TSTP Solution File: REL042+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL042+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 : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:44:27 EDT 2023
% Result : Theorem 34.39s 4.79s
% Output : Proof 34.39s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL042+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 : n003.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 20:48:53 EDT 2023
% 0.14/0.35 % CPUTime :
% 34.39/4.79 Command-line arguments: --no-flatten-goal
% 34.39/4.79
% 34.39/4.79 % SZS status Theorem
% 34.39/4.79
% 34.39/4.88 % SZS output start Proof
% 34.39/4.88 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 34.39/4.88 Axiom 2 (composition_identity): composition(X, one) = X.
% 34.39/4.88 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 34.39/4.88 Axiom 4 (def_zero): zero = meet(X, complement(X)).
% 34.39/4.88 Axiom 5 (def_top): top = join(X, complement(X)).
% 34.39/4.88 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 34.39/4.88 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 34.39/4.88 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 34.39/4.88 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 34.39/4.88 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 34.39/4.88 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 34.39/4.88 Axiom 12 (goals): meet(composition(x0, X), composition(x0, complement(X))) = zero.
% 34.39/4.88 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 34.39/4.88 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 34.39/4.88
% 34.39/4.88 Lemma 15: complement(top) = zero.
% 34.39/4.88 Proof:
% 34.39/4.88 complement(top)
% 34.39/4.88 = { by axiom 5 (def_top) }
% 34.39/4.88 complement(join(complement(X), complement(complement(X))))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 34.39/4.88 meet(X, complement(X))
% 34.39/4.88 = { by axiom 4 (def_zero) R->L }
% 34.39/4.88 zero
% 34.39/4.88
% 34.39/4.88 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 34.39/4.88 Proof:
% 34.39/4.88 converse(composition(converse(X), Y))
% 34.39/4.88 = { by axiom 6 (converse_multiplicativity) }
% 34.39/4.88 composition(converse(Y), converse(converse(X)))
% 34.39/4.88 = { by axiom 1 (converse_idempotence) }
% 34.39/4.88 composition(converse(Y), X)
% 34.39/4.88
% 34.39/4.88 Lemma 17: composition(converse(one), X) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 composition(converse(one), X)
% 34.39/4.88 = { by lemma 16 R->L }
% 34.39/4.88 converse(composition(converse(X), one))
% 34.39/4.88 = { by axiom 2 (composition_identity) }
% 34.39/4.88 converse(converse(X))
% 34.39/4.88 = { by axiom 1 (converse_idempotence) }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 18: converse(one) = one.
% 34.39/4.88 Proof:
% 34.39/4.88 converse(one)
% 34.39/4.88 = { by axiom 2 (composition_identity) R->L }
% 34.39/4.88 composition(converse(one), one)
% 34.39/4.88 = { by lemma 17 }
% 34.39/4.88 one
% 34.39/4.88
% 34.39/4.88 Lemma 19: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 join(meet(X, Y), complement(join(complement(X), Y)))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) }
% 34.39/4.88 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 34.39/4.88 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 20: join(zero, meet(X, X)) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 join(zero, meet(X, X))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) }
% 34.39/4.88 join(zero, complement(join(complement(X), complement(X))))
% 34.39/4.88 = { by axiom 4 (def_zero) }
% 34.39/4.88 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 34.39/4.88 = { by lemma 19 }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 21: composition(one, X) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 composition(one, X)
% 34.39/4.88 = { by lemma 17 R->L }
% 34.39/4.88 composition(converse(one), composition(one, X))
% 34.39/4.88 = { by axiom 7 (composition_associativity) }
% 34.39/4.88 composition(composition(converse(one), one), X)
% 34.39/4.88 = { by axiom 2 (composition_identity) }
% 34.39/4.88 composition(converse(one), X)
% 34.39/4.88 = { by lemma 17 }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 34.39/4.88 Proof:
% 34.39/4.88 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 34.39/4.88 = { by axiom 13 (converse_cancellativity) }
% 34.39/4.88 complement(X)
% 34.39/4.88
% 34.39/4.88 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 34.39/4.88 Proof:
% 34.39/4.88 join(complement(X), complement(X))
% 34.39/4.88 = { by lemma 17 R->L }
% 34.39/4.88 join(complement(X), composition(converse(one), complement(X)))
% 34.39/4.88 = { by lemma 21 R->L }
% 34.39/4.88 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 34.39/4.88 = { by lemma 22 }
% 34.39/4.88 complement(X)
% 34.39/4.88
% 34.39/4.88 Lemma 24: complement(complement(X)) = meet(X, X).
% 34.39/4.88 Proof:
% 34.39/4.88 complement(complement(X))
% 34.39/4.88 = { by lemma 23 R->L }
% 34.39/4.88 complement(join(complement(X), complement(X)))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 34.39/4.88 meet(X, X)
% 34.39/4.88
% 34.39/4.88 Lemma 25: meet(Y, X) = meet(X, Y).
% 34.39/4.88 Proof:
% 34.39/4.88 meet(Y, X)
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) }
% 34.39/4.88 complement(join(complement(Y), complement(X)))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 complement(join(complement(X), complement(Y)))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 34.39/4.88 meet(X, Y)
% 34.39/4.88
% 34.39/4.88 Lemma 26: complement(join(zero, complement(X))) = meet(X, top).
