TSTP Solution File: REL033+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL033+1 : 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 : n025.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 13:44:18 EDT 2023
% Result : Theorem 62.82s 8.49s
% Output : Proof 63.86s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL033+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n025.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 22:30:07 EDT 2023
% 0.20/0.34 % CPUTime :
% 62.82/8.49 Command-line arguments: --ground-connectedness --complete-subsets
% 62.82/8.49
% 62.82/8.49 % SZS status Theorem
% 62.82/8.49
% 63.36/8.55 % SZS output start Proof
% 63.36/8.55 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 63.36/8.55 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 63.36/8.55 Axiom 3 (composition_identity): composition(X, one) = X.
% 63.36/8.55 Axiom 4 (goals): composition(x0, top) = x0.
% 63.36/8.55 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 63.36/8.55 Axiom 6 (def_top): top = join(X, complement(X)).
% 63.36/8.55 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 63.36/8.55 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 63.36/8.55 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 63.36/8.55 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 63.36/8.55 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 63.36/8.55 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 63.36/8.55 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 63.36/8.55 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 63.36/8.55
% 63.36/8.55 Lemma 15: complement(top) = zero.
% 63.36/8.55 Proof:
% 63.36/8.55 complement(top)
% 63.36/8.55 = { by axiom 6 (def_top) }
% 63.36/8.55 complement(join(complement(X), complement(complement(X))))
% 63.36/8.55 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 63.36/8.55 meet(X, complement(X))
% 63.36/8.55 = { by axiom 5 (def_zero) R->L }
% 63.36/8.55 zero
% 63.36/8.55
% 63.36/8.55 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 63.36/8.55 Proof:
% 63.36/8.55 converse(composition(converse(X), Y))
% 63.36/8.55 = { by axiom 9 (converse_multiplicativity) }
% 63.36/8.55 composition(converse(Y), converse(converse(X)))
% 63.36/8.55 = { by axiom 1 (converse_idempotence) }
% 63.36/8.55 composition(converse(Y), X)
% 63.36/8.55
% 63.36/8.55 Lemma 17: composition(converse(one), X) = X.
% 63.36/8.55 Proof:
% 63.36/8.55 composition(converse(one), X)
% 63.36/8.55 = { by lemma 16 R->L }
% 63.36/8.55 converse(composition(converse(X), one))
% 63.36/8.55 = { by axiom 3 (composition_identity) }
% 63.36/8.55 converse(converse(X))
% 63.36/8.55 = { by axiom 1 (converse_idempotence) }
% 63.36/8.55 X
% 63.36/8.55
% 63.36/8.55 Lemma 18: composition(one, X) = X.
% 63.36/8.55 Proof:
% 63.36/8.55 composition(one, X)
% 63.36/8.55 = { by lemma 17 R->L }
% 63.36/8.55 composition(converse(one), composition(one, X))
% 63.36/8.55 = { by axiom 10 (composition_associativity) }
% 63.36/8.55 composition(composition(converse(one), one), X)
% 63.36/8.55 = { by axiom 3 (composition_identity) }
% 63.36/8.55 composition(converse(one), X)
% 63.36/8.55 = { by lemma 17 }
% 63.36/8.55 X
% 63.36/8.55
% 63.36/8.55 Lemma 19: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 63.36/8.55 Proof:
% 63.36/8.55 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.55 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 63.36/8.55 = { by axiom 13 (converse_cancellativity) }
% 63.36/8.55 complement(X)
% 63.36/8.55
% 63.36/8.55 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 63.36/8.55 Proof:
% 63.36/8.55 join(complement(X), complement(X))
% 63.36/8.55 = { by lemma 17 R->L }
% 63.36/8.55 join(complement(X), composition(converse(one), complement(X)))
% 63.36/8.55 = { by lemma 18 R->L }
% 63.36/8.55 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 63.36/8.55 = { by lemma 19 }
% 63.36/8.55 complement(X)
% 63.36/8.55
% 63.36/8.55 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 63.36/8.55 Proof:
% 63.36/8.55 join(meet(X, Y), complement(join(complement(X), Y)))
% 63.36/8.55 = { by axiom 11 (maddux4_definiton_of_meet) }
% 63.36/8.55 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 63.36/8.55 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 63.36/8.55 X
% 63.36/8.55
% 63.36/8.55 Lemma 22: join(zero, meet(X, X)) = X.
% 63.36/8.55 Proof:
% 63.36/8.55 join(zero, meet(X, X))
% 63.36/8.55 = { by axiom 11 (maddux4_definiton_of_meet) }
% 63.36/8.55 join(zero, complement(join(complement(X), complement(X))))
% 63.36/8.55 = { by axiom 5 (def_zero) }
% 63.36/8.55 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 63.36/8.55 = { by lemma 21 }
% 63.36/8.55 X
% 63.36/8.55
% 63.36/8.55 Lemma 23: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 63.36/8.55 Proof:
% 63.36/8.55 join(zero, join(X, meet(Y, Y)))
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.55 join(zero, join(meet(Y, Y), X))
% 63.36/8.55 = { by axiom 8 (maddux2_join_associativity) }
% 63.36/8.55 join(join(zero, meet(Y, Y)), X)
% 63.36/8.55 = { by lemma 22 }
% 63.36/8.55 join(Y, X)
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) }
% 63.36/8.55 join(X, Y)
% 63.36/8.55
% 63.36/8.55 Lemma 24: join(X, join(Y, complement(X))) = join(Y, top).
% 63.36/8.55 Proof:
% 63.36/8.55 join(X, join(Y, complement(X)))
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.55 join(X, join(complement(X), Y))
% 63.36/8.55 = { by axiom 8 (maddux2_join_associativity) }
% 63.36/8.55 join(join(X, complement(X)), Y)
% 63.36/8.55 = { by axiom 6 (def_top) R->L }
% 63.36/8.55 join(top, Y)
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) }
% 63.36/8.55 join(Y, top)
% 63.36/8.55
% 63.36/8.55 Lemma 25: join(top, complement(X)) = top.
% 63.36/8.55 Proof:
% 63.36/8.55 join(top, complement(X))
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.55 join(complement(X), top)
% 63.36/8.55 = { by lemma 24 R->L }
% 63.36/8.55 join(X, join(complement(X), complement(X)))
% 63.36/8.55 = { by lemma 20 }
% 63.36/8.55 join(X, complement(X))
% 63.36/8.55 = { by axiom 6 (def_top) R->L }
% 63.36/8.55 top
% 63.36/8.55
% 63.36/8.55 Lemma 26: join(Y, top) = join(X, top).
