% 131.64/17.19 = { by lemma 35 R->L }
% 131.64/17.19 join(zero, join(complement(complement(X)), complement(complement(X))))
% 131.64/17.19 = { by lemma 21 }
% 131.64/17.19 join(zero, complement(complement(X)))
% 131.64/17.19 = { by lemma 23 }
% 131.64/17.19 X
% 131.64/17.19
% 131.64/17.19 Lemma 37: join(X, zero) = X.
% 131.64/17.19 Proof:
% 131.64/17.19 join(X, zero)
% 131.64/17.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.19 join(zero, X)
% 131.64/17.19 = { by lemma 35 R->L }
% 131.64/17.19 join(zero, join(zero, complement(complement(X))))
% 131.64/17.19 = { by lemma 32 }
% 131.64/17.19 join(zero, complement(complement(X)))
% 131.64/17.19 = { by lemma 23 }
% 131.64/17.19 X
% 131.64/17.19
% 131.64/17.19 Lemma 38: join(one, x1) = one.
% 131.64/17.19 Proof:
% 131.64/17.19 join(one, x1)
% 131.64/17.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.19 join(x1, one)
% 131.64/17.19 = { by axiom 3 (goals) }
% 131.64/17.19 one
% 131.64/17.19
% 131.64/17.19 Lemma 39: join(one, x0) = one.
% 131.64/17.19 Proof:
% 131.64/17.19 join(one, x0)
% 131.64/17.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.19 join(x0, one)
% 131.64/17.19 = { by axiom 4 (goals_1) }
% 131.64/17.19 one
% 131.64/17.19
% 131.64/17.19 Lemma 40: meet(X, top) = X.
% 131.64/17.19 Proof:
% 131.64/17.19 meet(X, top)
% 131.64/17.19 = { by lemma 25 R->L }
% 131.64/17.19 complement(join(zero, complement(X)))
% 131.64/17.19 = { by lemma 32 R->L }
% 131.64/17.19 join(zero, complement(join(zero, complement(X))))
% 131.64/17.19 = { by lemma 25 }
% 131.64/17.19 join(zero, meet(X, top))
% 131.64/17.19 = { by lemma 31 }
% 131.64/17.19 X
% 131.64/17.19
% 131.64/17.19 Lemma 41: meet(top, X) = X.
% 131.64/17.19 Proof:
% 131.64/17.19 meet(top, X)
% 131.64/17.19 = { by lemma 24 }
% 131.64/17.19 meet(X, top)
% 131.64/17.19 = { by lemma 40 }
% 131.64/17.19 X
% 131.64/17.19
% 131.64/17.19 Lemma 42: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 131.64/17.19 Proof:
% 131.64/17.19 composition(join(one, Y), X)
% 131.64/17.20 = { by lemma 18 R->L }
% 131.64/17.20 composition(join(converse(one), Y), X)
% 131.64/17.20 = { by axiom 13 (composition_distributivity) }
% 131.64/17.20 join(composition(converse(one), X), composition(Y, X))
% 131.64/17.20 = { by lemma 17 }
% 131.64/17.20 join(X, composition(Y, X))
% 131.64/17.20
% 131.64/17.20 Lemma 43: join(X, composition(X, x1)) = X.
% 131.64/17.20 Proof:
% 131.64/17.20 join(X, composition(X, x1))
% 131.64/17.20 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.20 join(X, composition(converse(converse(X)), x1))
% 131.64/17.20 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.20 join(converse(converse(X)), composition(converse(converse(X)), x1))
% 131.64/17.20 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.20 join(converse(converse(X)), composition(converse(converse(X)), converse(converse(x1))))
% 131.64/17.20 = { by axiom 7 (converse_multiplicativity) R->L }
% 131.64/17.20 join(converse(converse(X)), converse(composition(converse(x1), converse(X))))
% 131.64/17.20 = { by axiom 6 (converse_additivity) R->L }
% 131.64/17.20 converse(join(converse(X), composition(converse(x1), converse(X))))
% 131.64/17.20 = { by lemma 42 R->L }
% 131.64/17.20 converse(composition(join(one, converse(x1)), converse(X)))
% 131.64/17.20 = { by lemma 18 R->L }
% 131.64/17.20 converse(composition(join(converse(one), converse(x1)), converse(X)))
% 131.64/17.20 = { by axiom 6 (converse_additivity) R->L }
% 131.64/17.20 converse(composition(converse(join(one, x1)), converse(X)))
% 131.64/17.20 = { by lemma 38 }
% 131.64/17.20 converse(composition(converse(one), converse(X)))
% 131.64/17.20 = { by lemma 18 }
% 131.64/17.20 converse(composition(one, converse(X)))
% 131.64/17.20 = { by lemma 19 }
% 131.64/17.20 converse(converse(X))
% 131.64/17.20 = { by axiom 1 (converse_idempotence) }
% 131.64/17.20 X
% 131.64/17.20
% 131.64/17.20 Lemma 44: join(X, composition(X, x0)) = X.
% 131.64/17.20 Proof:
% 131.64/17.20 join(X, composition(X, x0))
% 131.64/17.20 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.20 join(X, composition(converse(converse(X)), x0))
% 131.64/17.20 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.20 join(converse(converse(X)), composition(converse(converse(X)), x0))
% 131.64/17.20 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.20 join(converse(converse(X)), composition(converse(converse(X)), converse(converse(x0))))
% 131.64/17.20 = { by axiom 7 (converse_multiplicativity) R->L }
% 131.64/17.20 join(converse(converse(X)), converse(composition(converse(x0), converse(X))))
% 131.64/17.20 = { by axiom 6 (converse_additivity) R->L }
% 131.64/17.20 converse(join(converse(X), composition(converse(x0), converse(X))))
% 131.64/17.20 = { by lemma 42 R->L }
% 131.64/17.20 converse(composition(join(one, converse(x0)), converse(X)))
% 131.64/17.20 = { by lemma 18 R->L }
% 131.64/17.20 converse(composition(join(converse(one), converse(x0)), converse(X)))
% 131.64/17.20 = { by axiom 6 (converse_additivity) R->L }
% 131.64/17.20 converse(composition(converse(join(one, x0)), converse(X)))
% 131.64/17.20 = { by lemma 39 }
% 131.64/17.20 converse(composition(converse(one), converse(X)))
% 131.64/17.20 = { by lemma 18 }
% 131.64/17.20 converse(composition(one, converse(X)))
% 131.64/17.20 = { by lemma 19 }
% 131.64/17.20 converse(converse(X))
% 131.64/17.20 = { by axiom 1 (converse_idempotence) }
% 131.64/17.20 X
% 131.64/17.20
% 131.64/17.20 Lemma 45: join(X, composition(x0, X)) = X.