% 34.39/4.88 Proof:
% 34.39/4.88 complement(join(zero, complement(X)))
% 34.39/4.88 = { by lemma 15 R->L }
% 34.39/4.88 complement(join(complement(top), complement(X)))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 34.39/4.88 meet(top, X)
% 34.39/4.88 = { by lemma 25 R->L }
% 34.39/4.88 meet(X, top)
% 34.39/4.88
% 34.39/4.88 Lemma 27: join(X, join(Y, complement(X))) = join(Y, top).
% 34.39/4.88 Proof:
% 34.39/4.88 join(X, join(Y, complement(X)))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 join(X, join(complement(X), Y))
% 34.39/4.88 = { by axiom 9 (maddux2_join_associativity) }
% 34.39/4.88 join(join(X, complement(X)), Y)
% 34.39/4.88 = { by axiom 5 (def_top) R->L }
% 34.39/4.88 join(top, Y)
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.88 join(Y, top)
% 34.39/4.88
% 34.39/4.88 Lemma 28: join(top, complement(X)) = top.
% 34.39/4.88 Proof:
% 34.39/4.88 join(top, complement(X))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 join(complement(X), top)
% 34.39/4.88 = { by lemma 27 R->L }
% 34.39/4.88 join(X, join(complement(X), complement(X)))
% 34.39/4.88 = { by lemma 23 }
% 34.39/4.88 join(X, complement(X))
% 34.39/4.88 = { by axiom 5 (def_top) R->L }
% 34.39/4.88 top
% 34.39/4.88
% 34.39/4.88 Lemma 29: join(Y, top) = join(X, top).
% 34.39/4.88 Proof:
% 34.39/4.88 join(Y, top)
% 34.39/4.88 = { by lemma 28 R->L }
% 34.39/4.88 join(Y, join(top, complement(Y)))
% 34.39/4.88 = { by lemma 27 }
% 34.39/4.88 join(top, top)
% 34.39/4.88 = { by lemma 27 R->L }
% 34.39/4.88 join(X, join(top, complement(X)))
% 34.39/4.88 = { by lemma 28 }
% 34.39/4.88 join(X, top)
% 34.39/4.88
% 34.39/4.88 Lemma 30: join(X, top) = top.
% 34.39/4.88 Proof:
% 34.39/4.88 join(X, top)
% 34.39/4.88 = { by lemma 29 }
% 34.39/4.88 join(join(zero, zero), top)
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 join(top, join(zero, zero))
% 34.39/4.88 = { by lemma 15 R->L }
% 34.39/4.88 join(top, join(zero, complement(top)))
% 34.39/4.88 = { by lemma 15 R->L }
% 34.39/4.88 join(top, join(complement(top), complement(top)))
% 34.39/4.88 = { by lemma 23 }
% 34.39/4.88 join(top, complement(top))
% 34.39/4.88 = { by axiom 5 (def_top) R->L }
% 34.39/4.88 top
% 34.39/4.88
% 34.39/4.88 Lemma 31: join(X, join(complement(X), Y)) = top.
% 34.39/4.88 Proof:
% 34.39/4.88 join(X, join(complement(X), Y))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 join(X, join(Y, complement(X)))
% 34.39/4.88 = { by lemma 27 }
% 34.39/4.88 join(Y, top)
% 34.39/4.88 = { by lemma 29 R->L }
% 34.39/4.88 join(Z, top)
% 34.39/4.88 = { by lemma 30 }
% 34.39/4.88 top
% 34.39/4.88
% 34.39/4.88 Lemma 32: join(X, complement(zero)) = top.
% 34.39/4.88 Proof:
% 34.39/4.88 join(X, complement(zero))
% 34.39/4.88 = { by lemma 20 R->L }
% 34.39/4.88 join(join(zero, meet(X, X)), complement(zero))
% 34.39/4.88 = { by axiom 9 (maddux2_join_associativity) R->L }
% 34.39/4.88 join(zero, join(meet(X, X), complement(zero)))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.88 join(zero, join(complement(zero), meet(X, X)))
% 34.39/4.88 = { by lemma 31 }
% 34.39/4.88 top
% 34.39/4.88
% 34.39/4.88 Lemma 33: join(meet(X, Y), meet(X, complement(Y))) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 join(meet(X, Y), meet(X, complement(Y)))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 join(meet(X, complement(Y)), meet(X, Y))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) }
% 34.39/4.88 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 34.39/4.88 = { by lemma 19 }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 34: join(zero, meet(X, top)) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 join(zero, meet(X, top))
% 34.39/4.88 = { by lemma 32 R->L }
% 34.39/4.88 join(zero, meet(X, join(complement(zero), complement(zero))))
% 34.39/4.88 = { by lemma 23 }
% 34.39/4.88 join(zero, meet(X, complement(zero)))
% 34.39/4.88 = { by lemma 15 R->L }
% 34.39/4.88 join(complement(top), meet(X, complement(zero)))
% 34.39/4.88 = { by lemma 32 R->L }
% 34.39/4.88 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 34.39/4.88 join(meet(X, zero), meet(X, complement(zero)))
% 34.39/4.88 = { by lemma 33 }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 35: join(zero, complement(X)) = complement(X).