% 63.36/8.55 Proof:
% 63.36/8.55 join(Y, top)
% 63.36/8.55 = { by lemma 25 R->L }
% 63.36/8.55 join(Y, join(top, complement(Y)))
% 63.36/8.55 = { by lemma 24 }
% 63.36/8.55 join(top, top)
% 63.36/8.55 = { by lemma 24 R->L }
% 63.36/8.55 join(X, join(top, complement(X)))
% 63.36/8.55 = { by lemma 25 }
% 63.36/8.55 join(X, top)
% 63.36/8.55
% 63.36/8.55 Lemma 27: join(X, top) = top.
% 63.36/8.55 Proof:
% 63.36/8.55 join(X, top)
% 63.36/8.55 = { by lemma 26 }
% 63.36/8.55 join(zero, top)
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.55 join(top, zero)
% 63.36/8.55 = { by lemma 15 R->L }
% 63.36/8.55 join(top, complement(top))
% 63.36/8.55 = { by axiom 6 (def_top) R->L }
% 63.36/8.55 top
% 63.36/8.55
% 63.36/8.55 Lemma 28: join(X, join(complement(X), Y)) = top.
% 63.36/8.55 Proof:
% 63.36/8.55 join(X, join(complement(X), Y))
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.55 join(X, join(Y, complement(X)))
% 63.36/8.55 = { by lemma 24 }
% 63.36/8.55 join(Y, top)
% 63.36/8.55 = { by lemma 26 R->L }
% 63.36/8.55 join(Z, top)
% 63.36/8.55 = { by lemma 27 }
% 63.36/8.55 top
% 63.36/8.55
% 63.36/8.55 Lemma 29: join(X, complement(zero)) = top.
% 63.36/8.55 Proof:
% 63.36/8.55 join(X, complement(zero))
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.55 join(complement(zero), X)
% 63.36/8.55 = { by lemma 23 R->L }
% 63.36/8.55 join(zero, join(complement(zero), meet(X, X)))
% 63.36/8.55 = { by lemma 28 }
% 63.36/8.55 top
% 63.36/8.55
% 63.36/8.55 Lemma 30: complement(zero) = top.
% 63.36/8.55 Proof:
% 63.36/8.55 complement(zero)
% 63.36/8.55 = { by lemma 20 R->L }
% 63.36/8.55 join(complement(zero), complement(zero))
% 63.36/8.55 = { by lemma 29 }
% 63.36/8.55 top
% 63.36/8.55
% 63.36/8.55 Lemma 31: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 63.36/8.55 Proof:
% 63.36/8.55 converse(join(X, converse(Y)))
% 63.36/8.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.56 converse(join(converse(Y), X))
% 63.36/8.56 = { by axiom 7 (converse_additivity) }
% 63.36/8.56 join(converse(converse(Y)), converse(X))
% 63.36/8.56 = { by axiom 1 (converse_idempotence) }
% 63.36/8.56 join(Y, converse(X))
% 63.36/8.56
% 63.36/8.56 Lemma 32: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 63.36/8.56 Proof:
% 63.36/8.56 converse(join(converse(X), Y))
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.56 converse(join(Y, converse(X)))
% 63.36/8.56 = { by lemma 31 }
% 63.36/8.56 join(X, converse(Y))
% 63.36/8.56
% 63.36/8.56 Lemma 33: join(X, converse(top)) = top.
% 63.36/8.56 Proof:
% 63.36/8.56 join(X, converse(top))
% 63.36/8.56 = { by axiom 6 (def_top) }
% 63.36/8.56 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 63.36/8.56 = { by lemma 32 }
% 63.36/8.56 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 63.36/8.56 = { by lemma 28 }
% 63.36/8.56 top
% 63.36/8.56
% 63.36/8.56 Lemma 34: converse(top) = top.
% 63.36/8.56 Proof:
% 63.36/8.56 converse(top)
% 63.36/8.56 = { by lemma 27 R->L }
% 63.36/8.56 converse(join(X, top))
% 63.36/8.56 = { by axiom 7 (converse_additivity) }
% 63.36/8.56 join(converse(X), converse(top))
% 63.36/8.56 = { by lemma 33 }
% 63.36/8.56 top
% 63.36/8.56
% 63.36/8.56 Lemma 35: complement(complement(X)) = meet(X, X).
% 63.36/8.56 Proof:
% 63.36/8.56 complement(complement(X))
% 63.36/8.56 = { by lemma 20 R->L }
% 63.36/8.56 complement(join(complement(X), complement(X)))
% 63.36/8.56 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 63.36/8.56 meet(X, X)
% 63.36/8.56
% 63.36/8.56 Lemma 36: meet(Y, X) = meet(X, Y).
% 63.36/8.56 Proof:
% 63.36/8.56 meet(Y, X)
% 63.36/8.56 = { by axiom 11 (maddux4_definiton_of_meet) }
% 63.36/8.56 complement(join(complement(Y), complement(X)))
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.56 complement(join(complement(X), complement(Y)))
% 63.36/8.56 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 63.36/8.56 meet(X, Y)
% 63.36/8.56
% 63.36/8.56 Lemma 37: complement(join(zero, complement(X))) = meet(X, top).
% 63.36/8.56 Proof:
% 63.36/8.56 complement(join(zero, complement(X)))
% 63.36/8.56 = { by lemma 15 R->L }
% 63.36/8.56 complement(join(complement(top), complement(X)))
% 63.36/8.56 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 63.36/8.56 meet(top, X)
% 63.36/8.56 = { by lemma 36 R->L }
% 63.36/8.56 meet(X, top)
% 63.36/8.56
% 63.36/8.56 Lemma 38: join(meet(X, Y), meet(X, complement(Y))) = X.
% 63.36/8.56 Proof:
% 63.36/8.56 join(meet(X, Y), meet(X, complement(Y)))
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.56 join(meet(X, complement(Y)), meet(X, Y))
% 63.36/8.56 = { by axiom 11 (maddux4_definiton_of_meet) }
% 63.36/8.56 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 63.36/8.56 = { by lemma 21 }
% 63.36/8.56 X
% 63.36/8.56
% 63.36/8.56 Lemma 39: join(zero, meet(X, top)) = X.