% 131.64/17.20 Proof:
% 131.64/17.20 join(X, composition(x0, X))
% 131.64/17.20 = { by lemma 42 R->L }
% 131.64/17.20 composition(join(one, x0), X)
% 131.64/17.20 = { by lemma 39 }
% 131.64/17.20 composition(one, X)
% 131.64/17.20 = { by lemma 19 }
% 131.64/17.20 X
% 131.64/17.20
% 131.64/17.20 Lemma 46: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 131.64/17.20 Proof:
% 131.64/17.20 join(Y, join(X, Z))
% 131.64/17.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.20 join(join(X, Z), Y)
% 131.64/17.20 = { by axiom 10 (maddux2_join_associativity) R->L }
% 131.64/17.20 join(X, join(Z, Y))
% 131.64/17.20 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.20 join(X, join(Y, Z))
% 131.64/17.20
% 131.64/17.20 Lemma 47: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 131.64/17.20 Proof:
% 131.64/17.20 meet(X, join(complement(Y), complement(Z)))
% 131.64/17.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.20 meet(X, join(complement(Z), complement(Y)))
% 131.64/17.20 = { by lemma 24 }
% 131.64/17.20 meet(join(complement(Z), complement(Y)), X)
% 131.64/17.20 = { by axiom 12 (maddux4_definiton_of_meet) }
% 131.64/17.20 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 131.64/17.20 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 131.64/17.20 complement(join(meet(Z, Y), complement(X)))
% 131.64/17.20 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.20 complement(join(complement(X), meet(Z, Y)))
% 131.64/17.20 = { by lemma 24 R->L }
% 131.64/17.20 complement(join(complement(X), meet(Y, Z)))
% 131.64/17.20
% 131.64/17.20 Lemma 48: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 131.64/17.20 Proof:
% 131.64/17.20 join(complement(X), complement(Y))
% 131.64/17.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.20 join(complement(Y), complement(X))
% 131.64/17.20 = { by lemma 34 R->L }
% 131.64/17.20 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 131.64/17.20 = { by lemma 47 }
% 131.64/17.20 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 131.64/17.20 = { by lemma 32 }
% 131.64/17.20 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 131.64/17.20 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 131.64/17.20 complement(join(meet(Y, X), meet(Y, X)))
% 131.64/17.20 = { by lemma 36 }
% 131.64/17.20 complement(meet(Y, X))
% 131.64/17.20 = { by lemma 24 R->L }
% 131.64/17.20 complement(meet(X, Y))
% 131.64/17.20
% 131.64/17.20 Lemma 49: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 131.64/17.20 Proof:
% 131.64/17.20 complement(join(X, complement(Y)))
% 131.64/17.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.20 complement(join(complement(Y), X))
% 131.64/17.20 = { by lemma 41 R->L }
% 131.64/17.20 complement(join(complement(Y), meet(top, X)))
% 131.64/17.20 = { by lemma 47 R->L }
% 131.64/17.20 meet(Y, join(complement(top), complement(X)))
% 131.64/17.20 = { by lemma 16 }
% 131.64/17.20 meet(Y, join(zero, complement(X)))
% 131.64/17.20 = { by lemma 32 }
% 131.64/17.20 meet(Y, complement(X))
% 131.64/17.20
% 131.64/17.20 Lemma 50: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 131.64/17.20 Proof:
% 131.64/17.20 complement(join(complement(X), Y))
% 131.64/17.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.20 complement(join(Y, complement(X)))
% 131.64/17.20 = { by lemma 49 }
% 131.64/17.20 meet(X, complement(Y))
% 131.64/17.20
% 131.64/17.20 Lemma 51: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 131.64/17.20 Proof:
% 131.64/17.20 complement(meet(X, complement(Y)))
% 131.64/17.20 = { by lemma 37 R->L }
% 131.64/17.20 complement(join(meet(X, complement(Y)), zero))
% 131.64/17.20 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.20 complement(join(zero, meet(X, complement(Y))))
% 131.64/17.20 = { by lemma 49 R->L }
% 131.64/17.20 complement(join(zero, complement(join(Y, complement(X)))))
% 131.64/17.20 = { by lemma 25 }
% 131.64/17.20 meet(join(Y, complement(X)), top)
% 131.64/17.20 = { by lemma 40 }
% 131.64/17.20 join(Y, complement(X))
% 131.64/17.20
% 131.64/17.20 Lemma 52: meet(X, join(X, complement(Y))) = X.
% 131.64/17.20 Proof:
% 131.64/17.20 meet(X, join(X, complement(Y)))
% 131.64/17.20 = { by lemma 51 R->L }
% 131.64/17.20 meet(X, complement(meet(Y, complement(X))))
% 131.64/17.20 = { by lemma 50 R->L }
% 131.64/17.20 complement(join(complement(X), meet(Y, complement(X))))
% 131.64/17.20 = { by lemma 32 R->L }
% 131.64/17.20 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 131.64/17.20 = { by lemma 16 R->L }
% 131.64/17.20 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 131.64/17.20 = { by lemma 29 R->L }
% 131.64/17.20 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 131.64/17.20 = { by lemma 26 R->L }
% 131.64/17.20 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 131.64/17.20 = { by lemma 48 }
% 131.64/17.20 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 131.64/17.20 = { by lemma 24 R->L }
% 131.64/17.20 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 131.64/17.21 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 131.64/17.21 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 131.64/17.21 = { by lemma 24 R->L }
% 131.64/17.21 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 131.64/17.21 = { by lemma 22 }
% 131.64/17.21 X
% 131.64/17.21
% 131.64/17.21 Lemma 53: meet(X, join(Y, X)) = X.