% 34.39/4.88 Proof:
% 34.39/4.88 join(zero, complement(X))
% 34.39/4.88 = { by lemma 20 R->L }
% 34.39/4.88 join(zero, complement(join(zero, meet(X, X))))
% 34.39/4.88 = { by lemma 24 R->L }
% 34.39/4.88 join(zero, complement(join(zero, complement(complement(X)))))
% 34.39/4.88 = { by lemma 26 }
% 34.39/4.88 join(zero, meet(complement(X), top))
% 34.39/4.88 = { by lemma 34 }
% 34.39/4.88 complement(X)
% 34.39/4.88
% 34.39/4.88 Lemma 36: complement(complement(X)) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 complement(complement(X))
% 34.39/4.88 = { by lemma 35 R->L }
% 34.39/4.88 join(zero, complement(complement(X)))
% 34.39/4.88 = { by lemma 24 }
% 34.39/4.88 join(zero, meet(X, X))
% 34.39/4.88 = { by lemma 20 }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 37: join(X, X) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 join(X, X)
% 34.39/4.88 = { by lemma 36 R->L }
% 34.39/4.88 join(X, complement(complement(X)))
% 34.39/4.88 = { by lemma 36 R->L }
% 34.39/4.88 join(complement(complement(X)), complement(complement(X)))
% 34.39/4.88 = { by lemma 23 }
% 34.39/4.88 complement(complement(X))
% 34.39/4.88 = { by lemma 36 }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 38: join(X, zero) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 join(X, zero)
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 join(zero, X)
% 34.39/4.88 = { by lemma 36 R->L }
% 34.39/4.88 join(zero, complement(complement(X)))
% 34.39/4.88 = { by lemma 24 }
% 34.39/4.88 join(zero, meet(X, X))
% 34.39/4.88 = { by lemma 20 }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 39: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 34.39/4.88 Proof:
% 34.39/4.88 converse(join(X, converse(Y)))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 converse(join(converse(Y), X))
% 34.39/4.88 = { by axiom 8 (converse_additivity) }
% 34.39/4.88 join(converse(converse(Y)), converse(X))
% 34.39/4.88 = { by axiom 1 (converse_idempotence) }
% 34.39/4.88 join(Y, converse(X))
% 34.39/4.88
% 34.39/4.88 Lemma 40: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 34.39/4.88 Proof:
% 34.39/4.88 converse(join(converse(X), Y))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 converse(join(Y, converse(X)))
% 34.39/4.88 = { by lemma 39 }
% 34.39/4.88 join(X, converse(Y))
% 34.39/4.88
% 34.39/4.88 Lemma 41: meet(X, converse(complement(converse(complement(X))))) = X.
% 34.39/4.88 Proof:
% 34.39/4.88 meet(X, converse(complement(converse(complement(X)))))
% 34.39/4.88 = { by lemma 23 R->L }
% 34.39/4.88 meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X))))))
% 34.39/4.88 = { by lemma 38 R->L }
% 34.39/4.88 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), zero)
% 34.39/4.88 = { by lemma 15 R->L }
% 34.39/4.88 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(top))
% 34.39/4.88 = { by lemma 31 R->L }
% 34.39/4.88 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(converse(Y), join(complement(converse(Y)), converse(complement(converse(complement(converse(Y)))))))))
% 34.39/4.88 = { by lemma 40 R->L }
% 34.39/4.88 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(converse(Y), converse(join(converse(complement(converse(Y))), complement(converse(complement(converse(Y)))))))))
% 34.39/4.88 = { by axiom 5 (def_top) R->L }
% 34.39/4.88 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(converse(Y), converse(top))))
% 34.39/4.88 = { by axiom 8 (converse_additivity) R->L }
% 34.39/4.88 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(converse(join(Y, top))))
% 34.39/4.88 = { by lemma 30 }
% 34.39/4.88 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(converse(top)))
% 34.39/4.88 = { by lemma 31 R->L }
% 34.39/4.88 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(converse(join(converse(complement(X)), join(complement(converse(complement(X))), complement(converse(complement(X))))))))
% 34.39/4.88 = { by lemma 40 }
% 34.39/4.88 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(complement(X), converse(join(complement(converse(complement(X))), complement(converse(complement(X))))))))
% 34.39/4.88 = { by lemma 19 }
% 34.39/4.88 X
% 34.39/4.88
% 34.39/4.88 Lemma 42: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 34.39/4.88 Proof:
% 34.39/4.88 complement(join(complement(X), meet(Y, Z)))
% 34.39/4.88 = { by lemma 25 }
% 34.39/4.88 complement(join(complement(X), meet(Z, Y)))
% 34.39/4.88 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.88 complement(join(meet(Z, Y), complement(X)))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) }
% 34.39/4.88 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 34.39/4.88 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 34.39/4.88 meet(join(complement(Z), complement(Y)), X)
% 34.39/4.88 = { by lemma 25 R->L }
% 34.39/4.88 meet(X, join(complement(Z), complement(Y)))
% 34.39/4.89 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.89 meet(X, join(complement(Y), complement(Z)))
% 34.39/4.89
% 34.39/4.89 Lemma 43: meet(X, top) = X.
% 34.39/4.89 Proof:
% 34.39/4.89 meet(X, top)
% 34.39/4.89 = { by lemma 26 R->L }
% 34.39/4.89 complement(join(zero, complement(X)))
% 34.39/4.89 = { by lemma 35 R->L }
% 34.39/4.89 join(zero, complement(join(zero, complement(X))))
% 34.39/4.89 = { by lemma 26 }
% 34.39/4.89 join(zero, meet(X, top))
% 34.39/4.89 = { by lemma 34 }
% 34.39/4.89 X
% 34.39/4.89
% 34.39/4.89 Lemma 44: meet(top, X) = X.