% 63.36/8.56 Proof:
% 63.36/8.56 join(zero, meet(X, top))
% 63.36/8.56 = { by lemma 30 R->L }
% 63.36/8.56 join(zero, meet(X, complement(zero)))
% 63.36/8.56 = { by lemma 15 R->L }
% 63.36/8.56 join(complement(top), meet(X, complement(zero)))
% 63.36/8.56 = { by lemma 29 R->L }
% 63.36/8.56 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 63.36/8.56 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 63.36/8.56 join(meet(X, zero), meet(X, complement(zero)))
% 63.36/8.56 = { by lemma 38 }
% 63.36/8.56 X
% 63.36/8.56
% 63.36/8.56 Lemma 40: join(zero, complement(X)) = complement(X).
% 63.36/8.56 Proof:
% 63.36/8.56 join(zero, complement(X))
% 63.36/8.56 = { by lemma 22 R->L }
% 63.36/8.56 join(zero, complement(join(zero, meet(X, X))))
% 63.36/8.56 = { by lemma 35 R->L }
% 63.36/8.56 join(zero, complement(join(zero, complement(complement(X)))))
% 63.36/8.56 = { by lemma 37 }
% 63.36/8.56 join(zero, meet(complement(X), top))
% 63.36/8.56 = { by lemma 39 }
% 63.36/8.56 complement(X)
% 63.36/8.56
% 63.36/8.56 Lemma 41: complement(complement(X)) = X.
% 63.36/8.56 Proof:
% 63.36/8.56 complement(complement(X))
% 63.36/8.56 = { by lemma 40 R->L }
% 63.36/8.56 join(zero, complement(complement(X)))
% 63.36/8.56 = { by lemma 35 }
% 63.36/8.56 join(zero, meet(X, X))
% 63.36/8.56 = { by lemma 22 }
% 63.36/8.56 X
% 63.36/8.56
% 63.36/8.56 Lemma 42: join(X, zero) = X.
% 63.36/8.56 Proof:
% 63.36/8.56 join(X, zero)
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.56 join(zero, X)
% 63.36/8.56 = { by lemma 41 R->L }
% 63.36/8.56 join(zero, complement(complement(X)))
% 63.36/8.56 = { by lemma 35 }
% 63.36/8.56 join(zero, meet(X, X))
% 63.36/8.56 = { by lemma 22 }
% 63.36/8.56 X
% 63.36/8.56
% 63.36/8.56 Lemma 43: join(zero, X) = X.
% 63.36/8.56 Proof:
% 63.36/8.56 join(zero, X)
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.56 join(X, zero)
% 63.36/8.56 = { by lemma 42 }
% 63.36/8.56 X
% 63.36/8.56
% 63.36/8.56 Lemma 44: composition(converse(x0), complement(x0)) = zero.
% 63.36/8.56 Proof:
% 63.36/8.56 composition(converse(x0), complement(x0))
% 63.36/8.56 = { by lemma 43 R->L }
% 63.36/8.56 join(zero, composition(converse(x0), complement(x0)))
% 63.36/8.56 = { by lemma 15 R->L }
% 63.36/8.56 join(complement(top), composition(converse(x0), complement(x0)))
% 63.36/8.56 = { by axiom 4 (goals) R->L }
% 63.36/8.56 join(complement(top), composition(converse(x0), complement(composition(x0, top))))
% 63.36/8.56 = { by lemma 19 }
% 63.36/8.56 complement(top)
% 63.36/8.56 = { by lemma 15 }
% 63.36/8.56 zero
% 63.36/8.56
% 63.36/8.56 Lemma 45: join(top, X) = top.
% 63.36/8.56 Proof:
% 63.36/8.56 join(top, X)
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.56 join(X, top)
% 63.36/8.56 = { by lemma 26 R->L }
% 63.36/8.56 join(Y, top)
% 63.36/8.56 = { by lemma 27 }
% 63.36/8.56 top
% 63.36/8.56
% 63.36/8.56 Lemma 46: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 63.36/8.56 Proof:
% 63.36/8.56 composition(join(one, Y), X)
% 63.36/8.56 = { by axiom 12 (composition_distributivity) }
% 63.36/8.56 join(composition(one, X), composition(Y, X))
% 63.36/8.56 = { by lemma 18 }
% 63.36/8.56 join(X, composition(Y, X))
% 63.36/8.56
% 63.36/8.56 Lemma 47: join(X, composition(top, X)) = composition(top, X).
% 63.36/8.56 Proof:
% 63.36/8.56 join(X, composition(top, X))
% 63.36/8.56 = { by lemma 34 R->L }
% 63.36/8.56 join(X, composition(converse(top), X))
% 63.36/8.56 = { by lemma 46 R->L }
% 63.36/8.56 composition(join(one, converse(top)), X)
% 63.36/8.56 = { by lemma 33 }
% 63.36/8.56 composition(top, X)
% 63.36/8.56
% 63.36/8.56 Lemma 48: composition(top, zero) = zero.
% 63.36/8.56 Proof:
% 63.36/8.56 composition(top, zero)
% 63.36/8.56 = { by lemma 34 R->L }
% 63.36/8.56 composition(converse(top), zero)
% 63.36/8.56 = { by lemma 43 R->L }
% 63.36/8.56 join(zero, composition(converse(top), zero))
% 63.36/8.56 = { by lemma 15 R->L }
% 63.36/8.56 join(complement(top), composition(converse(top), zero))
% 63.36/8.56 = { by lemma 15 R->L }
% 63.36/8.56 join(complement(top), composition(converse(top), complement(top)))
% 63.36/8.56 = { by lemma 45 R->L }
% 63.36/8.56 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 63.36/8.56 = { by lemma 47 }
% 63.36/8.56 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 63.36/8.56 = { by lemma 19 }
% 63.36/8.56 complement(top)
% 63.36/8.56 = { by lemma 15 }
% 63.36/8.56 zero
% 63.36/8.56
% 63.36/8.56 Lemma 49: composition(converse(complement(x0)), composition(x0, X)) = converse(zero).