% 131.64/17.21 Proof:
% 131.64/17.21 meet(X, join(Y, X))
% 131.64/17.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.21 meet(X, join(X, Y))
% 131.64/17.21 = { by lemma 40 R->L }
% 131.64/17.21 meet(X, join(X, meet(Y, top)))
% 131.64/17.21 = { by lemma 25 R->L }
% 131.64/17.21 meet(X, join(X, complement(join(zero, complement(Y)))))
% 131.64/17.21 = { by lemma 52 }
% 131.64/17.21 X
% 131.64/17.21
% 131.64/17.21 Lemma 54: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 131.64/17.21 Proof:
% 131.64/17.21 meet(complement(X), join(Y, complement(Z)))
% 131.64/17.21 = { by lemma 24 }
% 131.64/17.21 meet(join(Y, complement(Z)), complement(X))
% 131.64/17.21 = { by lemma 49 R->L }
% 131.64/17.21 complement(join(X, complement(join(Y, complement(Z)))))
% 131.64/17.21 = { by lemma 49 }
% 131.64/17.21 complement(join(X, meet(Z, complement(Y))))
% 131.64/17.21
% 131.64/17.21 Lemma 55: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 131.64/17.21 Proof:
% 131.64/17.21 meet(complement(X), complement(Y))
% 131.64/17.21 = { by lemma 24 }
% 131.64/17.21 meet(complement(Y), complement(X))
% 131.64/17.21 = { by lemma 32 R->L }
% 131.64/17.21 meet(join(zero, complement(Y)), complement(X))
% 131.64/17.21 = { by lemma 49 R->L }
% 131.64/17.21 complement(join(X, complement(join(zero, complement(Y)))))
% 131.64/17.21 = { by lemma 25 }
% 131.64/17.21 complement(join(X, meet(Y, top)))
% 131.64/17.21 = { by lemma 40 }
% 131.64/17.21 complement(join(X, Y))
% 131.64/17.21
% 131.64/17.21 Lemma 56: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 131.64/17.21 Proof:
% 131.64/17.21 join(meet(X, Y), meet(Y, complement(X)))
% 131.64/17.21 = { by lemma 24 }
% 131.64/17.21 join(meet(Y, X), meet(Y, complement(X)))
% 131.64/17.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.21 join(meet(Y, complement(X)), meet(Y, X))
% 131.64/17.21 = { by axiom 12 (maddux4_definiton_of_meet) }
% 131.64/17.21 join(meet(Y, complement(X)), complement(join(complement(Y), complement(X))))
% 131.64/17.21 = { by lemma 22 }
% 131.64/17.21 Y
% 131.64/17.21
% 131.64/17.21 Lemma 57: meet(complement(Z), meet(X, Y)) = meet(X, meet(Y, complement(Z))).
% 131.64/17.21 Proof:
% 131.64/17.21 meet(complement(Z), meet(X, Y))
% 131.64/17.21 = { by lemma 24 }
% 131.64/17.21 meet(meet(X, Y), complement(Z))
% 131.64/17.21 = { by axiom 12 (maddux4_definiton_of_meet) }
% 131.64/17.21 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 131.64/17.21 = { by lemma 55 }
% 131.64/17.21 complement(join(join(complement(X), complement(Y)), Z))
% 131.64/17.21 = { by axiom 10 (maddux2_join_associativity) R->L }
% 131.64/17.21 complement(join(complement(X), join(complement(Y), Z)))
% 131.64/17.21 = { by lemma 50 }
% 131.64/17.21 meet(X, complement(join(complement(Y), Z)))
% 131.64/17.21 = { by lemma 50 }
% 131.64/17.21 meet(X, meet(Y, complement(Z)))
% 131.64/17.21
% 131.64/17.21 Lemma 58: meet(X, join(complement(X), Y)) = meet(X, Y).
% 131.64/17.21 Proof:
% 131.64/17.21 meet(X, join(complement(X), Y))
% 131.64/17.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.21 meet(X, join(Y, complement(X)))
% 131.64/17.21 = { by lemma 51 R->L }
% 131.64/17.21 meet(X, complement(meet(X, complement(Y))))
% 131.64/17.21 = { by lemma 50 R->L }
% 131.64/17.21 complement(join(complement(X), meet(X, complement(Y))))
% 131.64/17.21 = { by lemma 47 R->L }
% 131.64/17.21 meet(X, join(complement(X), complement(complement(Y))))
% 131.64/17.21 = { by lemma 51 R->L }
% 131.64/17.21 meet(X, complement(meet(complement(Y), complement(complement(X)))))
% 131.64/17.21 = { by lemma 50 R->L }
% 131.64/17.21 complement(join(complement(X), meet(complement(Y), complement(complement(X)))))
% 131.64/17.21 = { by lemma 54 R->L }
% 131.64/17.21 meet(complement(complement(X)), join(complement(X), complement(complement(Y))))
% 131.64/17.21 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.21 meet(complement(complement(X)), join(complement(complement(Y)), complement(X)))
% 131.64/17.21 = { by lemma 24 }
% 131.64/17.21 meet(join(complement(complement(Y)), complement(X)), complement(complement(X)))
% 131.64/17.21 = { by lemma 49 R->L }
% 131.64/17.21 complement(join(complement(X), complement(join(complement(complement(Y)), complement(X)))))
% 131.64/17.21 = { by lemma 55 R->L }
% 131.64/17.21 complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))
% 131.64/17.21 = { by lemma 56 R->L }
% 131.64/17.21 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X))))), complement(complement(complement(Y)))))
% 131.64/17.21 = { by lemma 24 R->L }
% 131.64/17.21 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(complement(Y))), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))))
% 131.64/17.21 = { by lemma 54 R->L }
% 131.64/17.21 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(complement(Y))), meet(complement(complement(X)), join(complement(X), complement(complement(complement(complement(Y))))))))
% 131.64/17.21 = { by lemma 24 }
% 131.64/17.22 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(complement(Y))), meet(join(complement(X), complement(complement(complement(complement(Y))))), complement(complement(X)))))
% 131.64/17.22 = { by lemma 57 R->L }
% 131.64/17.22 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(X)), meet(complement(complement(complement(Y))), join(complement(X), complement(complement(complement(complement(Y))))))))
