TSTP Solution File: REL021+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL021+2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n026.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 : Thu Aug 31 13:44:02 EDT 2023
% Result : Theorem 3.61s 0.89s
% Output : Proof 4.38s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL021+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n026.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 22:39:33 EDT 2023
% 0.14/0.36 % CPUTime :
% 3.61/0.89 Command-line arguments: --ground-connectedness --complete-subsets
% 3.61/0.89
% 3.61/0.89 % SZS status Theorem
% 3.61/0.89
% 4.38/0.95 % SZS output start Proof
% 4.38/0.95 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 4.38/0.95 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 4.38/0.95 Axiom 3 (composition_identity): composition(X, one) = X.
% 4.38/0.95 Axiom 4 (goals): composition(x0, top) = x0.
% 4.38/0.95 Axiom 5 (def_top): top = join(X, complement(X)).
% 4.38/0.95 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 4.38/0.95 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 4.38/0.95 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 4.38/0.95 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 4.38/0.95 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 4.38/0.95 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 4.38/0.95 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 4.38/0.95 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 4.38/0.95 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 4.38/0.95 Axiom 15 (modular_law_1): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 4.38/0.95
% 4.38/0.95 Lemma 16: complement(top) = zero.
% 4.38/0.95 Proof:
% 4.38/0.95 complement(top)
% 4.38/0.95 = { by axiom 5 (def_top) }
% 4.38/0.95 complement(join(complement(X), complement(complement(X))))
% 4.38/0.95 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 4.38/0.95 meet(X, complement(X))
% 4.38/0.95 = { by axiom 6 (def_zero) R->L }
% 4.38/0.95 zero
% 4.38/0.95
% 4.38/0.95 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 4.38/0.95 Proof:
% 4.38/0.95 join(X, join(Y, complement(X)))
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.95 join(X, join(complement(X), Y))
% 4.38/0.95 = { by axiom 8 (maddux2_join_associativity) }
% 4.38/0.95 join(join(X, complement(X)), Y)
% 4.38/0.95 = { by axiom 5 (def_top) R->L }
% 4.38/0.95 join(top, Y)
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) }
% 4.38/0.95 join(Y, top)
% 4.38/0.95
% 4.38/0.95 Lemma 18: composition(converse(one), X) = X.
% 4.38/0.95 Proof:
% 4.38/0.95 composition(converse(one), X)
% 4.38/0.95 = { by axiom 1 (converse_idempotence) R->L }
% 4.38/0.95 composition(converse(one), converse(converse(X)))
% 4.38/0.95 = { by axiom 9 (converse_multiplicativity) R->L }
% 4.38/0.95 converse(composition(converse(X), one))
% 4.38/0.95 = { by axiom 3 (composition_identity) }
% 4.38/0.95 converse(converse(X))
% 4.38/0.95 = { by axiom 1 (converse_idempotence) }
% 4.38/0.95 X
% 4.38/0.95
% 4.38/0.95 Lemma 19: composition(one, X) = X.
% 4.38/0.95 Proof:
% 4.38/0.95 composition(one, X)
% 4.38/0.95 = { by lemma 18 R->L }
% 4.38/0.95 composition(converse(one), composition(one, X))
% 4.38/0.95 = { by axiom 10 (composition_associativity) }
% 4.38/0.95 composition(composition(converse(one), one), X)
% 4.38/0.95 = { by axiom 3 (composition_identity) }
% 4.38/0.95 composition(converse(one), X)
% 4.38/0.95 = { by lemma 18 }
% 4.38/0.95 X
% 4.38/0.95
% 4.38/0.95 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 4.38/0.95 Proof:
% 4.38/0.95 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.95 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 4.38/0.95 = { by axiom 13 (converse_cancellativity) }
% 4.38/0.95 complement(X)
% 4.38/0.95
% 4.38/0.95 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 4.38/0.95 Proof:
% 4.38/0.95 join(complement(X), complement(X))
% 4.38/0.95 = { by lemma 18 R->L }
% 4.38/0.95 join(complement(X), composition(converse(one), complement(X)))
% 4.38/0.95 = { by lemma 19 R->L }
% 4.38/0.95 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 4.38/0.95 = { by lemma 20 }
% 4.38/0.95 complement(X)
% 4.38/0.95
% 4.38/0.95 Lemma 22: join(top, complement(X)) = top.
