TSTP Solution File: REL040+3 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL040+3 : 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 : n001.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 13:44:26 EDT 2023
% Result : Theorem 158.14s 20.81s
% Output : Proof 161.00s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL040+3 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n001.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Fri Aug 25 22:10:14 EDT 2023
% 0.15/0.35 % CPUTime :
% 158.14/20.81 Command-line arguments: --flatten
% 158.14/20.81
% 158.14/20.81 % SZS status Theorem
% 158.14/20.81
% 160.68/20.95 % SZS output start Proof
% 160.68/20.95 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 160.68/20.95 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 160.68/20.95 Axiom 3 (composition_identity): composition(X, one) = X.
% 160.68/20.95 Axiom 4 (def_top): top = join(X, complement(X)).
% 160.68/20.95 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 160.68/20.95 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 160.68/20.95 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 160.68/20.95 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 160.68/20.95 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 160.68/20.95 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 160.68/20.95 Axiom 11 (goals): join(composition(converse(x0), x0), one) = one.
% 160.68/20.95 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 160.68/20.95 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 160.68/20.95 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 160.68/20.95 Axiom 15 (modular_law_2): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 160.68/20.95
% 160.68/20.95 Lemma 16: complement(top) = zero.
% 160.68/20.95 Proof:
% 160.68/20.95 complement(top)
% 160.68/20.95 = { by axiom 4 (def_top) }
% 160.68/20.95 complement(join(complement(X), complement(complement(X))))
% 160.68/20.95 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 160.68/20.95 meet(X, complement(X))
% 160.68/20.95 = { by axiom 5 (def_zero) R->L }
% 160.68/20.95 zero
% 160.68/20.95
% 160.68/20.95 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 160.68/20.95 Proof:
% 160.68/20.95 join(X, join(Y, complement(X)))
% 160.68/20.95 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.95 join(X, join(complement(X), Y))
% 160.68/20.95 = { by axiom 7 (maddux2_join_associativity) }
% 160.68/20.95 join(join(X, complement(X)), Y)
% 160.68/20.95 = { by axiom 4 (def_top) R->L }
% 160.68/20.95 join(top, Y)
% 160.68/20.95 = { by axiom 1 (maddux1_join_commutativity) }
% 160.68/20.95 join(Y, top)
% 160.68/20.95
% 160.68/20.95 Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 160.68/20.95 Proof:
% 160.68/20.95 converse(composition(converse(X), Y))
% 160.68/20.95 = { by axiom 8 (converse_multiplicativity) }
% 160.68/20.95 composition(converse(Y), converse(converse(X)))
% 160.68/20.95 = { by axiom 2 (converse_idempotence) }
% 160.68/20.95 composition(converse(Y), X)
% 160.68/20.95
% 160.68/20.95 Lemma 19: composition(converse(join(composition(converse(x0), x0), one)), X) = X.
% 160.68/20.95 Proof:
% 160.68/20.95 composition(converse(join(composition(converse(x0), x0), one)), X)
% 160.68/20.95 = { by axiom 11 (goals) }
% 160.68/20.95 composition(converse(one), X)
% 160.68/20.95 = { by lemma 18 R->L }
% 160.68/20.95 converse(composition(converse(X), one))
% 160.68/20.95 = { by axiom 3 (composition_identity) }
% 160.68/20.95 converse(converse(X))
% 160.68/20.95 = { by axiom 2 (converse_idempotence) }
% 160.68/20.95 X
% 160.68/20.95
% 160.68/20.95 Lemma 20: composition(join(composition(converse(x0), x0), one), X) = X.