% 131.64/17.22 = { by lemma 24 }
% 131.64/17.22 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(X)), meet(join(complement(X), complement(complement(complement(complement(Y))))), complement(complement(complement(Y))))))
% 131.64/17.22 = { by lemma 57 }
% 131.64/17.22 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(join(complement(X), complement(complement(complement(complement(Y))))), meet(complement(complement(complement(Y))), complement(complement(X)))))
% 131.64/17.22 = { by lemma 49 R->L }
% 131.64/17.22 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(join(complement(X), complement(complement(complement(complement(Y))))), complement(join(complement(X), complement(complement(complement(complement(Y))))))))
% 131.64/17.22 = { by axiom 9 (def_zero) R->L }
% 131.64/17.22 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), zero)
% 131.64/17.22 = { by lemma 37 }
% 131.64/17.22 meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X))))))
% 131.64/17.22 = { by lemma 55 }
% 131.64/17.22 meet(complement(complement(Y)), complement(join(complement(X), complement(join(complement(complement(Y)), complement(X))))))
% 131.64/17.22 = { by lemma 49 }
% 131.64/17.22 meet(complement(complement(Y)), meet(join(complement(complement(Y)), complement(X)), complement(complement(X))))
% 131.64/17.22 = { by lemma 24 R->L }
% 131.64/17.22 meet(complement(complement(Y)), meet(complement(complement(X)), join(complement(complement(Y)), complement(X))))
% 131.64/17.22 = { by lemma 48 }
% 131.64/17.22 meet(complement(complement(Y)), meet(complement(complement(X)), complement(meet(complement(Y), X))))
% 131.64/17.22 = { by lemma 55 }
% 131.64/17.22 meet(complement(complement(Y)), complement(join(complement(X), meet(complement(Y), X))))
% 131.64/17.22 = { by lemma 55 }
% 131.64/17.22 complement(join(complement(Y), join(complement(X), meet(complement(Y), X))))
% 131.64/17.22 = { by lemma 46 }
% 131.64/17.22 complement(join(complement(X), join(complement(Y), meet(complement(Y), X))))
% 131.64/17.22 = { by lemma 50 }
% 131.64/17.22 meet(X, complement(join(complement(Y), meet(complement(Y), X))))
% 131.64/17.22 = { by axiom 12 (maddux4_definiton_of_meet) }
% 131.64/17.22 meet(X, complement(join(complement(Y), complement(join(complement(complement(Y)), complement(X))))))
% 131.64/17.22 = { by lemma 51 R->L }
% 131.64/17.22 meet(X, complement(complement(meet(join(complement(complement(Y)), complement(X)), complement(complement(Y))))))
% 131.64/17.22 = { by lemma 24 R->L }
% 131.64/17.22 meet(X, complement(complement(meet(complement(complement(Y)), join(complement(complement(Y)), complement(X))))))
% 131.64/17.22 = { by lemma 52 }
% 131.64/17.22 meet(X, complement(complement(complement(complement(Y)))))
% 131.64/17.22 = { by lemma 33 }
% 131.64/17.22 meet(X, complement(complement(Y)))
% 131.64/17.22 = { by lemma 33 }
% 131.64/17.22 meet(X, Y)
% 131.64/17.22
% 131.64/17.22 Lemma 59: join(composition(X, Y), join(Z, composition(W, Y))) = join(Z, composition(join(X, W), Y)).
% 131.64/17.22 Proof:
% 131.64/17.22 join(composition(X, Y), join(Z, composition(W, Y)))
% 131.64/17.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.22 join(composition(X, Y), join(composition(W, Y), Z))
% 131.64/17.22 = { by lemma 46 }
% 131.64/17.22 join(composition(W, Y), join(composition(X, Y), Z))
% 131.64/17.22 = { by axiom 10 (maddux2_join_associativity) }
% 131.64/17.22 join(join(composition(W, Y), composition(X, Y)), Z)
% 131.64/17.22 = { by axiom 13 (composition_distributivity) R->L }
% 131.64/17.22 join(composition(join(W, X), Y), Z)
% 131.64/17.22 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.22 join(Z, composition(join(W, X), Y))
% 131.64/17.22 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.22 join(Z, composition(join(X, W), Y))
% 131.64/17.22
% 131.64/17.22 Lemma 60: meet(X, composition(top, x1)) = composition(X, x1).
% 131.64/17.22 Proof:
% 131.64/17.22 meet(X, composition(top, x1))
% 131.64/17.22 = { by lemma 58 R->L }
% 131.64/17.22 meet(X, join(complement(X), composition(top, x1)))
% 131.64/17.22 = { by axiom 8 (def_top) }
% 131.64/17.22 meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), x1)))
% 131.64/17.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.22 meet(X, join(complement(X), composition(join(complement(complement(X)), complement(X)), x1)))
% 131.64/17.22 = { by lemma 59 R->L }
% 131.64/17.22 meet(X, join(composition(complement(complement(X)), x1), join(complement(X), composition(complement(X), x1))))
% 131.64/17.22 = { by lemma 43 }
% 131.64/17.22 meet(X, join(composition(complement(complement(X)), x1), complement(X)))
% 131.64/17.22 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.22 meet(X, join(complement(X), composition(complement(complement(X)), x1)))
% 131.64/17.22 = { by lemma 58 }
% 131.64/17.22 meet(X, composition(complement(complement(X)), x1))
% 131.64/17.22 = { by lemma 33 }
% 131.64/17.22 meet(X, composition(X, x1))
% 131.64/17.22 = { by lemma 24 }
% 131.64/17.22 meet(composition(X, x1), X)
% 131.64/17.22 = { by lemma 43 R->L }
% 131.64/17.22 meet(composition(X, x1), join(X, composition(X, x1)))
% 131.64/17.22 = { by lemma 53 }
% 131.64/17.23 composition(X, x1)
% 131.64/17.23
% 131.64/17.23 Lemma 61: meet(X, composition(top, x0)) = composition(X, x0).