% 4.38/0.95 Proof:
% 4.38/0.95 join(top, complement(X))
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.95 join(complement(X), top)
% 4.38/0.95 = { by lemma 17 R->L }
% 4.38/0.95 join(X, join(complement(X), complement(X)))
% 4.38/0.95 = { by lemma 21 }
% 4.38/0.95 join(X, complement(X))
% 4.38/0.95 = { by axiom 5 (def_top) R->L }
% 4.38/0.95 top
% 4.38/0.95
% 4.38/0.95 Lemma 23: join(Y, top) = join(X, top).
% 4.38/0.95 Proof:
% 4.38/0.95 join(Y, top)
% 4.38/0.95 = { by lemma 22 R->L }
% 4.38/0.95 join(Y, join(top, complement(Y)))
% 4.38/0.95 = { by lemma 17 }
% 4.38/0.95 join(top, top)
% 4.38/0.95 = { by lemma 17 R->L }
% 4.38/0.95 join(X, join(top, complement(X)))
% 4.38/0.95 = { by lemma 22 }
% 4.38/0.95 join(X, top)
% 4.38/0.95
% 4.38/0.95 Lemma 24: join(X, top) = top.
% 4.38/0.95 Proof:
% 4.38/0.95 join(X, top)
% 4.38/0.95 = { by lemma 23 }
% 4.38/0.95 join(zero, top)
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.95 join(top, zero)
% 4.38/0.95 = { by lemma 16 R->L }
% 4.38/0.95 join(top, complement(top))
% 4.38/0.95 = { by axiom 5 (def_top) R->L }
% 4.38/0.95 top
% 4.38/0.95
% 4.38/0.95 Lemma 25: join(X, join(complement(X), Y)) = top.
% 4.38/0.95 Proof:
% 4.38/0.95 join(X, join(complement(X), Y))
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.95 join(X, join(Y, complement(X)))
% 4.38/0.95 = { by lemma 17 }
% 4.38/0.95 join(Y, top)
% 4.38/0.95 = { by lemma 23 R->L }
% 4.38/0.95 join(Z, top)
% 4.38/0.95 = { by lemma 24 }
% 4.38/0.95 top
% 4.38/0.95
% 4.38/0.95 Lemma 26: join(X, converse(top)) = top.
% 4.38/0.95 Proof:
% 4.38/0.95 join(X, converse(top))
% 4.38/0.95 = { by axiom 5 (def_top) }
% 4.38/0.95 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 4.38/0.95 = { by axiom 7 (converse_additivity) }
% 4.38/0.95 join(X, join(converse(converse(complement(X))), converse(complement(converse(complement(X))))))
% 4.38/0.95 = { by axiom 1 (converse_idempotence) }
% 4.38/0.95 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 4.38/0.95 = { by lemma 25 }
% 4.38/0.95 top
% 4.38/0.95
% 4.38/0.95 Lemma 27: converse(top) = top.
% 4.38/0.95 Proof:
% 4.38/0.95 converse(top)
% 4.38/0.95 = { by lemma 24 R->L }
% 4.38/0.95 converse(join(X, top))
% 4.38/0.95 = { by axiom 7 (converse_additivity) }
% 4.38/0.95 join(converse(X), converse(top))
% 4.38/0.95 = { by lemma 26 }
% 4.38/0.95 top
% 4.38/0.95
% 4.38/0.95 Lemma 28: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 4.38/0.95 Proof:
% 4.38/0.95 join(meet(X, Y), complement(join(complement(X), Y)))
% 4.38/0.95 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.95 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 4.38/0.95 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 4.38/0.95 X
% 4.38/0.95
% 4.38/0.95 Lemma 29: join(zero, meet(X, X)) = X.
% 4.38/0.95 Proof:
% 4.38/0.95 join(zero, meet(X, X))
% 4.38/0.95 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.95 join(zero, complement(join(complement(X), complement(X))))
% 4.38/0.95 = { by axiom 6 (def_zero) }
% 4.38/0.95 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 4.38/0.95 = { by lemma 28 }
% 4.38/0.95 X
% 4.38/0.95
% 4.38/0.95 Lemma 30: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 4.38/0.95 Proof:
% 4.38/0.95 join(zero, join(X, complement(complement(Y))))
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.95 join(zero, join(complement(complement(Y)), X))
% 4.38/0.95 = { by lemma 21 R->L }
% 4.38/0.95 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 4.38/0.95 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 4.38/0.95 join(zero, join(meet(Y, Y), X))
% 4.38/0.95 = { by axiom 8 (maddux2_join_associativity) }
% 4.38/0.95 join(join(zero, meet(Y, Y)), X)
% 4.38/0.95 = { by lemma 29 }
% 4.38/0.95 join(Y, X)
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) }
% 4.38/0.95 join(X, Y)
% 4.38/0.95
% 4.38/0.95 Lemma 31: join(zero, complement(complement(X))) = X.