% 63.36/8.56 Proof:
% 63.36/8.56 composition(converse(complement(x0)), composition(x0, X))
% 63.36/8.56 = { by lemma 16 R->L }
% 63.36/8.56 converse(composition(converse(composition(x0, X)), complement(x0)))
% 63.36/8.56 = { by axiom 9 (converse_multiplicativity) }
% 63.36/8.56 converse(composition(composition(converse(X), converse(x0)), complement(x0)))
% 63.36/8.56 = { by axiom 10 (composition_associativity) R->L }
% 63.36/8.56 converse(composition(converse(X), composition(converse(x0), complement(x0))))
% 63.36/8.56 = { by lemma 44 }
% 63.36/8.56 converse(composition(converse(X), zero))
% 63.36/8.56 = { by lemma 43 R->L }
% 63.36/8.56 converse(join(zero, composition(converse(X), zero)))
% 63.36/8.56 = { by lemma 48 R->L }
% 63.36/8.56 converse(join(composition(top, zero), composition(converse(X), zero)))
% 63.36/8.56 = { by axiom 12 (composition_distributivity) R->L }
% 63.36/8.56 converse(composition(join(top, converse(X)), zero))
% 63.36/8.56 = { by lemma 45 }
% 63.36/8.56 converse(composition(top, zero))
% 63.36/8.56 = { by lemma 48 }
% 63.36/8.56 converse(zero)
% 63.36/8.56
% 63.36/8.56 Lemma 50: join(converse(zero), X) = X.
% 63.36/8.56 Proof:
% 63.36/8.56 join(converse(zero), X)
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.56 join(X, converse(zero))
% 63.36/8.56 = { by lemma 49 R->L }
% 63.36/8.56 join(X, composition(converse(complement(x0)), composition(x0, composition(Y, X))))
% 63.36/8.56 = { by axiom 10 (composition_associativity) }
% 63.36/8.56 join(X, composition(converse(complement(x0)), composition(composition(x0, Y), X)))
% 63.36/8.56 = { by axiom 10 (composition_associativity) }
% 63.36/8.56 join(X, composition(composition(converse(complement(x0)), composition(x0, Y)), X))
% 63.36/8.56 = { by lemma 49 }
% 63.36/8.56 join(X, composition(converse(zero), X))
% 63.36/8.56 = { by lemma 46 R->L }
% 63.36/8.56 composition(join(one, converse(zero)), X)
% 63.36/8.56 = { by lemma 17 R->L }
% 63.36/8.56 composition(join(composition(converse(one), one), converse(zero)), X)
% 63.36/8.56 = { by axiom 3 (composition_identity) }
% 63.36/8.56 composition(join(converse(one), converse(zero)), X)
% 63.36/8.56 = { by axiom 7 (converse_additivity) R->L }
% 63.36/8.56 composition(converse(join(one, zero)), X)
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) }
% 63.36/8.56 composition(converse(join(zero, one)), X)
% 63.36/8.56 = { by lemma 43 }
% 63.36/8.56 composition(converse(one), X)
% 63.36/8.56 = { by lemma 17 }
% 63.36/8.56 X
% 63.36/8.56
% 63.36/8.56 Lemma 51: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 63.36/8.56 Proof:
% 63.36/8.56 complement(join(complement(X), meet(Y, Z)))
% 63.36/8.56 = { by lemma 36 }
% 63.36/8.56 complement(join(complement(X), meet(Z, Y)))
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.36/8.56 complement(join(meet(Z, Y), complement(X)))
% 63.36/8.56 = { by axiom 11 (maddux4_definiton_of_meet) }
% 63.36/8.56 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 63.36/8.56 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 63.36/8.56 meet(join(complement(Z), complement(Y)), X)
% 63.36/8.56 = { by lemma 36 R->L }
% 63.36/8.56 meet(X, join(complement(Z), complement(Y)))
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) }
% 63.36/8.56 meet(X, join(complement(Y), complement(Z)))
% 63.36/8.56
% 63.36/8.56 Lemma 52: meet(X, top) = X.
% 63.36/8.56 Proof:
% 63.36/8.56 meet(X, top)
% 63.36/8.56 = { by lemma 37 R->L }
% 63.36/8.56 complement(join(zero, complement(X)))
% 63.36/8.56 = { by lemma 40 R->L }
% 63.36/8.56 join(zero, complement(join(zero, complement(X))))
% 63.36/8.56 = { by lemma 37 }
% 63.36/8.56 join(zero, meet(X, top))
% 63.36/8.56 = { by lemma 39 }
% 63.36/8.56 X
% 63.36/8.56
% 63.36/8.56 Lemma 53: meet(top, X) = X.
% 63.36/8.56 Proof:
% 63.36/8.56 meet(top, X)
% 63.36/8.56 = { by lemma 36 }
% 63.36/8.56 meet(X, top)
% 63.36/8.56 = { by lemma 52 }
% 63.36/8.56 X
% 63.36/8.56
% 63.36/8.56 Lemma 54: complement(join(zero, meet(X, Y))) = join(complement(X), complement(Y)).
% 63.36/8.56 Proof:
% 63.36/8.56 complement(join(zero, meet(X, Y)))
% 63.36/8.56 = { by lemma 36 }
% 63.36/8.56 complement(join(zero, meet(Y, X)))
% 63.36/8.56 = { by lemma 15 R->L }
% 63.36/8.56 complement(join(complement(top), meet(Y, X)))
% 63.36/8.56 = { by lemma 51 }
% 63.36/8.56 meet(top, join(complement(Y), complement(X)))
% 63.36/8.56 = { by lemma 53 }
% 63.36/8.56 join(complement(Y), complement(X))
% 63.36/8.56 = { by axiom 2 (maddux1_join_commutativity) }
% 63.86/8.56 join(complement(X), complement(Y))
% 63.86/8.56
% 63.86/8.56 Lemma 55: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 63.86/8.56 Proof:
% 63.86/8.56 join(complement(X), complement(Y))
% 63.86/8.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.86/8.56 join(complement(Y), complement(X))
% 63.86/8.56 = { by lemma 54 R->L }
% 63.86/8.56 complement(join(zero, meet(Y, X)))
% 63.86/8.56 = { by axiom 2 (maddux1_join_commutativity) }
% 63.86/8.56 complement(join(meet(Y, X), zero))
% 63.86/8.56 = { by lemma 42 }
% 63.86/8.56 complement(meet(Y, X))
% 63.86/8.56 = { by lemma 36 R->L }
% 63.86/8.56 complement(meet(X, Y))
% 63.86/8.56
% 63.86/8.56 Lemma 56: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 63.86/8.56 Proof:
% 63.86/8.56 complement(meet(X, complement(Y)))
% 63.86/8.56 = { by lemma 36 }
% 63.86/8.56 complement(meet(complement(Y), X))
% 63.86/8.56 = { by lemma 40 R->L }
% 63.86/8.56 complement(meet(join(zero, complement(Y)), X))
% 63.86/8.56 = { by lemma 55 R->L }
% 63.86/8.56 join(complement(join(zero, complement(Y))), complement(X))
% 63.86/8.56 = { by lemma 37 }
% 63.86/8.56 join(meet(Y, top), complement(X))
% 63.86/8.56 = { by lemma 52 }
% 63.86/8.56 join(Y, complement(X))
% 63.86/8.56
% 63.86/8.56 Lemma 57: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 63.86/8.56 Proof:
% 63.86/8.56 complement(meet(complement(X), Y))
% 63.86/8.56 = { by lemma 36 }
% 63.86/8.56 complement(meet(Y, complement(X)))
% 63.86/8.56 = { by lemma 56 }
% 63.86/8.56 join(X, complement(Y))
% 63.86/8.56
% 63.86/8.56 Lemma 58: join(X, complement(meet(X, Y))) = top.