% 34.39/4.89 Proof:
% 34.39/4.89 meet(top, X)
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 meet(X, top)
% 34.39/4.89 = { by lemma 43 }
% 34.39/4.89 X
% 34.39/4.89
% 34.39/4.89 Lemma 45: complement(join(zero, meet(X, Y))) = join(complement(X), complement(Y)).
% 34.39/4.89 Proof:
% 34.39/4.89 complement(join(zero, meet(X, Y)))
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 complement(join(zero, meet(Y, X)))
% 34.39/4.89 = { by lemma 15 R->L }
% 34.39/4.89 complement(join(complement(top), meet(Y, X)))
% 34.39/4.89 = { by lemma 42 }
% 34.39/4.89 meet(top, join(complement(Y), complement(X)))
% 34.39/4.89 = { by lemma 44 }
% 34.39/4.89 join(complement(Y), complement(X))
% 34.39/4.89 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.89 join(complement(X), complement(Y))
% 34.39/4.89
% 34.39/4.89 Lemma 46: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 34.39/4.89 Proof:
% 34.39/4.89 join(complement(X), complement(Y))
% 34.39/4.89 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.89 join(complement(Y), complement(X))
% 34.39/4.89 = { by lemma 45 R->L }
% 34.39/4.89 complement(join(zero, meet(Y, X)))
% 34.39/4.89 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.89 complement(join(meet(Y, X), zero))
% 34.39/4.89 = { by lemma 38 }
% 34.39/4.89 complement(meet(Y, X))
% 34.39/4.89 = { by lemma 25 R->L }
% 34.39/4.89 complement(meet(X, Y))
% 34.39/4.89
% 34.39/4.89 Lemma 47: meet(X, X) = X.
% 34.39/4.89 Proof:
% 34.39/4.89 meet(X, X)
% 34.39/4.89 = { by lemma 24 R->L }
% 34.39/4.89 complement(complement(X))
% 34.39/4.89 = { by lemma 36 }
% 34.39/4.89 X
% 34.39/4.89
% 34.39/4.89 Lemma 48: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 34.39/4.89 Proof:
% 34.39/4.89 complement(meet(X, complement(Y)))
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 complement(meet(complement(Y), X))
% 34.39/4.89 = { by lemma 35 R->L }
% 34.39/4.89 complement(meet(join(zero, complement(Y)), X))
% 34.39/4.89 = { by lemma 46 R->L }
% 34.39/4.89 join(complement(join(zero, complement(Y))), complement(X))
% 34.39/4.89 = { by lemma 26 }
% 34.39/4.89 join(meet(Y, top), complement(X))
% 34.39/4.89 = { by lemma 43 }
% 34.39/4.89 join(Y, complement(X))
% 34.39/4.89
% 34.39/4.89 Lemma 49: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 34.39/4.89 Proof:
% 34.39/4.89 meet(meet(X, Y), Z)
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 meet(Z, meet(X, Y))
% 34.39/4.89 = { by lemma 47 R->L }
% 34.39/4.89 meet(meet(Z, meet(X, Y)), meet(Z, meet(X, Y)))
% 34.39/4.89 = { by lemma 24 R->L }
% 34.39/4.89 complement(complement(meet(Z, meet(X, Y))))
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 complement(complement(meet(Z, meet(Y, X))))
% 34.39/4.89 = { by lemma 46 R->L }
% 34.39/4.89 complement(join(complement(Z), complement(meet(Y, X))))
% 34.39/4.89 = { by lemma 46 R->L }
% 34.39/4.89 complement(join(complement(Z), join(complement(Y), complement(X))))
% 34.39/4.89 = { by axiom 9 (maddux2_join_associativity) }
% 34.39/4.89 complement(join(join(complement(Z), complement(Y)), complement(X)))
% 34.39/4.89 = { by lemma 48 R->L }
% 34.39/4.89 complement(complement(meet(X, complement(join(complement(Z), complement(Y))))))
% 34.39/4.89 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 34.39/4.89 complement(complement(meet(X, meet(Z, Y))))
% 34.39/4.89 = { by lemma 25 R->L }
% 34.39/4.89 complement(complement(meet(X, meet(Y, Z))))
% 34.39/4.89 = { by lemma 36 }
% 34.39/4.89 meet(X, meet(Y, Z))
% 34.39/4.89 = { by lemma 25 R->L }
% 34.39/4.89 meet(X, meet(Z, Y))
% 34.39/4.89
% 34.39/4.89 Lemma 50: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 34.39/4.89 Proof:
% 34.39/4.89 join(meet(X, Y), meet(Y, complement(X)))
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 join(meet(Y, X), meet(Y, complement(X)))
% 34.39/4.89 = { by lemma 33 }
% 34.39/4.89 Y
% 34.39/4.89
% 34.39/4.89 Lemma 51: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 34.39/4.89 Proof:
% 34.39/4.89 join(meet(X, Y), meet(complement(X), Y))
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 join(meet(X, Y), meet(Y, complement(X)))
% 34.39/4.89 = { by lemma 50 }
% 34.39/4.89 Y
% 34.39/4.89
% 34.39/4.89 Lemma 52: complement(converse(X)) = converse(complement(X)).