% 4.38/0.95 Proof:
% 4.38/0.95 join(zero, complement(complement(X)))
% 4.38/0.95 = { by axiom 6 (def_zero) }
% 4.38/0.95 join(meet(X, complement(X)), complement(complement(X)))
% 4.38/0.95 = { by lemma 21 R->L }
% 4.38/0.95 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 4.38/0.95 = { by lemma 28 }
% 4.38/0.95 X
% 4.38/0.95
% 4.38/0.95 Lemma 32: join(zero, complement(X)) = complement(X).
% 4.38/0.95 Proof:
% 4.38/0.95 join(zero, complement(X))
% 4.38/0.95 = { by lemma 31 R->L }
% 4.38/0.95 join(zero, join(zero, complement(complement(complement(X)))))
% 4.38/0.95 = { by lemma 21 R->L }
% 4.38/0.95 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 4.38/0.95 = { by lemma 30 }
% 4.38/0.95 join(zero, join(complement(complement(complement(X))), complement(X)))
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) }
% 4.38/0.95 join(zero, join(complement(X), complement(complement(complement(X)))))
% 4.38/0.95 = { by lemma 30 }
% 4.38/0.95 join(complement(X), complement(X))
% 4.38/0.95 = { by lemma 21 }
% 4.38/0.95 complement(X)
% 4.38/0.95
% 4.38/0.95 Lemma 33: join(X, zero) = X.
% 4.38/0.95 Proof:
% 4.38/0.95 join(X, zero)
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.95 join(zero, X)
% 4.38/0.95 = { by lemma 30 R->L }
% 4.38/0.95 join(zero, join(zero, complement(complement(X))))
% 4.38/0.95 = { by lemma 32 }
% 4.38/0.95 join(zero, complement(complement(X)))
% 4.38/0.95 = { by lemma 31 }
% 4.38/0.95 X
% 4.38/0.95
% 4.38/0.95 Lemma 34: join(top, X) = top.
% 4.38/0.95 Proof:
% 4.38/0.95 join(top, X)
% 4.38/0.95 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.95 join(X, top)
% 4.38/0.95 = { by lemma 23 R->L }
% 4.38/0.95 join(Y, top)
% 4.38/0.96 = { by lemma 24 }
% 4.38/0.96 top
% 4.38/0.96
% 4.38/0.96 Lemma 35: join(zero, X) = X.
% 4.38/0.96 Proof:
% 4.38/0.96 join(zero, X)
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.96 join(X, zero)
% 4.38/0.96 = { by lemma 33 }
% 4.38/0.96 X
% 4.38/0.96
% 4.38/0.96 Lemma 36: meet(Y, X) = meet(X, Y).
% 4.38/0.96 Proof:
% 4.38/0.96 meet(Y, X)
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.96 complement(join(complement(Y), complement(X)))
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.96 complement(join(complement(X), complement(Y)))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 4.38/0.96 meet(X, Y)
% 4.38/0.96
% 4.38/0.96 Lemma 37: complement(join(zero, complement(X))) = meet(X, top).
% 4.38/0.96 Proof:
% 4.38/0.96 complement(join(zero, complement(X)))
% 4.38/0.96 = { by lemma 16 R->L }
% 4.38/0.96 complement(join(complement(top), complement(X)))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 4.38/0.96 meet(top, X)
% 4.38/0.96 = { by lemma 36 R->L }
% 4.38/0.96 meet(X, top)
% 4.38/0.96
% 4.38/0.96 Lemma 38: join(X, complement(zero)) = top.
% 4.38/0.96 Proof:
% 4.38/0.96 join(X, complement(zero))
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.96 join(complement(zero), X)
% 4.38/0.96 = { by lemma 30 R->L }
% 4.38/0.96 join(zero, join(complement(zero), complement(complement(X))))
% 4.38/0.96 = { by lemma 25 }
% 4.38/0.96 top
% 4.38/0.96
% 4.38/0.96 Lemma 39: meet(X, zero) = zero.
% 4.38/0.96 Proof:
% 4.38/0.96 meet(X, zero)
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.96 complement(join(complement(X), complement(zero)))
% 4.38/0.96 = { by lemma 38 }
% 4.38/0.96 complement(top)
% 4.38/0.96 = { by lemma 16 }
% 4.38/0.96 zero
% 4.38/0.96
% 4.38/0.96 Lemma 40: meet(X, top) = X.