% 131.64/17.23 Proof:
% 131.64/17.23 meet(X, composition(top, x0))
% 131.64/17.23 = { by lemma 58 R->L }
% 131.64/17.23 meet(X, join(complement(X), composition(top, x0)))
% 131.64/17.23 = { by axiom 8 (def_top) }
% 131.64/17.23 meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), x0)))
% 131.64/17.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.23 meet(X, join(complement(X), composition(join(complement(complement(X)), complement(X)), x0)))
% 131.64/17.23 = { by lemma 59 R->L }
% 131.64/17.23 meet(X, join(composition(complement(complement(X)), x0), join(complement(X), composition(complement(X), x0))))
% 131.64/17.23 = { by lemma 44 }
% 131.64/17.23 meet(X, join(composition(complement(complement(X)), x0), complement(X)))
% 131.64/17.23 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.23 meet(X, join(complement(X), composition(complement(complement(X)), x0)))
% 131.64/17.23 = { by lemma 58 }
% 131.64/17.23 meet(X, composition(complement(complement(X)), x0))
% 131.64/17.23 = { by lemma 33 }
% 131.64/17.23 meet(X, composition(X, x0))
% 131.64/17.23 = { by lemma 24 }
% 131.64/17.23 meet(composition(X, x0), X)
% 131.64/17.23 = { by lemma 44 R->L }
% 131.64/17.23 meet(composition(X, x0), join(X, composition(X, x0)))
% 131.64/17.23 = { by lemma 53 }
% 131.64/17.23 composition(X, x0)
% 131.64/17.23
% 131.64/17.23 Lemma 62: meet(X, composition(x0, top)) = composition(x0, X).
% 131.64/17.23 Proof:
% 131.64/17.23 meet(X, composition(x0, top))
% 131.64/17.23 = { by lemma 58 R->L }
% 131.64/17.23 meet(X, join(complement(X), composition(x0, top)))
% 131.64/17.23 = { by axiom 8 (def_top) }
% 131.64/17.23 meet(X, join(complement(X), composition(x0, join(complement(X), complement(complement(X))))))
% 131.64/17.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.23 meet(X, join(complement(X), composition(x0, join(complement(complement(X)), complement(X)))))
% 131.64/17.23 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.23 meet(X, join(complement(X), composition(x0, join(complement(complement(X)), converse(converse(complement(X)))))))
% 131.64/17.23 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.23 meet(X, join(complement(X), composition(x0, join(converse(converse(complement(complement(X)))), converse(converse(complement(X)))))))
% 131.64/17.23 = { by axiom 1 (converse_idempotence) R->L }
% 131.64/17.23 meet(X, join(complement(X), composition(converse(converse(x0)), join(converse(converse(complement(complement(X)))), converse(converse(complement(X)))))))
% 131.64/17.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.23 meet(X, join(complement(X), composition(converse(converse(x0)), join(converse(converse(complement(X))), converse(converse(complement(complement(X))))))))
% 131.64/17.23 = { by axiom 6 (converse_additivity) R->L }
% 131.64/17.23 meet(X, join(complement(X), composition(converse(converse(x0)), converse(join(converse(complement(X)), converse(complement(complement(X))))))))
% 131.64/17.23 = { by axiom 7 (converse_multiplicativity) R->L }
% 131.64/17.23 meet(X, join(complement(X), converse(composition(join(converse(complement(X)), converse(complement(complement(X)))), converse(x0)))))
% 131.64/17.23 = { by axiom 13 (composition_distributivity) }
% 131.64/17.23 meet(X, join(complement(X), converse(join(composition(converse(complement(X)), converse(x0)), composition(converse(complement(complement(X))), converse(x0))))))
% 131.64/17.23 = { by axiom 6 (converse_additivity) }
% 131.64/17.23 meet(X, join(complement(X), join(converse(composition(converse(complement(X)), converse(x0))), converse(composition(converse(complement(complement(X))), converse(x0))))))
% 131.64/17.23 = { by axiom 7 (converse_multiplicativity) }
% 131.64/17.23 meet(X, join(complement(X), join(composition(converse(converse(x0)), converse(converse(complement(X)))), converse(composition(converse(complement(complement(X))), converse(x0))))))
% 131.64/17.23 = { by axiom 7 (converse_multiplicativity) }
% 131.64/17.23 meet(X, join(complement(X), join(composition(converse(converse(x0)), converse(converse(complement(X)))), composition(converse(converse(x0)), converse(converse(complement(complement(X))))))))
% 131.64/17.23 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.23 meet(X, join(complement(X), join(composition(converse(converse(x0)), converse(converse(complement(complement(X))))), composition(converse(converse(x0)), converse(converse(complement(X)))))))
% 131.64/17.23 = { by axiom 1 (converse_idempotence) }
% 131.64/17.23 meet(X, join(complement(X), join(composition(x0, converse(converse(complement(complement(X))))), composition(converse(converse(x0)), converse(converse(complement(X)))))))
% 131.64/17.24 = { by axiom 1 (converse_idempotence) }
% 131.64/17.24 meet(X, join(complement(X), join(composition(x0, converse(converse(complement(complement(X))))), composition(x0, converse(converse(complement(X)))))))
% 131.64/17.24 = { by axiom 1 (converse_idempotence) }
% 131.64/17.24 meet(X, join(complement(X), join(composition(x0, complement(complement(X))), composition(x0, converse(converse(complement(X)))))))
% 131.64/17.24 = { by axiom 1 (converse_idempotence) }
% 131.64/17.24 meet(X, join(complement(X), join(composition(x0, complement(complement(X))), composition(x0, complement(X)))))
% 131.64/17.24 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 131.64/17.24 meet(X, join(complement(X), join(composition(x0, complement(X)), composition(x0, complement(complement(X))))))
% 131.64/17.24 = { by axiom 10 (maddux2_join_associativity) }
% 131.64/17.24 meet(X, join(join(complement(X), composition(x0, complement(X))), composition(x0, complement(complement(X)))))
% 131.64/17.24 = { by lemma 45 }
% 131.64/17.24 meet(X, join(complement(X), composition(x0, complement(complement(X)))))
% 131.64/17.24 = { by lemma 58 }
% 131.64/17.24 meet(X, composition(x0, complement(complement(X))))
% 131.64/17.24 = { by lemma 33 }
% 131.64/17.24 meet(X, composition(x0, X))
% 131.64/17.24 = { by lemma 24 }
% 131.64/17.24 meet(composition(x0, X), X)
% 131.64/17.24 = { by lemma 45 R->L }
% 131.64/17.24 meet(composition(x0, X), join(X, composition(x0, X)))
% 131.64/17.24 = { by lemma 53 }
% 131.64/17.24 composition(x0, X)
% 131.64/17.24
% 131.64/17.24 Lemma 63: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 131.64/17.24 Proof:
% 131.64/17.24 meet(Y, meet(X, Z))
% 131.64/17.24 = { by lemma 40 R->L }
% 131.64/17.24 meet(meet(Y, top), meet(X, Z))
% 131.64/17.24 = { by lemma 25 R->L }
% 131.64/17.24 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 131.64/17.24 = { by lemma 57 }
% 131.64/17.24 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 131.64/17.24 = { by lemma 25 }
% 131.64/17.24 meet(X, meet(Z, meet(Y, top)))
% 131.64/17.24 = { by lemma 40 }
% 131.64/17.24 meet(X, meet(Z, Y))
% 131.64/17.24 = { by lemma 24 R->L }
% 131.64/17.24 meet(X, meet(Y, Z))
% 131.64/17.24
% 131.64/17.24 Lemma 64: meet(Y, composition(x0, X)) = meet(X, composition(x0, Y)).