% 160.68/20.95 Proof:
% 160.68/20.95 composition(join(composition(converse(x0), x0), one), X)
% 160.68/20.95 = { by lemma 19 R->L }
% 160.68/20.95 composition(converse(join(composition(converse(x0), x0), one)), composition(join(composition(converse(x0), x0), one), X))
% 160.68/20.95 = { by axiom 11 (goals) }
% 160.68/20.95 composition(converse(join(composition(converse(x0), x0), one)), composition(one, X))
% 160.68/20.95 = { by axiom 9 (composition_associativity) }
% 160.68/20.95 composition(composition(converse(join(composition(converse(x0), x0), one)), one), X)
% 160.68/20.95 = { by axiom 3 (composition_identity) }
% 160.68/20.95 composition(converse(join(composition(converse(x0), x0), one)), X)
% 160.68/20.95 = { by lemma 19 }
% 160.68/20.95 X
% 160.68/20.95
% 160.68/20.95 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 160.68/20.95 Proof:
% 160.68/20.95 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 160.68/20.95 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.95 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 160.68/20.95 = { by axiom 13 (converse_cancellativity) }
% 160.68/20.95 complement(X)
% 160.68/20.95
% 160.68/20.95 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 160.68/20.95 Proof:
% 160.68/20.95 join(complement(X), complement(X))
% 160.68/20.95 = { by lemma 19 R->L }
% 160.68/20.95 join(complement(X), composition(converse(join(composition(converse(x0), x0), one)), complement(X)))
% 160.68/20.95 = { by lemma 20 R->L }
% 160.68/20.95 join(complement(X), composition(converse(join(composition(converse(x0), x0), one)), complement(composition(join(composition(converse(x0), x0), one), X))))
% 160.68/20.95 = { by lemma 21 }
% 160.68/20.95 complement(X)
% 160.68/20.95
% 160.68/20.95 Lemma 23: join(top, complement(X)) = top.
% 160.68/20.95 Proof:
% 160.68/20.95 join(top, complement(X))
% 160.68/20.95 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.95 join(complement(X), top)
% 160.68/20.95 = { by lemma 17 R->L }
% 160.68/20.95 join(X, join(complement(X), complement(X)))
% 160.68/20.95 = { by lemma 22 }
% 160.68/20.95 join(X, complement(X))
% 160.68/20.95 = { by axiom 4 (def_top) R->L }
% 160.68/20.95 top
% 160.68/20.95
% 160.68/20.95 Lemma 24: join(Y, top) = join(X, top).
% 160.68/20.95 Proof:
% 160.68/20.95 join(Y, top)
% 160.68/20.95 = { by lemma 23 R->L }
% 160.68/20.95 join(Y, join(top, complement(Y)))
% 160.68/20.95 = { by lemma 17 }
% 160.68/20.95 join(top, top)
% 160.68/20.95 = { by lemma 17 R->L }
% 160.68/20.95 join(X, join(top, complement(X)))
% 160.68/20.95 = { by lemma 23 }
% 160.68/20.95 join(X, top)
% 160.68/20.95
% 160.68/20.95 Lemma 25: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 160.68/20.95 Proof:
% 160.68/20.95 join(meet(X, Y), complement(join(complement(X), Y)))
% 160.68/20.95 = { by axiom 10 (maddux4_definiton_of_meet) }
% 160.68/20.95 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 160.68/20.95 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 160.68/20.95 X
% 160.68/20.95
% 160.68/20.95 Lemma 26: join(zero, meet(X, X)) = X.
% 160.68/20.95 Proof:
% 160.68/20.95 join(zero, meet(X, X))
% 160.68/20.95 = { by axiom 10 (maddux4_definiton_of_meet) }
% 160.68/20.95 join(zero, complement(join(complement(X), complement(X))))
% 160.68/20.95 = { by axiom 5 (def_zero) }
% 160.68/20.95 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 160.68/20.95 = { by lemma 25 }
% 160.68/20.95 X
% 160.68/20.95
% 160.68/20.95 Lemma 27: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 160.68/20.95 Proof:
% 160.68/20.95 join(zero, join(X, complement(complement(Y))))
% 160.68/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.96 join(zero, join(complement(complement(Y)), X))
% 160.68/20.96 = { by lemma 22 R->L }
% 160.68/20.96 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 160.68/20.96 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 160.68/20.96 join(zero, join(meet(Y, Y), X))
% 160.68/20.96 = { by axiom 7 (maddux2_join_associativity) }
% 160.68/20.96 join(join(zero, meet(Y, Y)), X)
% 160.68/20.96 = { by lemma 26 }
% 160.68/20.96 join(Y, X)
% 160.68/20.96 = { by axiom 1 (maddux1_join_commutativity) }
% 160.68/20.96 join(X, Y)
% 160.68/20.96
% 160.68/20.96 Lemma 28: join(zero, complement(complement(X))) = X.