% 63.86/8.56 Proof:
% 63.86/8.56 join(X, complement(meet(X, Y)))
% 63.86/8.56 = { by lemma 36 }
% 63.86/8.56 join(X, complement(meet(Y, X)))
% 63.86/8.56 = { by lemma 55 R->L }
% 63.86/8.56 join(X, join(complement(Y), complement(X)))
% 63.86/8.56 = { by lemma 24 }
% 63.86/8.56 join(complement(Y), top)
% 63.86/8.56 = { by lemma 27 }
% 63.86/8.56 top
% 63.86/8.56
% 63.86/8.56 Lemma 59: meet(X, meet(Y, complement(X))) = zero.
% 63.86/8.56 Proof:
% 63.86/8.56 meet(X, meet(Y, complement(X)))
% 63.86/8.56 = { by lemma 36 }
% 63.86/8.56 meet(X, meet(complement(X), Y))
% 63.86/8.56 = { by axiom 11 (maddux4_definiton_of_meet) }
% 63.86/8.56 complement(join(complement(X), complement(meet(complement(X), Y))))
% 63.86/8.56 = { by lemma 58 }
% 63.86/8.56 complement(top)
% 63.86/8.56 = { by lemma 15 }
% 63.86/8.56 zero
% 63.86/8.56
% 63.86/8.56 Lemma 60: meet(X, join(X, complement(Y))) = X.
% 63.86/8.56 Proof:
% 63.86/8.56 meet(X, join(X, complement(Y)))
% 63.86/8.56 = { by lemma 56 R->L }
% 63.86/8.56 meet(X, complement(meet(Y, complement(X))))
% 63.86/8.56 = { by lemma 55 R->L }
% 63.86/8.56 meet(X, join(complement(Y), complement(complement(X))))
% 63.86/8.56 = { by lemma 51 R->L }
% 63.86/8.56 complement(join(complement(X), meet(Y, complement(X))))
% 63.86/8.56 = { by lemma 40 R->L }
% 63.86/8.56 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 63.86/8.56 = { by lemma 59 R->L }
% 63.86/8.56 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 63.86/8.56 = { by lemma 21 }
% 63.86/8.56 X
% 63.86/8.56
% 63.86/8.56 Lemma 61: converse(zero) = zero.
% 63.86/8.56 Proof:
% 63.86/8.56 converse(zero)
% 63.86/8.56 = { by lemma 50 R->L }
% 63.86/8.56 join(converse(zero), converse(zero))
% 63.86/8.56 = { by lemma 23 R->L }
% 63.86/8.56 join(zero, join(converse(zero), meet(converse(zero), converse(zero))))
% 63.86/8.56 = { by axiom 11 (maddux4_definiton_of_meet) }
% 63.86/8.56 join(zero, join(converse(zero), complement(join(complement(converse(zero)), complement(converse(zero))))))
% 63.86/8.56 = { by lemma 57 R->L }
% 63.86/8.56 join(zero, complement(meet(complement(converse(zero)), join(complement(converse(zero)), complement(converse(zero))))))
% 63.86/8.56 = { by lemma 60 }
% 63.86/8.56 join(zero, complement(complement(converse(zero))))
% 63.86/8.56 = { by lemma 41 }
% 63.86/8.56 join(zero, converse(zero))
% 63.86/8.56 = { by axiom 2 (maddux1_join_commutativity) }
% 63.86/8.56 join(converse(zero), zero)
% 63.86/8.56 = { by lemma 50 }
% 63.86/8.56 zero
% 63.86/8.56
% 63.86/8.56 Lemma 62: meet(X, X) = X.
% 63.86/8.56 Proof:
% 63.86/8.56 meet(X, X)
% 63.86/8.56 = { by lemma 35 R->L }
% 63.86/8.56 complement(complement(X))
% 63.86/8.56 = { by lemma 41 }
% 63.86/8.56 X
% 63.86/8.56
% 63.86/8.56 Lemma 63: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 63.86/8.56 Proof:
% 63.86/8.56 converse(composition(X, converse(Y)))
% 63.86/8.56 = { by axiom 9 (converse_multiplicativity) }
% 63.86/8.56 composition(converse(converse(Y)), converse(X))
% 63.86/8.56 = { by axiom 1 (converse_idempotence) }
% 63.86/8.56 composition(Y, converse(X))
% 63.86/8.56
% 63.86/8.56 Lemma 64: composition(complement(x0), top) = complement(x0).