% 4.38/0.96 Proof:
% 4.38/0.96 meet(X, top)
% 4.38/0.96 = { by lemma 37 R->L }
% 4.38/0.96 complement(join(zero, complement(X)))
% 4.38/0.96 = { by lemma 32 R->L }
% 4.38/0.96 join(zero, complement(join(zero, complement(X))))
% 4.38/0.96 = { by lemma 37 }
% 4.38/0.96 join(zero, meet(X, top))
% 4.38/0.96 = { by lemma 38 R->L }
% 4.38/0.96 join(zero, meet(X, join(complement(zero), complement(zero))))
% 4.38/0.96 = { by lemma 21 }
% 4.38/0.96 join(zero, meet(X, complement(zero)))
% 4.38/0.96 = { by lemma 39 R->L }
% 4.38/0.96 join(meet(X, zero), meet(X, complement(zero)))
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.96 join(meet(X, complement(zero)), meet(X, zero))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.96 join(meet(X, complement(zero)), complement(join(complement(X), complement(zero))))
% 4.38/0.96 = { by lemma 28 }
% 4.38/0.96 X
% 4.38/0.96
% 4.38/0.96 Lemma 41: meet(zero, X) = zero.
% 4.38/0.96 Proof:
% 4.38/0.96 meet(zero, X)
% 4.38/0.96 = { by lemma 36 }
% 4.38/0.96 meet(X, zero)
% 4.38/0.96 = { by lemma 39 }
% 4.38/0.96 zero
% 4.38/0.96
% 4.38/0.96 Lemma 42: composition(top, zero) = zero.
% 4.38/0.96 Proof:
% 4.38/0.96 composition(top, zero)
% 4.38/0.96 = { by lemma 27 R->L }
% 4.38/0.96 composition(converse(top), zero)
% 4.38/0.96 = { by lemma 35 R->L }
% 4.38/0.96 join(zero, composition(converse(top), zero))
% 4.38/0.96 = { by lemma 16 R->L }
% 4.38/0.96 join(complement(top), composition(converse(top), zero))
% 4.38/0.96 = { by lemma 16 R->L }
% 4.38/0.96 join(complement(top), composition(converse(top), complement(top)))
% 4.38/0.96 = { by lemma 34 R->L }
% 4.38/0.96 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 4.38/0.96 = { by lemma 27 R->L }
% 4.38/0.96 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 4.38/0.96 = { by lemma 19 R->L }
% 4.38/0.96 join(complement(top), composition(converse(top), complement(join(composition(one, top), composition(converse(top), top)))))
% 4.38/0.96 = { by axiom 12 (composition_distributivity) R->L }
% 4.38/0.96 join(complement(top), composition(converse(top), complement(composition(join(one, converse(top)), top))))
% 4.38/0.96 = { by lemma 26 }
% 4.38/0.96 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 4.38/0.96 = { by lemma 20 }
% 4.38/0.96 complement(top)
% 4.38/0.96 = { by lemma 16 }
% 4.38/0.96 zero
% 4.38/0.96
% 4.38/0.96 Lemma 43: composition(X, zero) = zero.
% 4.38/0.96 Proof:
% 4.38/0.96 composition(X, zero)
% 4.38/0.96 = { by lemma 35 R->L }
% 4.38/0.96 join(zero, composition(X, zero))
% 4.38/0.96 = { by lemma 42 R->L }
% 4.38/0.96 join(composition(top, zero), composition(X, zero))
% 4.38/0.96 = { by axiom 12 (composition_distributivity) R->L }
% 4.38/0.96 composition(join(top, X), zero)
% 4.38/0.96 = { by lemma 34 }
% 4.38/0.96 composition(top, zero)
% 4.38/0.96 = { by lemma 42 }
% 4.38/0.96 zero
% 4.38/0.96
% 4.38/0.96 Lemma 44: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 4.38/0.96 Proof:
% 4.38/0.96 meet(X, join(complement(Y), complement(Z)))
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.96 meet(X, join(complement(Z), complement(Y)))
% 4.38/0.96 = { by lemma 36 }
% 4.38/0.96 meet(join(complement(Z), complement(Y)), X)
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.96 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 4.38/0.96 complement(join(meet(Z, Y), complement(X)))
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) }
% 4.38/0.96 complement(join(complement(X), meet(Z, Y)))
% 4.38/0.96 = { by lemma 36 R->L }
% 4.38/0.96 complement(join(complement(X), meet(Y, Z)))
% 4.38/0.96
% 4.38/0.96 Lemma 45: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 4.38/0.96 Proof:
% 4.38/0.96 complement(join(X, complement(Y)))
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.96 complement(join(complement(Y), X))
% 4.38/0.96 = { by lemma 40 R->L }
% 4.38/0.96 complement(join(complement(Y), meet(X, top)))
% 4.38/0.96 = { by lemma 36 R->L }
% 4.38/0.96 complement(join(complement(Y), meet(top, X)))
% 4.38/0.96 = { by lemma 44 R->L }
% 4.38/0.96 meet(Y, join(complement(top), complement(X)))
% 4.38/0.96 = { by lemma 16 }