% 34.39/4.89 Proof:
% 34.39/4.89 complement(converse(X))
% 34.39/4.89 = { by lemma 41 R->L }
% 34.39/4.89 complement(converse(meet(X, converse(complement(converse(complement(X)))))))
% 34.39/4.89 = { by lemma 38 R->L }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), zero)))
% 34.39/4.89 = { by axiom 4 (def_zero) }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), complement(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))))))))
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), complement(meet(converse(complement(converse(complement(X)))), converse(converse(complement(X)))))))))
% 34.39/4.89 = { by lemma 46 R->L }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(converse(complement(X)))))))))
% 34.39/4.89 = { by lemma 41 R->L }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(meet(converse(complement(X)), converse(complement(converse(complement(converse(complement(X))))))))))))))
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 34.39/4.89 = { by axiom 1 (converse_idempotence) R->L }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(converse(complement(converse(complement(converse(complement(X))))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 34.39/4.89 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(converse(complement(converse(complement(converse(complement(X))))))))))))
% 34.39/4.89 = { by lemma 19 R->L }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(join(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))), complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))))))))
% 34.39/4.89 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(join(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))), complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 34.39/4.89 = { by axiom 8 (converse_additivity) }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(join(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))), converse(complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 34.39/4.89 = { by axiom 9 (maddux2_join_associativity) R->L }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))), join(converse(complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))))))))))
% 34.39/4.89 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))))))))
% 34.39/4.89 = { by lemma 31 }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), top))))
% 34.39/4.89 = { by lemma 49 }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(converse(complement(X))), meet(top, converse(complement(converse(complement(X)))))))))
% 34.39/4.89 = { by lemma 44 }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))))))
% 34.39/4.89 = { by axiom 1 (converse_idempotence) }
% 34.39/4.89 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(converse(complement(X))))))))
% 34.39/4.89 = { by lemma 51 }
% 34.39/4.89 complement(converse(converse(complement(converse(complement(X))))))
% 34.39/4.89 = { by axiom 1 (converse_idempotence) }
% 34.39/4.89 complement(complement(converse(complement(X))))
% 34.39/4.89 = { by lemma 24 }
% 34.39/4.89 meet(converse(complement(X)), converse(complement(X)))
% 34.39/4.89 = { by lemma 47 }
% 34.39/4.89 converse(complement(X))
% 34.39/4.89
% 34.39/4.89 Lemma 53: join(one, converse(X)) = converse(join(X, one)).
% 34.39/4.89 Proof:
% 34.39/4.89 join(one, converse(X))
% 34.39/4.89 = { by lemma 18 R->L }
% 34.39/4.89 join(converse(one), converse(X))
% 34.39/4.89 = { by axiom 8 (converse_additivity) R->L }
% 34.39/4.89 converse(join(one, X))
% 34.39/4.89 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.89 converse(join(X, one))
% 34.39/4.89
% 34.39/4.89 Lemma 54: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 34.39/4.89 Proof:
% 34.39/4.89 converse(composition(X, converse(Y)))
% 34.39/4.89 = { by axiom 6 (converse_multiplicativity) }
% 34.39/4.89 composition(converse(converse(Y)), converse(X))
% 34.39/4.89 = { by axiom 1 (converse_idempotence) }
% 34.39/4.89 composition(Y, converse(X))
% 34.39/4.89
% 34.39/4.89 Lemma 55: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 34.39/4.89 Proof:
% 34.39/4.89 join(X, composition(Y, X))
% 34.39/4.89 = { by lemma 21 R->L }
% 34.39/4.89 join(composition(one, X), composition(Y, X))
% 34.39/4.89 = { by axiom 11 (composition_distributivity) R->L }
% 34.39/4.89 composition(join(one, Y), X)
% 34.39/4.89 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.89 composition(join(Y, one), X)
% 34.39/4.89
% 34.39/4.89 Lemma 56: join(X, composition(X, Y)) = composition(X, join(Y, one)).
% 34.39/4.89 Proof:
% 34.39/4.89 join(X, composition(X, Y))
% 34.39/4.89 = { by axiom 1 (converse_idempotence) R->L }
% 34.39/4.89 join(X, composition(X, converse(converse(Y))))
% 34.39/4.89 = { by lemma 54 R->L }
% 34.39/4.89 join(X, converse(composition(converse(Y), converse(X))))
% 34.39/4.89 = { by lemma 40 R->L }
% 34.39/4.89 converse(join(converse(X), composition(converse(Y), converse(X))))
% 34.39/4.89 = { by lemma 55 }
% 34.39/4.89 converse(composition(join(converse(Y), one), converse(X)))
% 34.39/4.89 = { by lemma 54 }
% 34.39/4.89 composition(X, converse(join(converse(Y), one)))
% 34.39/4.89 = { by lemma 40 }
% 34.39/4.89 composition(X, join(Y, converse(one)))
% 34.39/4.89 = { by lemma 18 }
% 34.39/4.89 composition(X, join(Y, one))
% 34.39/4.89
% 34.39/4.89 Lemma 57: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 34.39/4.89 Proof:
% 34.39/4.89 complement(meet(complement(X), Y))
% 34.39/4.89 = { by lemma 25 }
% 34.39/4.89 complement(meet(Y, complement(X)))
% 34.39/4.89 = { by lemma 48 }
% 34.39/4.89 join(X, complement(Y))
% 34.39/4.89
% 34.39/4.89 Lemma 58: meet(X, join(X, complement(Y))) = X.