% 160.68/20.96 Proof:
% 160.68/20.96 join(zero, complement(complement(X)))
% 160.68/20.96 = { by axiom 5 (def_zero) }
% 160.68/20.96 join(meet(X, complement(X)), complement(complement(X)))
% 160.68/20.96 = { by lemma 22 R->L }
% 160.68/20.96 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 160.68/20.96 = { by lemma 25 }
% 160.68/20.96 X
% 160.68/20.96
% 160.68/20.96 Lemma 29: join(X, zero) = join(X, X).
% 160.68/20.96 Proof:
% 160.68/20.96 join(X, zero)
% 160.68/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.96 join(zero, X)
% 160.68/20.96 = { by lemma 28 R->L }
% 160.68/20.96 join(zero, join(zero, complement(complement(X))))
% 160.68/20.96 = { by lemma 22 R->L }
% 160.68/20.96 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 160.68/20.96 = { by lemma 27 }
% 160.68/20.96 join(zero, join(complement(complement(X)), X))
% 160.68/20.96 = { by axiom 1 (maddux1_join_commutativity) }
% 160.68/20.96 join(zero, join(X, complement(complement(X))))
% 160.68/20.96 = { by lemma 27 }
% 160.68/20.96 join(X, X)
% 160.68/20.96
% 160.68/20.96 Lemma 30: join(zero, complement(X)) = complement(X).
% 160.68/20.96 Proof:
% 160.68/20.96 join(zero, complement(X))
% 160.68/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.96 join(complement(X), zero)
% 160.68/20.96 = { by lemma 29 }
% 160.68/20.96 join(complement(X), complement(X))
% 160.68/20.96 = { by lemma 22 }
% 160.68/20.96 complement(X)
% 160.68/20.96
% 160.68/20.96 Lemma 31: join(X, zero) = X.
% 160.68/20.96 Proof:
% 160.68/20.96 join(X, zero)
% 160.68/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.96 join(zero, X)
% 160.68/20.96 = { by lemma 27 R->L }
% 160.68/20.96 join(zero, join(zero, complement(complement(X))))
% 160.68/20.96 = { by lemma 30 }
% 160.68/20.96 join(zero, complement(complement(X)))
% 160.68/20.96 = { by lemma 28 }
% 160.68/20.96 X
% 160.68/20.96
% 160.68/20.96 Lemma 32: join(X, top) = top.
% 160.68/20.96 Proof:
% 160.68/20.96 join(X, top)
% 160.68/20.96 = { by lemma 24 }
% 160.68/20.96 join(join(zero, zero), top)
% 160.68/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.96 join(top, join(zero, zero))
% 160.68/20.96 = { by lemma 31 }
% 160.68/20.96 join(top, zero)
% 160.68/20.96 = { by lemma 31 }
% 160.68/20.96 top
% 160.68/20.96
% 160.68/20.96 Lemma 33: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 160.68/20.96 Proof:
% 160.68/20.96 converse(join(X, converse(Y)))
% 160.68/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.96 converse(join(converse(Y), X))
% 160.68/20.96 = { by axiom 6 (converse_additivity) }
% 160.68/20.96 join(converse(converse(Y)), converse(X))
% 160.68/20.96 = { by axiom 2 (converse_idempotence) }
% 160.68/20.96 join(Y, converse(X))
% 160.68/20.96
% 160.68/20.96 Lemma 34: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 160.68/20.96 Proof:
% 160.68/20.96 converse(join(converse(X), Y))
% 160.68/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 160.68/20.96 converse(join(Y, converse(X)))
% 160.68/20.96 = { by lemma 33 }
% 160.68/20.96 join(X, converse(Y))
% 160.68/20.96
% 160.68/20.96 Lemma 35: join(X, converse(complement(converse(X)))) = converse(top).