% 4.38/0.96 meet(Y, join(zero, complement(X)))
% 4.38/0.96 = { by lemma 32 }
% 4.38/0.96 meet(Y, complement(X))
% 4.38/0.96
% 4.38/0.96 Lemma 46: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 4.38/0.96 Proof:
% 4.38/0.96 complement(meet(X, complement(Y)))
% 4.38/0.96 = { by lemma 35 R->L }
% 4.38/0.96 complement(join(zero, meet(X, complement(Y))))
% 4.38/0.96 = { by lemma 45 R->L }
% 4.38/0.96 complement(join(zero, complement(join(Y, complement(X)))))
% 4.38/0.96 = { by lemma 37 }
% 4.38/0.96 meet(join(Y, complement(X)), top)
% 4.38/0.96 = { by lemma 40 }
% 4.38/0.96 join(Y, complement(X))
% 4.38/0.96
% 4.38/0.96 Lemma 47: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 4.38/0.96 Proof:
% 4.38/0.96 complement(join(complement(X), Y))
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.96 complement(join(Y, complement(X)))
% 4.38/0.96 = { by lemma 45 }
% 4.38/0.96 meet(X, complement(Y))
% 4.38/0.96
% 4.38/0.96 Lemma 48: meet(X, join(X, complement(Y))) = X.
% 4.38/0.96 Proof:
% 4.38/0.96 meet(X, join(X, complement(Y)))
% 4.38/0.96 = { by lemma 46 R->L }
% 4.38/0.96 meet(X, complement(meet(Y, complement(X))))
% 4.38/0.96 = { by lemma 47 R->L }
% 4.38/0.96 complement(join(complement(X), meet(Y, complement(X))))
% 4.38/0.96 = { by lemma 32 R->L }
% 4.38/0.96 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 16 R->L }
% 4.38/0.96 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 24 R->L }
% 4.38/0.96 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 17 R->L }
% 4.38/0.96 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.96 join(complement(join(complement(X), join(complement(complement(X)), complement(Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 29 R->L }
% 4.38/0.96 join(complement(join(complement(X), join(zero, meet(join(complement(complement(X)), complement(Y)), join(complement(complement(X)), complement(Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 44 }
% 4.38/0.96 join(complement(join(complement(X), join(zero, complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 32 }
% 4.38/0.96 join(complement(join(complement(X), complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 4.38/0.96 join(complement(join(complement(X), complement(join(meet(complement(X), Y), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 36 }
% 4.38/0.96 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 36 }
% 4.38/0.96 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(Y, complement(X)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.96 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), complement(join(complement(Y), complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.96 join(complement(join(complement(X), complement(join(complement(join(complement(Y), complement(complement(X)))), complement(join(complement(Y), complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 21 }
% 4.38/0.96 join(complement(join(complement(X), complement(complement(join(complement(Y), complement(complement(X))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 4.38/0.96 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 36 R->L }
% 4.38/0.96 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 4.38/0.96 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 36 R->L }
% 4.38/0.96 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 4.38/0.96 = { by lemma 28 }
% 4.38/0.96 X
% 4.38/0.96
% 4.38/0.96 Lemma 49: join(X, meet(X, Y)) = X.
% 4.38/0.96 Proof:
% 4.38/0.96 join(X, meet(X, Y))
% 4.38/0.96 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.96 join(X, complement(join(complement(X), complement(Y))))
% 4.38/0.96 = { by lemma 46 R->L }
% 4.38/0.96 complement(meet(join(complement(X), complement(Y)), complement(X)))
% 4.38/0.96 = { by lemma 36 R->L }
% 4.38/0.96 complement(meet(complement(X), join(complement(X), complement(Y))))
% 4.38/0.96 = { by lemma 48 }
% 4.38/0.96 complement(complement(X))
% 4.38/0.96 = { by lemma 32 R->L }
% 4.38/0.96 join(zero, complement(complement(X)))
% 4.38/0.96 = { by lemma 31 }
% 4.38/0.96 X
% 4.38/0.96
% 4.38/0.96 Lemma 50: join(Z, join(Y, X)) = join(X, join(Y, Z)).