% 131.64/17.24 Proof:
% 131.64/17.24 meet(Y, composition(x0, X))
% 131.64/17.24 = { by lemma 62 R->L }
% 131.64/17.24 meet(Y, meet(X, composition(x0, top)))
% 131.64/17.24 = { by lemma 63 R->L }
% 131.64/17.24 meet(X, meet(Y, composition(x0, top)))
% 131.64/17.24 = { by lemma 62 }
% 131.64/17.24 meet(X, composition(x0, Y))
% 131.64/17.24
% 131.64/17.24 Lemma 65: meet(X, composition(x0, composition(x1, X))) = composition(x0, composition(x1, X)).
% 131.64/17.24 Proof:
% 131.64/17.24 meet(X, composition(x0, composition(x1, X)))
% 131.64/17.24 = { by lemma 24 }
% 131.64/17.24 meet(composition(x0, composition(x1, X)), X)
% 131.64/17.24 = { by lemma 19 R->L }
% 131.64/17.24 meet(composition(x0, composition(x1, X)), composition(one, X))
% 131.64/17.24 = { by lemma 38 R->L }
% 131.64/17.24 meet(composition(x0, composition(x1, X)), composition(join(one, x1), X))
% 131.64/17.24 = { by lemma 45 R->L }
% 131.64/17.24 meet(composition(x0, composition(x1, X)), composition(join(one, join(x1, composition(x0, x1))), X))
% 131.64/17.24 = { by axiom 10 (maddux2_join_associativity) }
% 131.64/17.24 meet(composition(x0, composition(x1, X)), composition(join(join(one, x1), composition(x0, x1)), X))
% 131.64/17.24 = { by lemma 38 }
% 131.64/17.24 meet(composition(x0, composition(x1, X)), composition(join(one, composition(x0, x1)), X))
% 131.64/17.24 = { by lemma 42 }
% 131.64/17.24 meet(composition(x0, composition(x1, X)), join(X, composition(composition(x0, x1), X)))
% 131.64/17.24 = { by axiom 11 (composition_associativity) R->L }
% 131.64/17.24 meet(composition(x0, composition(x1, X)), join(X, composition(x0, composition(x1, X))))
% 131.64/17.24 = { by lemma 53 }
% 131.64/17.24 composition(x0, composition(x1, X))
% 131.64/17.24
% 131.64/17.24 Lemma 66: meet(X, composition(meet(one, Y), X)) = composition(meet(Y, one), X).
% 131.64/17.24 Proof:
% 131.64/17.24 meet(X, composition(meet(one, Y), X))
% 131.64/17.24 = { by lemma 24 }
% 131.64/17.24 meet(X, composition(meet(Y, one), X))
% 131.64/17.24 = { by lemma 24 }
% 131.64/17.24 meet(composition(meet(Y, one), X), X)
% 131.64/17.24 = { by axiom 12 (maddux4_definiton_of_meet) }
% 131.64/17.24 complement(join(complement(composition(meet(Y, one), X)), complement(X)))
% 131.64/17.24 = { by lemma 41 R->L }
% 131.64/17.24 complement(meet(top, join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by lemma 29 R->L }
% 131.64/17.24 complement(meet(join(composition(meet(one, complement(Y)), X), top), join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by lemma 26 R->L }
% 131.64/17.24 complement(meet(join(composition(meet(Y, one), X), join(composition(meet(one, complement(Y)), X), complement(composition(meet(Y, one), X)))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.24 complement(meet(join(composition(meet(Y, one), X), join(complement(composition(meet(Y, one), X)), composition(meet(one, complement(Y)), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by lemma 59 }
% 131.64/17.24 complement(meet(join(complement(composition(meet(Y, one), X)), composition(join(meet(Y, one), meet(one, complement(Y))), X)), join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by lemma 56 }
% 131.64/17.24 complement(meet(join(complement(composition(meet(Y, one), X)), composition(one, X)), join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.24 complement(meet(join(composition(one, X), complement(composition(meet(Y, one), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by lemma 24 R->L }
% 131.64/17.24 complement(meet(join(composition(one, X), complement(composition(meet(one, Y), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by lemma 19 }
% 131.64/17.24 complement(meet(join(X, complement(composition(meet(one, Y), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by lemma 24 R->L }
% 131.64/17.24 complement(meet(join(X, complement(composition(meet(Y, one), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 131.64/17.24 = { by lemma 33 R->L }
% 131.64/17.24 complement(meet(join(X, complement(composition(meet(Y, one), X))), join(complement(composition(meet(Y, one), X)), complement(complement(complement(X))))))
% 131.64/17.24 = { by lemma 24 }
% 131.64/17.24 complement(meet(join(complement(composition(meet(Y, one), X)), complement(complement(complement(X)))), join(X, complement(composition(meet(Y, one), X)))))
% 131.64/17.24 = { by lemma 48 R->L }
% 131.64/17.24 join(complement(join(complement(composition(meet(Y, one), X)), complement(complement(complement(X))))), complement(join(X, complement(composition(meet(Y, one), X)))))
% 131.64/17.24 = { by lemma 49 }
% 131.64/17.24 join(meet(complement(complement(X)), complement(complement(composition(meet(Y, one), X)))), complement(join(X, complement(composition(meet(Y, one), X)))))
% 131.64/17.24 = { by axiom 2 (maddux1_join_commutativity) }
% 131.64/17.24 join(complement(join(X, complement(composition(meet(Y, one), X)))), meet(complement(complement(X)), complement(complement(composition(meet(Y, one), X)))))
% 131.64/17.24 = { by lemma 24 R->L }
% 131.64/17.25 join(complement(join(X, complement(composition(meet(Y, one), X)))), meet(complement(complement(composition(meet(Y, one), X))), complement(complement(X))))
% 131.64/17.25 = { by lemma 55 R->L }
% 131.64/17.25 join(meet(complement(X), complement(complement(composition(meet(Y, one), X)))), meet(complement(complement(composition(meet(Y, one), X))), complement(complement(X))))
% 131.64/17.25 = { by lemma 56 }
% 131.64/17.25 complement(complement(composition(meet(Y, one), X)))
% 131.64/17.25 = { by lemma 33 }
% 131.64/17.25 composition(meet(Y, one), X)
% 131.64/17.25
% 131.64/17.25 Lemma 67: meet(X, composition(x1, composition(x0, X))) = composition(x1, composition(x0, X)).