% 34.39/4.89 Proof:
% 34.39/4.89 meet(X, join(X, complement(Y)))
% 34.39/4.89 = { by lemma 48 R->L }
% 34.39/4.89 meet(X, complement(meet(Y, complement(X))))
% 34.39/4.89 = { by lemma 46 R->L }
% 34.39/4.89 meet(X, join(complement(Y), complement(complement(X))))
% 34.39/4.89 = { by lemma 42 R->L }
% 34.39/4.89 complement(join(complement(X), meet(Y, complement(X))))
% 34.39/4.89 = { by lemma 35 R->L }
% 34.39/4.89 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 34.39/4.89 = { by lemma 15 R->L }
% 34.39/4.89 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 34.39/4.89 = { by lemma 30 R->L }
% 34.39/4.89 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 34.39/4.90 = { by lemma 27 R->L }
% 34.39/4.90 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 34.39/4.90 = { by lemma 46 }
% 34.39/4.90 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 34.39/4.90 = { by lemma 25 R->L }
% 34.39/4.90 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 34.39/4.90 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 34.39/4.90 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 34.39/4.90 = { by lemma 25 R->L }
% 34.39/4.90 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 34.39/4.90 = { by lemma 19 }
% 34.39/4.90 X
% 34.39/4.90
% 34.39/4.90 Lemma 59: meet(X, join(X, Y)) = X.
% 34.39/4.90 Proof:
% 34.39/4.90 meet(X, join(X, Y))
% 34.39/4.90 = { by lemma 47 R->L }
% 34.39/4.90 meet(X, join(X, meet(Y, Y)))
% 34.39/4.90 = { by lemma 24 R->L }
% 34.39/4.90 meet(X, join(X, complement(complement(Y))))
% 34.39/4.90 = { by lemma 58 }
% 34.39/4.90 X
% 34.39/4.90
% 34.39/4.90 Lemma 60: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 34.39/4.90 Proof:
% 34.39/4.90 join(complement(one), composition(converse(X), complement(X)))
% 34.39/4.90 = { by axiom 2 (composition_identity) R->L }
% 34.39/4.90 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 34.39/4.90 = { by lemma 22 }
% 34.39/4.90 complement(one)
% 34.39/4.90
% 34.39/4.90 Lemma 61: join(complement(one), converse(complement(one))) = complement(one).
% 34.39/4.90 Proof:
% 34.39/4.90 join(complement(one), converse(complement(one)))
% 34.39/4.90 = { by axiom 2 (composition_identity) R->L }
% 34.39/4.90 join(complement(one), composition(converse(complement(one)), one))
% 34.39/4.90 = { by lemma 43 R->L }
% 34.39/4.90 join(complement(one), composition(converse(complement(one)), meet(one, top)))
% 34.39/4.90 = { by lemma 35 R->L }
% 34.39/4.90 join(complement(one), composition(converse(join(zero, complement(one))), meet(one, top)))
% 34.39/4.90 = { by lemma 26 R->L }
% 34.39/4.90 join(complement(one), composition(converse(join(zero, complement(one))), complement(join(zero, complement(one)))))
% 34.39/4.90 = { by lemma 60 }
% 34.39/4.90 complement(one)
% 34.39/4.90
% 34.39/4.90 Lemma 62: join(converse(X), composition(converse(Y), Z)) = converse(join(X, composition(converse(Z), Y))).
% 34.39/4.90 Proof:
% 34.39/4.90 join(converse(X), composition(converse(Y), Z))
% 34.39/4.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.90 join(composition(converse(Y), Z), converse(X))
% 34.39/4.90 = { by lemma 16 R->L }
% 34.39/4.90 join(converse(composition(converse(Z), Y)), converse(X))
% 34.39/4.90 = { by axiom 8 (converse_additivity) R->L }
% 34.39/4.90 converse(join(composition(converse(Z), Y), X))
% 34.39/4.90 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.90 converse(join(X, composition(converse(Z), Y)))
% 34.39/4.90
% 34.39/4.90 Goal 1 (goals_1): join(composition(converse(x0), x0), one) = one.
% 34.39/4.90 Proof:
% 34.39/4.90 join(composition(converse(x0), x0), one)
% 34.39/4.90 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.90 join(one, composition(converse(x0), x0))
% 34.39/4.90 = { by lemma 18 R->L }
% 34.39/4.90 join(converse(one), composition(converse(x0), x0))
% 34.39/4.90 = { by lemma 62 }
% 34.39/4.90 converse(join(one, composition(converse(x0), x0)))
% 34.39/4.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.90 converse(join(composition(converse(x0), x0), one))
% 34.39/4.90 = { by lemma 53 R->L }
% 34.39/4.90 join(one, converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 21 R->L }
% 34.39/4.90 join(composition(one, one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 37 R->L }
% 34.39/4.90 join(composition(join(one, one), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 50 R->L }
% 34.39/4.90 join(composition(join(meet(one, join(one, one)), meet(join(one, one), complement(one))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 59 }
% 34.39/4.90 join(composition(join(one, meet(join(one, one), complement(one))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 25 }
% 34.39/4.90 join(composition(join(one, meet(complement(one), join(one, one))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 59 R->L }
% 34.39/4.90 join(composition(join(meet(one, join(one, join(one, one))), meet(complement(one), join(one, one))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), one)), meet(complement(one), join(one, one))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 36 R->L }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), meet(complement(one), join(one, one))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 58 R->L }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), meet(complement(one), meet(join(one, one), join(join(one, one), complement(complement(one)))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 57 R->L }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), meet(complement(one), meet(join(one, one), complement(meet(complement(join(one, one)), complement(one)))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 49 R->L }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), join(one, one))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 38 R->L }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), join(meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), join(one, one)), zero)), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 15 R->L }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), join(meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), join(one, one)), complement(top))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by axiom 5 (def_top) }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), join(meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), join(one, one)), complement(join(join(complement(complement(join(one, one))), complement(complement(one))), complement(join(complement(complement(join(one, one))), complement(complement(one)))))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by axiom 9 (maddux2_join_associativity) R->L }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), join(meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), join(one, one)), complement(join(complement(complement(join(one, one))), join(complement(complement(one)), complement(join(complement(complement(join(one, one))), complement(complement(one))))))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 46 }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), join(meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), join(one, one)), complement(join(complement(complement(join(one, one))), complement(meet(complement(one), join(complement(complement(join(one, one))), complement(complement(one))))))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), join(meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), join(one, one)), meet(complement(join(one, one)), meet(complement(one), join(complement(complement(join(one, one))), complement(complement(one))))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 46 }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), join(meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), join(one, one)), meet(complement(join(one, one)), meet(complement(one), complement(meet(complement(join(one, one)), complement(one))))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 25 }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), join(meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), join(one, one)), meet(meet(complement(one), complement(meet(complement(join(one, one)), complement(one)))), complement(join(one, one))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 33 }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), meet(complement(one), complement(meet(complement(join(one, one)), complement(one))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 57 }
% 34.39/4.90 join(composition(join(meet(one, join(join(one, one), complement(complement(one)))), meet(complement(one), join(join(one, one), complement(complement(one))))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 51 }
% 34.39/4.90 join(composition(join(join(one, one), complement(complement(one))), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 36 }
% 34.39/4.90 join(composition(join(join(one, one), one), one), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by lemma 55 R->L }
% 34.39/4.90 join(join(one, composition(join(one, one), one)), converse(composition(converse(x0), x0)))
% 34.39/4.90 = { by axiom 9 (maddux2_join_associativity) R->L }
% 34.39/4.90 join(one, join(composition(join(one, one), one), converse(composition(converse(x0), x0))))
% 34.39/4.90 = { by lemma 37 }
% 34.39/4.90 join(one, join(composition(one, one), converse(composition(converse(x0), x0))))
% 34.39/4.90 = { by lemma 21 }
% 34.39/4.90 join(one, join(one, converse(composition(converse(x0), x0))))
% 34.39/4.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.90 join(one, join(converse(composition(converse(x0), x0)), one))
% 34.39/4.90 = { by axiom 9 (maddux2_join_associativity) }
% 34.39/4.90 join(join(one, converse(composition(converse(x0), x0))), one)
% 34.39/4.90 = { by lemma 53 }
% 34.39/4.90 join(converse(join(composition(converse(x0), x0), one)), one)
% 34.39/4.90 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.90 join(converse(join(one, composition(converse(x0), x0))), one)
% 34.39/4.90 = { by lemma 21 R->L }
% 34.39/4.90 composition(one, join(converse(join(one, composition(converse(x0), x0))), one))
% 34.39/4.90 = { by lemma 56 R->L }
% 34.39/4.90 join(one, composition(one, converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.90 = { by lemma 36 R->L }
% 34.39/4.90 join(one, composition(complement(complement(one)), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.90 = { by lemma 60 R->L }
% 34.39/4.90 join(one, composition(complement(join(complement(one), composition(converse(complement(x0)), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.90 = { by lemma 38 R->L }
% 34.39/4.90 join(one, composition(complement(join(complement(one), composition(converse(complement(join(x0, zero))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.90 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.90 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, x0))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.90 = { by lemma 19 R->L }
% 34.39/4.90 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, join(meet(x0, join(complement(composition(x0, complement(one))), meet(x0, composition(x0, complement(one))))), complement(join(complement(x0), join(complement(composition(x0, complement(one))), meet(x0, composition(x0, complement(one)))))))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.90 = { by axiom 9 (maddux2_join_associativity) }
% 34.39/4.90 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, join(meet(x0, join(complement(composition(x0, complement(one))), meet(x0, composition(x0, complement(one))))), complement(join(join(complement(x0), complement(composition(x0, complement(one)))), meet(x0, composition(x0, complement(one))))))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.90 = { by axiom 10 (maddux4_definiton_of_meet) }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, join(meet(x0, join(complement(composition(x0, complement(one))), meet(x0, composition(x0, complement(one))))), complement(join(join(complement(x0), complement(composition(x0, complement(one)))), complement(join(complement(x0), complement(composition(x0, complement(one))))))))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 5 (def_top) R->L }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, join(meet(x0, join(complement(composition(x0, complement(one))), meet(x0, composition(x0, complement(one))))), complement(top))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 15 }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, join(meet(x0, join(complement(composition(x0, complement(one))), meet(x0, composition(x0, complement(one))))), zero)))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 38 }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, meet(x0, join(complement(composition(x0, complement(one))), meet(x0, composition(x0, complement(one)))))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 2 (composition_identity) R->L }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, meet(x0, join(complement(composition(x0, complement(one))), meet(composition(x0, one), composition(x0, complement(one)))))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 12 (goals) }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, meet(x0, join(complement(composition(x0, complement(one))), zero))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 38 }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(complement(join(zero, meet(x0, complement(composition(x0, complement(one))))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 45 }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(join(complement(x0), complement(complement(composition(x0, complement(one)))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 46 }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(complement(meet(x0, complement(composition(x0, complement(one)))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 48 }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(join(composition(x0, complement(one)), complement(x0))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 8 (converse_additivity) }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(join(converse(composition(x0, complement(one))), converse(complement(x0))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.91 join(one, composition(complement(join(composition(join(converse(composition(x0, complement(one))), converse(complement(x0))), complement(complement(x0))), complement(one))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 11 (composition_distributivity) }