% 63.86/8.56 Proof:
% 63.86/8.56 composition(complement(x0), top)
% 63.86/8.57 = { by lemma 34 R->L }
% 63.86/8.57 composition(complement(x0), converse(top))
% 63.86/8.57 = { by lemma 63 R->L }
% 63.86/8.57 converse(composition(top, converse(complement(x0))))
% 63.86/8.57 = { by lemma 47 R->L }
% 63.86/8.57 converse(join(converse(complement(x0)), composition(top, converse(complement(x0)))))
% 63.86/8.57 = { by lemma 32 }
% 63.86/8.57 join(complement(x0), converse(composition(top, converse(complement(x0)))))
% 63.86/8.57 = { by lemma 63 }
% 63.86/8.57 join(complement(x0), composition(complement(x0), converse(top)))
% 63.86/8.57 = { by lemma 34 }
% 63.86/8.57 join(complement(x0), composition(complement(x0), top))
% 63.86/8.57 = { by lemma 30 R->L }
% 63.86/8.57 join(complement(x0), composition(complement(x0), complement(zero)))
% 63.86/8.57 = { by lemma 61 R->L }
% 63.86/8.57 join(complement(x0), composition(complement(x0), complement(converse(zero))))
% 63.86/8.57 = { by axiom 1 (converse_idempotence) R->L }
% 63.86/8.57 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(zero))))
% 63.86/8.57 = { by lemma 44 R->L }
% 63.86/8.57 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(composition(converse(x0), complement(x0))))))
% 63.86/8.57 = { by lemma 16 }
% 63.86/8.57 join(complement(x0), composition(converse(converse(complement(x0))), complement(composition(converse(complement(x0)), x0))))
% 63.86/8.57 = { by lemma 19 }
% 63.86/8.57 complement(x0)
% 63.86/8.57
% 63.86/8.57 Lemma 65: join(meet(X, Y), meet(complement(Y), X)) = X.
% 63.86/8.57 Proof:
% 63.86/8.57 join(meet(X, Y), meet(complement(Y), X))
% 63.86/8.57 = { by lemma 36 }
% 63.86/8.57 join(meet(X, Y), meet(X, complement(Y)))
% 63.86/8.57 = { by lemma 38 }
% 63.86/8.57 X
% 63.86/8.57
% 63.86/8.57 Lemma 66: meet(X, join(Y, complement(X))) = meet(X, Y).
% 63.86/8.57 Proof:
% 63.86/8.57 meet(X, join(Y, complement(X)))
% 63.86/8.57 = { by lemma 57 R->L }
% 63.86/8.57 meet(X, complement(meet(complement(Y), X)))
% 63.86/8.57 = { by lemma 65 R->L }
% 63.86/8.57 join(meet(meet(X, complement(meet(complement(Y), X))), Y), meet(complement(Y), meet(X, complement(meet(complement(Y), X)))))
% 63.86/8.57 = { by lemma 55 R->L }
% 63.86/8.57 join(meet(meet(X, complement(meet(complement(Y), X))), Y), meet(complement(Y), meet(X, join(complement(complement(Y)), complement(X)))))
% 63.86/8.57 = { by axiom 11 (maddux4_definiton_of_meet) }
% 63.86/8.57 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(join(complement(complement(Y)), complement(meet(X, join(complement(complement(Y)), complement(X)))))))
% 63.86/8.57 = { by lemma 55 R->L }
% 63.86/8.57 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(join(complement(complement(Y)), join(complement(X), complement(join(complement(complement(Y)), complement(X)))))))
% 63.86/8.57 = { by axiom 8 (maddux2_join_associativity) }
% 63.86/8.57 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(join(join(complement(complement(Y)), complement(X)), complement(join(complement(complement(Y)), complement(X))))))
% 63.86/8.57 = { by axiom 6 (def_top) R->L }
% 63.86/8.57 join(meet(meet(X, complement(meet(complement(Y), X))), Y), complement(top))
% 63.86/8.57 = { by lemma 15 }
% 63.86/8.57 join(meet(meet(X, complement(meet(complement(Y), X))), Y), zero)
% 63.86/8.57 = { by lemma 42 }
% 63.86/8.57 meet(meet(X, complement(meet(complement(Y), X))), Y)
% 63.86/8.57 = { by lemma 36 }
% 63.86/8.57 meet(Y, meet(X, complement(meet(complement(Y), X))))
% 63.86/8.57 = { by lemma 62 R->L }
% 63.86/8.57 meet(meet(Y, meet(X, complement(meet(complement(Y), X)))), meet(Y, meet(X, complement(meet(complement(Y), X)))))
% 63.86/8.57 = { by lemma 35 R->L }
% 63.86/8.57 complement(complement(meet(Y, meet(X, complement(meet(complement(Y), X))))))
% 63.86/8.57 = { by lemma 36 }
% 63.86/8.57 complement(complement(meet(Y, meet(complement(meet(complement(Y), X)), X))))
% 63.86/8.57 = { by lemma 55 R->L }
% 63.86/8.57 complement(join(complement(Y), complement(meet(complement(meet(complement(Y), X)), X))))
% 63.86/8.57 = { by lemma 55 R->L }
% 63.86/8.57 complement(join(complement(Y), join(complement(complement(meet(complement(Y), X))), complement(X))))
% 63.86/8.57 = { by axiom 8 (maddux2_join_associativity) }
% 63.86/8.57 complement(join(join(complement(Y), complement(complement(meet(complement(Y), X)))), complement(X)))
% 63.86/8.57 = { by lemma 56 R->L }
% 63.86/8.57 complement(complement(meet(X, complement(join(complement(Y), complement(complement(meet(complement(Y), X))))))))
% 63.86/8.57 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 63.86/8.57 complement(complement(meet(X, meet(Y, complement(meet(complement(Y), X))))))
% 63.86/8.57 = { by lemma 36 R->L }
% 63.86/8.57 complement(complement(meet(X, meet(complement(meet(complement(Y), X)), Y))))
% 63.86/8.57 = { by lemma 41 }
% 63.86/8.57 meet(X, meet(complement(meet(complement(Y), X)), Y))
% 63.86/8.57 = { by lemma 36 R->L }
% 63.86/8.57 meet(X, meet(Y, complement(meet(complement(Y), X))))
% 63.86/8.57 = { by lemma 57 }
% 63.86/8.57 meet(X, meet(Y, join(Y, complement(X))))
% 63.86/8.57 = { by lemma 60 }
% 63.86/8.57 meet(X, Y)
% 63.86/8.57
% 63.86/8.57 Lemma 67: join(meet(Y, X), complement(Y)) = join(X, complement(Y)).