% 131.64/17.25 Proof:
% 131.64/17.25 meet(X, composition(x1, composition(x0, X)))
% 131.64/17.25 = { by lemma 24 }
% 131.64/17.25 meet(composition(x1, composition(x0, X)), X)
% 131.64/17.25 = { by lemma 19 R->L }
% 131.64/17.25 meet(composition(x1, composition(x0, X)), composition(one, X))
% 131.64/17.25 = { by lemma 39 R->L }
% 131.64/17.25 meet(composition(x1, composition(x0, X)), composition(join(one, x0), X))
% 131.64/17.25 = { by lemma 19 R->L }
% 132.91/17.25 meet(composition(x1, composition(x0, X)), composition(join(one, composition(one, x0)), X))
% 132.91/17.25 = { by lemma 38 R->L }
% 132.91/17.25 meet(composition(x1, composition(x0, X)), composition(join(one, composition(join(one, x1), x0)), X))
% 132.91/17.25 = { by lemma 42 }
% 132.91/17.25 meet(composition(x1, composition(x0, X)), composition(join(one, join(x0, composition(x1, x0))), X))
% 132.91/17.25 = { by axiom 10 (maddux2_join_associativity) }
% 132.91/17.25 meet(composition(x1, composition(x0, X)), composition(join(join(one, x0), composition(x1, x0)), X))
% 132.91/17.25 = { by lemma 39 }
% 132.91/17.25 meet(composition(x1, composition(x0, X)), composition(join(one, composition(x1, x0)), X))
% 132.91/17.25 = { by lemma 42 }
% 132.91/17.25 meet(composition(x1, composition(x0, X)), join(X, composition(composition(x1, x0), X)))
% 132.91/17.25 = { by axiom 11 (composition_associativity) R->L }
% 132.91/17.25 meet(composition(x1, composition(x0, X)), join(X, composition(x1, composition(x0, X))))
% 132.91/17.25 = { by lemma 53 }
% 132.91/17.25 composition(x1, composition(x0, X))
% 132.91/17.25
% 132.91/17.25 Lemma 68: composition(meet(x1, x0), X) = composition(x1, composition(x0, X)).
% 132.91/17.25 Proof:
% 132.91/17.25 composition(meet(x1, x0), X)
% 132.91/17.25 = { by lemma 19 R->L }
% 132.91/17.25 composition(meet(x1, composition(one, x0)), X)
% 132.91/17.25 = { by lemma 61 R->L }
% 132.91/17.25 composition(meet(x1, meet(one, composition(top, x0))), X)
% 132.91/17.25 = { by lemma 63 R->L }
% 132.91/17.25 composition(meet(one, meet(x1, composition(top, x0))), X)
% 132.91/17.25 = { by lemma 61 }
% 132.91/17.25 composition(meet(one, composition(x1, x0)), X)
% 132.91/17.25 = { by lemma 24 R->L }
% 132.91/17.25 composition(meet(composition(x1, x0), one), X)
% 132.91/17.25 = { by axiom 5 (composition_identity) R->L }
% 132.91/17.25 composition(meet(composition(x1, composition(x0, one)), one), X)
% 132.91/17.25 = { by lemma 66 R->L }
% 132.91/17.25 meet(X, composition(meet(one, composition(x1, composition(x0, one))), X))
% 132.91/17.25 = { by lemma 67 }
% 132.91/17.25 meet(X, composition(composition(x1, composition(x0, one)), X))
% 132.91/17.25 = { by axiom 11 (composition_associativity) R->L }
% 132.91/17.25 meet(X, composition(x1, composition(composition(x0, one), X)))
% 132.91/17.25 = { by axiom 11 (composition_associativity) R->L }
% 132.91/17.25 meet(X, composition(x1, composition(x0, composition(one, X))))
% 132.91/17.25 = { by lemma 19 }
% 132.91/17.25 meet(X, composition(x1, composition(x0, X)))
% 132.91/17.25 = { by lemma 67 }
% 132.91/17.25 composition(x1, composition(x0, X))
% 132.91/17.25
% 132.91/17.25 Lemma 69: composition(x0, composition(x1, X)) = composition(x1, composition(x0, X)).