% 34.39/4.91 join(one, composition(complement(join(join(composition(converse(composition(x0, complement(one))), complement(complement(x0))), composition(converse(complement(x0)), complement(complement(x0)))), complement(one))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 9 (maddux2_join_associativity) R->L }
% 34.39/4.91 join(one, composition(complement(join(composition(converse(composition(x0, complement(one))), complement(complement(x0))), join(composition(converse(complement(x0)), complement(complement(x0))), complement(one)))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.91 join(one, composition(complement(join(composition(converse(composition(x0, complement(one))), complement(complement(x0))), join(complement(one), composition(converse(complement(x0)), complement(complement(x0)))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 60 }
% 34.39/4.91 join(one, composition(complement(join(composition(converse(composition(x0, complement(one))), complement(complement(x0))), complement(one))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(composition(x0, complement(one))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 61 R->L }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(composition(x0, join(complement(one), converse(complement(one))))), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 6 (converse_multiplicativity) }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(composition(converse(join(complement(one), converse(complement(one)))), converse(x0)), complement(complement(x0))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 7 (composition_associativity) R->L }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(converse(join(complement(one), converse(complement(one)))), composition(converse(x0), complement(complement(x0)))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 39 }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(join(complement(one), converse(complement(one))), composition(converse(x0), complement(complement(x0)))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 61 }
% 34.39/4.91 join(one, composition(complement(join(complement(one), composition(complement(one), composition(converse(x0), complement(complement(x0)))))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 56 }
% 34.39/4.91 join(one, composition(complement(composition(complement(one), join(composition(converse(x0), complement(complement(x0))), one))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 36 }
% 34.39/4.91 join(one, composition(complement(composition(complement(one), join(composition(converse(x0), x0), one))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 3 (maddux1_join_commutativity) }
% 34.39/4.91 join(one, composition(complement(composition(complement(one), join(one, composition(converse(x0), x0)))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 35 R->L }
% 34.39/4.91 join(one, composition(complement(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0)))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 43 R->L }
% 34.39/4.91 join(meet(one, top), composition(complement(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0)))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by lemma 26 R->L }
% 34.39/4.91 join(complement(join(zero, complement(one))), composition(complement(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0)))), converse(join(one, composition(converse(x0), x0)))))
% 34.39/4.91 = { by axiom 1 (converse_idempotence) R->L }
% 34.39/4.91 converse(converse(join(complement(join(zero, complement(one))), composition(complement(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0)))), converse(join(one, composition(converse(x0), x0)))))))
% 34.39/4.91 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 34.39/4.91 converse(converse(join(composition(complement(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0)))), converse(join(one, composition(converse(x0), x0)))), complement(join(zero, complement(one))))))
% 34.39/4.91 = { by axiom 8 (converse_additivity) }
% 34.39/4.91 converse(join(converse(composition(complement(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0)))), converse(join(one, composition(converse(x0), x0))))), converse(complement(join(zero, complement(one))))))
% 34.39/4.91 = { by lemma 54 }
% 34.39/4.91 converse(join(composition(join(one, composition(converse(x0), x0)), converse(complement(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0)))))), converse(complement(join(zero, complement(one))))))
% 34.39/4.91 = { by lemma 52 R->L }
% 34.39/4.91 converse(join(composition(join(one, composition(converse(x0), x0)), complement(converse(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0)))))), converse(complement(join(zero, complement(one))))))
% 34.39/4.91 = { by lemma 39 R->L }
% 34.39/4.91 converse(converse(join(complement(join(zero, complement(one))), converse(composition(join(one, composition(converse(x0), x0)), complement(converse(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0))))))))))
% 34.39/4.91 = { by axiom 6 (converse_multiplicativity) }
% 34.39/4.91 converse(converse(join(complement(join(zero, complement(one))), composition(converse(complement(converse(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0)))))), converse(join(one, composition(converse(x0), x0)))))))
% 34.39/4.91 = { by lemma 62 R->L }
% 34.39/4.91 converse(join(converse(complement(join(zero, complement(one)))), composition(converse(converse(join(one, composition(converse(x0), x0)))), complement(converse(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0))))))))
% 34.39/4.91 = { by lemma 52 R->L }
% 34.39/4.91 converse(join(complement(converse(join(zero, complement(one)))), composition(converse(converse(join(one, composition(converse(x0), x0)))), complement(converse(composition(join(zero, complement(one)), join(one, composition(converse(x0), x0))))))))
% 34.39/4.91 = { by axiom 6 (converse_multiplicativity) }
% 34.39/4.91 converse(join(complement(converse(join(zero, complement(one)))), composition(converse(converse(join(one, composition(converse(x0), x0)))), complement(composition(converse(join(one, composition(converse(x0), x0))), converse(join(zero, complement(one))))))))
% 34.39/4.91 = { by lemma 22 }
% 34.39/4.91 converse(complement(converse(join(zero, complement(one)))))
% 34.39/4.91 = { by lemma 52 }
% 34.39/4.91 converse(converse(complement(join(zero, complement(one)))))
% 34.39/4.91 = { by axiom 1 (converse_idempotence) }
% 34.39/4.91 complement(join(zero, complement(one)))
% 34.39/4.91 = { by lemma 26 }
% 34.39/4.91 meet(one, top)
% 34.39/4.91 = { by lemma 43 }
% 34.39/4.91 one
% 34.39/4.91 % SZS output end Proof
% 34.39/4.91
% 34.39/4.91 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------