% 63.86/8.57 Proof:
% 63.86/8.57 join(meet(Y, X), complement(Y))
% 63.86/8.57 = { by lemma 56 R->L }
% 63.86/8.57 complement(meet(Y, complement(meet(Y, X))))
% 63.86/8.57 = { by lemma 55 R->L }
% 63.86/8.57 join(complement(Y), complement(complement(meet(Y, X))))
% 63.86/8.57 = { by lemma 54 R->L }
% 63.86/8.57 complement(join(zero, meet(Y, complement(meet(Y, X)))))
% 63.86/8.57 = { by lemma 43 R->L }
% 63.86/8.57 complement(join(zero, meet(Y, complement(join(zero, meet(Y, X))))))
% 63.86/8.57 = { by lemma 54 }
% 63.86/8.57 complement(join(zero, meet(Y, join(complement(Y), complement(X)))))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) }
% 63.86/8.57 complement(join(zero, meet(Y, join(complement(X), complement(Y)))))
% 63.86/8.57 = { by lemma 66 }
% 63.86/8.57 complement(join(zero, meet(Y, complement(X))))
% 63.86/8.57 = { by lemma 54 }
% 63.86/8.57 join(complement(Y), complement(complement(X)))
% 63.86/8.57 = { by lemma 55 }
% 63.86/8.57 complement(meet(Y, complement(X)))
% 63.86/8.57 = { by lemma 56 }
% 63.86/8.57 join(X, complement(Y))
% 63.86/8.57
% 63.86/8.57 Lemma 68: meet(complement(X), join(Y, X)) = meet(Y, complement(X)).
% 63.86/8.57 Proof:
% 63.86/8.57 meet(complement(X), join(Y, X))
% 63.86/8.57 = { by lemma 62 R->L }
% 63.86/8.57 meet(complement(X), join(Y, meet(X, X)))
% 63.86/8.57 = { by lemma 35 R->L }
% 63.86/8.57 meet(complement(X), join(Y, complement(complement(X))))
% 63.86/8.57 = { by lemma 66 }
% 63.86/8.57 meet(complement(X), Y)
% 63.86/8.57 = { by lemma 36 R->L }
% 63.86/8.57 meet(Y, complement(X))
% 63.86/8.57
% 63.86/8.57 Lemma 69: join(complement(x0), composition(join(X, complement(x0)), Y)) = join(complement(x0), composition(X, Y)).
% 63.86/8.57 Proof:
% 63.86/8.57 join(complement(x0), composition(join(X, complement(x0)), Y))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.86/8.57 join(complement(x0), composition(join(complement(x0), X), Y))
% 63.86/8.57 = { by lemma 64 R->L }
% 63.86/8.57 join(composition(complement(x0), top), composition(join(complement(x0), X), Y))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.86/8.57 join(composition(complement(x0), top), composition(join(X, complement(x0)), Y))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.86/8.57 join(composition(join(X, complement(x0)), Y), composition(complement(x0), top))
% 63.86/8.57 = { by axiom 12 (composition_distributivity) }
% 63.86/8.57 join(join(composition(X, Y), composition(complement(x0), Y)), composition(complement(x0), top))
% 63.86/8.57 = { by axiom 8 (maddux2_join_associativity) R->L }
% 63.86/8.57 join(composition(X, Y), join(composition(complement(x0), Y), composition(complement(x0), top)))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) }
% 63.86/8.57 join(composition(X, Y), join(composition(complement(x0), top), composition(complement(x0), Y)))
% 63.86/8.57 = { by axiom 1 (converse_idempotence) R->L }
% 63.86/8.57 join(composition(X, Y), join(composition(complement(x0), top), composition(complement(x0), converse(converse(Y)))))
% 63.86/8.57 = { by axiom 1 (converse_idempotence) R->L }
% 63.86/8.57 join(composition(X, Y), converse(converse(join(composition(complement(x0), top), composition(complement(x0), converse(converse(Y)))))))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.86/8.57 join(composition(X, Y), converse(converse(join(composition(complement(x0), converse(converse(Y))), composition(complement(x0), top)))))
% 63.86/8.57 = { by axiom 7 (converse_additivity) }
% 63.86/8.57 join(composition(X, Y), converse(join(converse(composition(complement(x0), converse(converse(Y)))), converse(composition(complement(x0), top)))))
% 63.86/8.57 = { by lemma 63 }
% 63.86/8.57 join(composition(X, Y), converse(join(composition(converse(Y), converse(complement(x0))), converse(composition(complement(x0), top)))))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) }
% 63.86/8.57 join(composition(X, Y), converse(join(converse(composition(complement(x0), top)), composition(converse(Y), converse(complement(x0))))))
% 63.86/8.57 = { by axiom 9 (converse_multiplicativity) }
% 63.86/8.57 join(composition(X, Y), converse(join(composition(converse(top), converse(complement(x0))), composition(converse(Y), converse(complement(x0))))))
% 63.86/8.57 = { by axiom 12 (composition_distributivity) R->L }
% 63.86/8.57 join(composition(X, Y), converse(composition(join(converse(top), converse(Y)), converse(complement(x0)))))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.86/8.57 join(composition(X, Y), converse(composition(join(converse(Y), converse(top)), converse(complement(x0)))))
% 63.86/8.57 = { by lemma 31 R->L }
% 63.86/8.57 join(composition(X, Y), converse(composition(converse(join(top, converse(converse(Y)))), converse(complement(x0)))))
% 63.86/8.57 = { by axiom 9 (converse_multiplicativity) R->L }
% 63.86/8.57 join(composition(X, Y), converse(converse(composition(complement(x0), join(top, converse(converse(Y)))))))
% 63.86/8.57 = { by axiom 1 (converse_idempotence) }
% 63.86/8.57 join(composition(X, Y), composition(complement(x0), join(top, converse(converse(Y)))))
% 63.86/8.57 = { by axiom 1 (converse_idempotence) }
% 63.86/8.57 join(composition(X, Y), composition(complement(x0), join(top, Y)))
% 63.86/8.57 = { by lemma 45 }
% 63.86/8.57 join(composition(X, Y), composition(complement(x0), top))
% 63.86/8.57 = { by lemma 64 }
% 63.86/8.57 join(composition(X, Y), complement(x0))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) }
% 63.86/8.57 join(complement(x0), composition(X, Y))
% 63.86/8.57
% 63.86/8.57 Goal 1 (goals_1): composition(meet(x0, x1), x2) = meet(x0, composition(x1, x2)).