% 4.38/0.96 Proof:
% 4.38/0.96 join(Z, join(Y, X))
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.96 join(join(Y, X), Z)
% 4.38/0.96 = { by axiom 2 (maddux1_join_commutativity) }
% 4.38/0.96 join(join(X, Y), Z)
% 4.38/0.96 = { by axiom 8 (maddux2_join_associativity) R->L }
% 4.38/0.96 join(X, join(Y, Z))
% 4.38/0.96
% 4.38/0.96 Lemma 51: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 4.38/0.96 Proof:
% 4.38/0.96 meet(complement(X), complement(Y))
% 4.38/0.96 = { by lemma 36 }
% 4.38/0.96 meet(complement(Y), complement(X))
% 4.38/0.96 = { by lemma 32 R->L }
% 4.38/0.96 meet(join(zero, complement(Y)), complement(X))
% 4.38/0.96 = { by lemma 45 R->L }
% 4.38/0.96 complement(join(X, complement(join(zero, complement(Y)))))
% 4.38/0.96 = { by lemma 37 }
% 4.38/0.96 complement(join(X, meet(Y, top)))
% 4.38/0.96 = { by lemma 40 }
% 4.38/0.96 complement(join(X, Y))
% 4.38/0.96
% 4.38/0.96 Lemma 52: composition(converse(x0), complement(x0)) = zero.
% 4.38/0.96 Proof:
% 4.38/0.96 composition(converse(x0), complement(x0))
% 4.38/0.96 = { by lemma 35 R->L }
% 4.38/0.96 join(zero, composition(converse(x0), complement(x0)))
% 4.38/0.96 = { by lemma 16 R->L }
% 4.38/0.97 join(complement(top), composition(converse(x0), complement(x0)))
% 4.38/0.97 = { by axiom 4 (goals) R->L }
% 4.38/0.97 join(complement(top), composition(converse(x0), complement(composition(x0, top))))
% 4.38/0.97 = { by lemma 20 }
% 4.38/0.97 complement(top)
% 4.38/0.97 = { by lemma 16 }
% 4.38/0.97 zero
% 4.38/0.97
% 4.38/0.97 Lemma 53: join(x0, composition(x0, X)) = x0.
% 4.38/0.97 Proof:
% 4.38/0.97 join(x0, composition(x0, X))
% 4.38/0.97 = { by lemma 40 R->L }
% 4.38/0.97 meet(join(x0, composition(x0, X)), top)
% 4.38/0.97 = { by lemma 37 R->L }
% 4.38/0.97 complement(join(zero, complement(join(x0, composition(x0, X)))))
% 4.38/0.97 = { by lemma 51 R->L }
% 4.38/0.97 complement(join(zero, meet(complement(x0), complement(composition(x0, X)))))
% 4.38/0.97 = { by lemma 47 R->L }
% 4.38/0.97 complement(join(zero, complement(join(complement(complement(x0)), composition(x0, X)))))
% 4.38/0.97 = { by lemma 32 R->L }
% 4.38/0.97 complement(join(zero, join(zero, complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 41 R->L }
% 4.38/0.97 complement(join(zero, join(meet(zero, complement(x0)), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 43 R->L }
% 4.38/0.97 complement(join(zero, join(meet(composition(x0, zero), complement(x0)), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 39 R->L }
% 4.38/0.97 complement(join(zero, join(meet(composition(x0, meet(X, zero)), complement(x0)), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 52 R->L }
% 4.38/0.97 complement(join(zero, join(meet(composition(x0, meet(X, composition(converse(x0), complement(x0)))), complement(x0)), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by axiom 15 (modular_law_1) R->L }
% 4.38/0.97 complement(join(zero, join(join(meet(composition(x0, X), complement(x0)), meet(composition(x0, meet(X, composition(converse(x0), complement(x0)))), complement(x0))), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 52 }
% 4.38/0.97 complement(join(zero, join(join(meet(composition(x0, X), complement(x0)), meet(composition(x0, meet(X, zero)), complement(x0))), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 39 }
% 4.38/0.97 complement(join(zero, join(join(meet(composition(x0, X), complement(x0)), meet(composition(x0, zero), complement(x0))), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 43 }
% 4.38/0.97 complement(join(zero, join(join(meet(composition(x0, X), complement(x0)), meet(zero, complement(x0))), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 41 }
% 4.38/0.97 complement(join(zero, join(join(meet(composition(x0, X), complement(x0)), zero), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 33 }
% 4.38/0.97 complement(join(zero, join(meet(composition(x0, X), complement(x0)), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 36 R->L }
% 4.38/0.97 complement(join(zero, join(meet(complement(x0), composition(x0, X)), complement(join(complement(complement(x0)), composition(x0, X))))))
% 4.38/0.97 = { by lemma 28 }
% 4.38/0.97 complement(join(zero, complement(x0)))
% 4.38/0.97 = { by lemma 37 }
% 4.38/0.97 meet(x0, top)
% 4.38/0.97 = { by lemma 40 }
% 4.38/0.97 x0
% 4.38/0.97
% 4.38/0.97 Lemma 54: join(meet(X, Y), X) = X.