% 132.91/17.25 Proof:
% 132.91/17.25 composition(x0, composition(x1, X))
% 132.91/17.25 = { by lemma 65 R->L }
% 132.91/17.25 meet(X, composition(x0, composition(x1, X)))
% 132.91/17.25 = { by lemma 19 R->L }
% 132.91/17.25 meet(X, composition(x0, composition(x1, composition(one, X))))
% 132.91/17.25 = { by axiom 11 (composition_associativity) }
% 132.91/17.25 meet(X, composition(x0, composition(composition(x1, one), X)))
% 132.91/17.25 = { by axiom 11 (composition_associativity) }
% 132.91/17.25 meet(X, composition(composition(x0, composition(x1, one)), X))
% 132.91/17.25 = { by lemma 65 R->L }
% 132.91/17.25 meet(X, composition(meet(one, composition(x0, composition(x1, one))), X))
% 132.91/17.25 = { by lemma 66 }
% 132.91/17.25 composition(meet(composition(x0, composition(x1, one)), one), X)
% 132.91/17.25 = { by axiom 5 (composition_identity) }
% 132.91/17.25 composition(meet(composition(x0, x1), one), X)
% 132.91/17.25 = { by lemma 24 }
% 132.91/17.25 composition(meet(one, composition(x0, x1)), X)
% 132.91/17.25 = { by lemma 60 R->L }
% 132.91/17.25 composition(meet(one, meet(x0, composition(top, x1))), X)
% 132.91/17.25 = { by lemma 63 }
% 132.91/17.25 composition(meet(x0, meet(one, composition(top, x1))), X)
% 132.91/17.25 = { by lemma 60 }
% 132.91/17.25 composition(meet(x0, composition(one, x1)), X)
% 132.91/17.25 = { by lemma 19 }
% 132.91/17.25 composition(meet(x0, x1), X)
% 132.91/17.25 = { by lemma 24 R->L }
% 132.91/17.25 composition(meet(x1, x0), X)
% 132.91/17.25 = { by lemma 68 }
% 132.91/17.25 composition(x1, composition(x0, X))
% 132.91/17.25
% 132.91/17.26 Goal 1 (goals_2): tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))) = tuple(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))).
% 132.91/17.26 Proof:
% 132.91/17.26 tuple(join(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))))
% 132.91/17.26 = { by axiom 2 (maddux1_join_commutativity) }
% 132.91/17.26 tuple(join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))))
% 132.91/17.26 = { by lemma 24 R->L }
% 132.91/17.26 tuple(join(composition(meet(x1, x0), x2), meet(composition(x0, x2), composition(x1, x2))), join(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2))))
% 132.91/17.26 = { by lemma 24 R->L }
% 132.91/17.26 tuple(join(composition(meet(x1, x0), x2), meet(composition(x0, x2), composition(x1, x2))), join(composition(meet(x1, x0), x2), meet(composition(x0, x2), composition(x1, x2))))
% 132.91/17.26 = { by lemma 24 R->L }
% 132.91/17.26 tuple(join(composition(meet(x1, x0), x2), meet(composition(x1, x2), composition(x0, x2))), join(composition(meet(x1, x0), x2), meet(composition(x0, x2), composition(x1, x2))))
% 132.91/17.26 = { by lemma 24 R->L }
% 132.91/17.26 tuple(join(composition(meet(x1, x0), x2), meet(composition(x1, x2), composition(x0, x2))), join(composition(meet(x1, x0), x2), meet(composition(x1, x2), composition(x0, x2))))
% 132.91/17.26 = { by lemma 68 }
% 132.91/17.26 tuple(join(composition(x1, composition(x0, x2)), meet(composition(x1, x2), composition(x0, x2))), join(composition(meet(x1, x0), x2), meet(composition(x1, x2), composition(x0, x2))))
% 132.91/17.26 = { by lemma 68 }
% 132.91/17.26 tuple(join(composition(x1, composition(x0, x2)), meet(composition(x1, x2), composition(x0, x2))), join(composition(x1, composition(x0, x2)), meet(composition(x1, x2), composition(x0, x2))))
% 132.91/17.26 = { by lemma 64 R->L }
% 132.91/17.26 tuple(join(composition(x1, composition(x0, x2)), meet(x2, composition(x0, composition(x1, x2)))), join(composition(x1, composition(x0, x2)), meet(composition(x1, x2), composition(x0, x2))))
% 132.91/17.26 = { by lemma 64 R->L }
% 132.91/17.26 tuple(join(composition(x1, composition(x0, x2)), meet(x2, composition(x0, composition(x1, x2)))), join(composition(x1, composition(x0, x2)), meet(x2, composition(x0, composition(x1, x2)))))
% 132.91/17.26 = { by lemma 65 }
% 132.91/17.26 tuple(join(composition(x1, composition(x0, x2)), composition(x0, composition(x1, x2))), join(composition(x1, composition(x0, x2)), meet(x2, composition(x0, composition(x1, x2)))))
% 132.91/17.26 = { by lemma 65 }
% 132.91/17.26 tuple(join(composition(x1, composition(x0, x2)), composition(x0, composition(x1, x2))), join(composition(x1, composition(x0, x2)), composition(x0, composition(x1, x2))))
% 132.91/17.26 = { by lemma 69 }
% 132.91/17.26 tuple(join(composition(x1, composition(x0, x2)), composition(x1, composition(x0, x2))), join(composition(x1, composition(x0, x2)), composition(x0, composition(x1, x2))))
% 132.91/17.26 = { by lemma 69 }
% 132.91/17.26 tuple(join(composition(x1, composition(x0, x2)), composition(x1, composition(x0, x2))), join(composition(x1, composition(x0, x2)), composition(x1, composition(x0, x2))))
% 132.91/17.26 = { by lemma 36 }
% 132.91/17.26 tuple(composition(x1, composition(x0, x2)), join(composition(x1, composition(x0, x2)), composition(x1, composition(x0, x2))))
% 132.91/17.26 = { by lemma 36 }
% 132.91/17.26 tuple(composition(x1, composition(x0, x2)), composition(x1, composition(x0, x2)))
% 132.91/17.26 = { by lemma 69 R->L }
% 132.91/17.26 tuple(composition(x1, composition(x0, x2)), composition(x0, composition(x1, x2)))
% 132.91/17.26 = { by lemma 65 R->L }
% 132.91/17.26 tuple(composition(x1, composition(x0, x2)), meet(x2, composition(x0, composition(x1, x2))))
% 132.91/17.26 = { by lemma 64 }
% 132.91/17.26 tuple(composition(x1, composition(x0, x2)), meet(composition(x1, x2), composition(x0, x2)))
% 132.91/17.26 = { by lemma 68 R->L }
% 132.91/17.26 tuple(composition(meet(x1, x0), x2), meet(composition(x1, x2), composition(x0, x2)))
% 132.91/17.26 = { by lemma 24 }
% 132.91/17.26 tuple(composition(meet(x1, x0), x2), meet(composition(x0, x2), composition(x1, x2)))
% 132.91/17.26 = { by lemma 24 }
% 132.91/17.26 tuple(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))
% 132.91/17.26 % SZS output end Proof
% 132.91/17.26
% 132.91/17.26 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------