% 63.86/8.57 Proof:
% 63.86/8.57 composition(meet(x0, x1), x2)
% 63.86/8.57 = { by lemma 53 R->L }
% 63.86/8.57 meet(top, composition(meet(x0, x1), x2))
% 63.86/8.57 = { by lemma 36 }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), top)
% 63.86/8.57 = { by lemma 27 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), top))
% 63.86/8.57 = { by lemma 58 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(x0, complement(meet(x0, composition(x0, x2))))))
% 63.86/8.57 = { by lemma 36 }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(x0, complement(meet(composition(x0, x2), x0)))))
% 63.86/8.57 = { by lemma 55 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(x0, join(complement(composition(x0, x2)), complement(x0)))))
% 63.86/8.57 = { by lemma 64 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(x0, join(complement(composition(x0, x2)), composition(complement(x0), top)))))
% 63.86/8.57 = { by lemma 30 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(x0, join(complement(composition(x0, x2)), composition(complement(x0), complement(zero))))))
% 63.86/8.57 = { by lemma 61 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(x0, join(complement(composition(x0, x2)), composition(complement(x0), complement(converse(zero)))))))
% 63.86/8.57 = { by lemma 49 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(x0, join(complement(composition(x0, x2)), composition(complement(x0), complement(composition(converse(complement(x0)), composition(x0, x2))))))))
% 63.86/8.57 = { by axiom 1 (converse_idempotence) R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(x0, join(complement(composition(x0, x2)), composition(converse(converse(complement(x0))), complement(composition(converse(complement(x0)), composition(x0, x2))))))))
% 63.86/8.57 = { by lemma 19 }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(x0, complement(composition(x0, x2)))))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(composition(meet(x0, x1), x2)), join(complement(composition(x0, x2)), x0)))
% 63.86/8.57 = { by axiom 8 (maddux2_join_associativity) }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(join(complement(composition(meet(x0, x1), x2)), complement(composition(x0, x2))), x0))
% 63.86/8.57 = { by lemma 55 }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(complement(meet(composition(meet(x0, x1), x2), composition(x0, x2))), x0))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(x0, complement(meet(composition(meet(x0, x1), x2), composition(x0, x2)))))
% 63.86/8.57 = { by lemma 38 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(x0, complement(meet(composition(meet(x0, x1), x2), composition(join(meet(x0, x1), meet(x0, complement(x1))), x2)))))
% 63.86/8.57 = { by axiom 12 (composition_distributivity) }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(x0, complement(meet(composition(meet(x0, x1), x2), join(composition(meet(x0, x1), x2), composition(meet(x0, complement(x1)), x2))))))
% 63.86/8.57 = { by lemma 62 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(x0, complement(meet(composition(meet(x0, x1), x2), join(composition(meet(x0, x1), x2), meet(composition(meet(x0, complement(x1)), x2), composition(meet(x0, complement(x1)), x2)))))))
% 63.86/8.57 = { by lemma 35 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(x0, complement(meet(composition(meet(x0, x1), x2), join(composition(meet(x0, x1), x2), complement(complement(composition(meet(x0, complement(x1)), x2))))))))
% 63.86/8.57 = { by lemma 60 }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), join(x0, complement(composition(meet(x0, x1), x2))))
% 63.86/8.57 = { by lemma 66 }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), x0)
% 63.86/8.57 = { by lemma 41 R->L }
% 63.86/8.57 meet(composition(meet(x0, x1), x2), complement(complement(x0)))
% 63.86/8.57 = { by lemma 68 R->L }
% 63.86/8.57 meet(complement(complement(x0)), join(composition(meet(x0, x1), x2), complement(x0)))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 63.86/8.57 meet(complement(complement(x0)), join(complement(x0), composition(meet(x0, x1), x2)))
% 63.86/8.57 = { by lemma 69 R->L }
% 63.86/8.57 meet(complement(complement(x0)), join(complement(x0), composition(join(meet(x0, x1), complement(x0)), x2)))
% 63.86/8.57 = { by lemma 67 }
% 63.86/8.57 meet(complement(complement(x0)), join(complement(x0), composition(join(x1, complement(x0)), x2)))
% 63.86/8.57 = { by lemma 69 }
% 63.86/8.57 meet(complement(complement(x0)), join(complement(x0), composition(x1, x2)))
% 63.86/8.57 = { by axiom 2 (maddux1_join_commutativity) }
% 63.86/8.57 meet(complement(complement(x0)), join(composition(x1, x2), complement(x0)))
% 63.86/8.57 = { by lemma 67 R->L }
% 63.86/8.57 meet(complement(complement(x0)), join(meet(x0, composition(x1, x2)), complement(x0)))
% 63.86/8.57 = { by lemma 68 }
% 63.86/8.57 meet(meet(x0, composition(x1, x2)), complement(complement(x0)))
% 63.86/8.57 = { by lemma 41 }
% 63.86/8.57 meet(meet(x0, composition(x1, x2)), x0)
% 63.86/8.58 = { by lemma 36 }
% 63.86/8.58 meet(x0, meet(x0, composition(x1, x2)))
% 63.86/8.58 = { by lemma 36 }
% 63.86/8.58 meet(x0, meet(composition(x1, x2), x0))
% 63.86/8.58 = { by lemma 36 }
% 63.86/8.58 meet(meet(composition(x1, x2), x0), x0)
% 63.86/8.58 = { by lemma 41 R->L }
% 63.86/8.58 meet(meet(composition(x1, x2), complement(complement(x0))), x0)
% 63.86/8.58 = { by lemma 42 R->L }
% 63.86/8.58 join(meet(meet(composition(x1, x2), complement(complement(x0))), x0), zero)
% 63.86/8.58 = { by lemma 59 R->L }
% 63.86/8.58 join(meet(meet(composition(x1, x2), complement(complement(x0))), x0), meet(complement(x0), meet(composition(x1, x2), complement(complement(x0)))))
% 63.86/8.58 = { by lemma 65 }
% 63.86/8.58 meet(composition(x1, x2), complement(complement(x0)))
% 63.86/8.58 = { by lemma 41 }
% 63.86/8.58 meet(composition(x1, x2), x0)
% 63.86/8.58 = { by lemma 36 R->L }
% 63.86/8.58 meet(x0, composition(x1, x2))
% 63.86/8.58 % SZS output end Proof
% 63.86/8.58
% 63.86/8.58 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------