% 4.38/0.97 Proof:
% 4.38/0.97 join(meet(X, Y), X)
% 4.38/0.97 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.97 join(X, meet(X, Y))
% 4.38/0.97 = { by lemma 49 }
% 4.38/0.97 X
% 4.38/0.97
% 4.38/0.97 Lemma 55: join(meet(X, Y), Y) = Y.
% 4.38/0.97 Proof:
% 4.38/0.97 join(meet(X, Y), Y)
% 4.38/0.97 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.97 join(Y, meet(X, Y))
% 4.38/0.97 = { by lemma 36 R->L }
% 4.38/0.97 join(Y, meet(Y, X))
% 4.38/0.97 = { by lemma 49 }
% 4.38/0.97 Y
% 4.38/0.97
% 4.38/0.97 Lemma 56: meet(X, join(X, Y)) = X.
% 4.38/0.97 Proof:
% 4.38/0.97 meet(X, join(X, Y))
% 4.38/0.97 = { by lemma 40 R->L }
% 4.38/0.97 meet(X, join(X, meet(Y, top)))
% 4.38/0.97 = { by lemma 37 R->L }
% 4.38/0.97 meet(X, join(X, complement(join(zero, complement(Y)))))
% 4.38/0.97 = { by lemma 48 }
% 4.38/0.97 X
% 4.38/0.97
% 4.38/0.97 Goal 1 (goals_1): join(composition(meet(x0, one), x1), meet(x0, x1)) = meet(x0, x1).
% 4.38/0.97 Proof:
% 4.38/0.97 join(composition(meet(x0, one), x1), meet(x0, x1))
% 4.38/0.97 = { by lemma 56 R->L }
% 4.38/0.97 meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(join(composition(meet(x0, one), x1), meet(x0, x1)), x1))
% 4.38/0.97 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.97 meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(x1, join(composition(meet(x0, one), x1), meet(x0, x1))))
% 4.38/0.97 = { by lemma 50 }
% 4.38/0.97 meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(meet(x0, x1), join(composition(meet(x0, one), x1), x1)))
% 4.38/0.97 = { by lemma 19 R->L }
% 4.38/0.97 meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(meet(x0, x1), join(composition(meet(x0, one), x1), composition(one, x1))))
% 4.38/0.97 = { by axiom 12 (composition_distributivity) R->L }
% 4.38/0.97 meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(meet(x0, x1), composition(join(meet(x0, one), one), x1)))
% 4.38/0.97 = { by lemma 55 }
% 4.38/0.97 meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(meet(x0, x1), composition(one, x1)))
% 4.38/0.97 = { by lemma 19 }
% 4.38/0.97 meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(meet(x0, x1), x1))
% 4.38/0.97 = { by lemma 55 }
% 4.38/0.97 meet(join(composition(meet(x0, one), x1), meet(x0, x1)), x1)
% 4.38/0.97 = { by lemma 36 }
% 4.38/0.97 meet(x1, join(composition(meet(x0, one), x1), meet(x0, x1)))
% 4.38/0.97 = { by lemma 56 R->L }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(join(composition(meet(x0, one), x1), meet(x0, x1)), x0)))
% 4.38/0.97 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(x0, join(composition(meet(x0, one), x1), meet(x0, x1)))))
% 4.38/0.97 = { by lemma 50 }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(meet(x0, x1), join(composition(meet(x0, one), x1), x0))))
% 4.38/0.97 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(meet(x0, x1), join(x0, composition(meet(x0, one), x1)))))
% 4.38/0.97 = { by axiom 8 (maddux2_join_associativity) }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(join(meet(x0, x1), x0), composition(meet(x0, one), x1))))
% 4.38/0.97 = { by lemma 54 }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(x0, composition(meet(x0, one), x1))))
% 4.38/0.97 = { by lemma 53 R->L }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(join(x0, composition(x0, x1)), composition(meet(x0, one), x1))))
% 4.38/0.97 = { by axiom 8 (maddux2_join_associativity) R->L }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(x0, join(composition(x0, x1), composition(meet(x0, one), x1)))))
% 4.38/0.97 = { by axiom 2 (maddux1_join_commutativity) }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(x0, join(composition(meet(x0, one), x1), composition(x0, x1)))))
% 4.38/0.97 = { by axiom 12 (composition_distributivity) R->L }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(x0, composition(join(meet(x0, one), x0), x1))))
% 4.38/0.97 = { by lemma 54 }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), join(x0, composition(x0, x1))))
% 4.38/0.97 = { by lemma 53 }
% 4.38/0.97 meet(x1, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), x0))
% 4.38/0.97 = { by lemma 36 }
% 4.38/0.97 meet(x1, meet(x0, join(composition(meet(x0, one), x1), meet(x0, x1))))
% 4.38/0.97 = { by lemma 40 R->L }
% 4.38/0.97 meet(x1, meet(x0, meet(join(composition(meet(x0, one), x1), meet(x0, x1)), top)))
% 4.38/0.97 = { by lemma 37 R->L }
% 4.38/0.97 meet(x1, meet(x0, complement(join(zero, complement(join(composition(meet(x0, one), x1), meet(x0, x1)))))))
% 4.38/0.97 = { by lemma 47 R->L }
% 4.38/0.97 meet(x1, complement(join(complement(x0), join(zero, complement(join(composition(meet(x0, one), x1), meet(x0, x1)))))))
% 4.38/0.97 = { by lemma 47 R->L }
% 4.38/0.97 complement(join(complement(x1), join(complement(x0), join(zero, complement(join(composition(meet(x0, one), x1), meet(x0, x1)))))))
% 4.38/0.97 = { by axiom 8 (maddux2_join_associativity) }
% 4.38/0.97 complement(join(join(complement(x1), complement(x0)), join(zero, complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by lemma 51 R->L }
% 4.38/0.97 meet(complement(join(complement(x1), complement(x0))), complement(join(zero, complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 4.38/0.97 meet(meet(x1, x0), complement(join(zero, complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by lemma 36 R->L }
% 4.38/0.97 meet(complement(join(zero, complement(join(composition(meet(x0, one), x1), meet(x0, x1))))), meet(x1, x0))
% 4.38/0.97 = { by lemma 36 R->L }
% 4.38/0.97 meet(complement(join(zero, complement(join(composition(meet(x0, one), x1), meet(x0, x1))))), meet(x0, x1))
% 4.38/0.97 = { by lemma 37 }
% 4.38/0.97 meet(meet(join(composition(meet(x0, one), x1), meet(x0, x1)), top), meet(x0, x1))
% 4.38/0.97 = { by lemma 40 }
% 4.38/0.97 meet(join(composition(meet(x0, one), x1), meet(x0, x1)), meet(x0, x1))
% 4.38/0.97 = { by lemma 36 }
% 4.38/0.97 meet(meet(x0, x1), join(composition(meet(x0, one), x1), meet(x0, x1)))
% 4.38/0.97 = { by axiom 11 (maddux4_definiton_of_meet) }
% 4.38/0.97 complement(join(complement(meet(x0, x1)), complement(join(composition(meet(x0, one), x1), meet(x0, x1)))))
% 4.38/0.97 = { by lemma 32 R->L }
% 4.38/0.97 join(zero, complement(join(complement(meet(x0, x1)), complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by lemma 16 R->L }
% 4.38/0.97 join(complement(top), complement(join(complement(meet(x0, x1)), complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by lemma 24 R->L }
% 4.38/0.97 join(complement(join(composition(meet(x0, one), x1), top)), complement(join(complement(meet(x0, x1)), complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by lemma 17 R->L }
% 4.38/0.97 join(complement(join(meet(x0, x1), join(composition(meet(x0, one), x1), complement(meet(x0, x1))))), complement(join(complement(meet(x0, x1)), complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by lemma 50 R->L }
% 4.38/0.97 join(complement(join(complement(meet(x0, x1)), join(composition(meet(x0, one), x1), meet(x0, x1)))), complement(join(complement(meet(x0, x1)), complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by axiom 2 (maddux1_join_commutativity) }
% 4.38/0.97 join(complement(join(join(composition(meet(x0, one), x1), meet(x0, x1)), complement(meet(x0, x1)))), complement(join(complement(meet(x0, x1)), complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by lemma 45 }
% 4.38/0.97 join(meet(meet(x0, x1), complement(join(composition(meet(x0, one), x1), meet(x0, x1)))), complement(join(complement(meet(x0, x1)), complement(join(composition(meet(x0, one), x1), meet(x0, x1))))))
% 4.38/0.97 = { by lemma 28 }
% 4.38/0.97 meet(x0, x1)
% 4.38/0.97 % SZS output end Proof
% 4.38/0.97
% 4.38/0.97 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------