% 160.68/20.96 Proof:
% 160.68/20.96 join(X, converse(complement(converse(X))))
% 160.68/20.96 = { by lemma 34 R->L }
% 160.68/20.96 converse(join(converse(X), complement(converse(X))))
% 161.00/20.96 = { by axiom 4 (def_top) R->L }
% 161.00/20.96 converse(top)
% 161.00/20.96
% 161.00/20.96 Lemma 36: join(X, join(complement(X), Y)) = top.
% 161.00/20.96 Proof:
% 161.00/20.96 join(X, join(complement(X), Y))
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 join(X, join(Y, complement(X)))
% 161.00/20.96 = { by lemma 17 }
% 161.00/20.96 join(Y, top)
% 161.00/20.96 = { by lemma 24 R->L }
% 161.00/20.96 join(Z, top)
% 161.00/20.96 = { by lemma 32 }
% 161.00/20.96 top
% 161.00/20.96
% 161.00/20.96 Lemma 37: join(X, converse(top)) = top.
% 161.00/20.96 Proof:
% 161.00/20.96 join(X, converse(top))
% 161.00/20.96 = { by lemma 35 R->L }
% 161.00/20.96 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 161.00/20.96 = { by lemma 36 }
% 161.00/20.96 top
% 161.00/20.96
% 161.00/20.96 Lemma 38: converse(top) = top.
% 161.00/20.96 Proof:
% 161.00/20.96 converse(top)
% 161.00/20.96 = { by lemma 32 R->L }
% 161.00/20.96 converse(join(X, top))
% 161.00/20.96 = { by axiom 6 (converse_additivity) }
% 161.00/20.96 join(converse(X), converse(top))
% 161.00/20.96 = { by lemma 37 }
% 161.00/20.96 top
% 161.00/20.96
% 161.00/20.96 Lemma 39: join(zero, X) = X.
% 161.00/20.96 Proof:
% 161.00/20.96 join(zero, X)
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 join(X, zero)
% 161.00/20.96 = { by lemma 31 }
% 161.00/20.96 X
% 161.00/20.96
% 161.00/20.96 Lemma 40: meet(Y, X) = meet(X, Y).
% 161.00/20.96 Proof:
% 161.00/20.96 meet(Y, X)
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) }
% 161.00/20.96 complement(join(complement(Y), complement(X)))
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 complement(join(complement(X), complement(Y)))
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 161.00/20.96 meet(X, Y)
% 161.00/20.96
% 161.00/20.96 Lemma 41: complement(join(zero, complement(X))) = meet(X, top).
% 161.00/20.96 Proof:
% 161.00/20.96 complement(join(zero, complement(X)))
% 161.00/20.96 = { by lemma 16 R->L }
% 161.00/20.96 complement(join(complement(top), complement(X)))
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 161.00/20.96 meet(top, X)
% 161.00/20.96 = { by lemma 40 R->L }
% 161.00/20.96 meet(X, top)
% 161.00/20.96
% 161.00/20.96 Lemma 42: join(X, complement(zero)) = top.
% 161.00/20.96 Proof:
% 161.00/20.96 join(X, complement(zero))
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 join(complement(zero), X)
% 161.00/20.96 = { by lemma 27 R->L }
% 161.00/20.96 join(zero, join(complement(zero), complement(complement(X))))
% 161.00/20.96 = { by lemma 36 }
% 161.00/20.96 top
% 161.00/20.96
% 161.00/20.96 Lemma 43: meet(X, zero) = zero.
% 161.00/20.96 Proof:
% 161.00/20.96 meet(X, zero)
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) }
% 161.00/20.96 complement(join(complement(X), complement(zero)))
% 161.00/20.96 = { by lemma 42 }
% 161.00/20.96 complement(top)
% 161.00/20.96 = { by lemma 16 }
% 161.00/20.96 zero
% 161.00/20.96
% 161.00/20.96 Lemma 44: join(meet(X, Y), meet(X, complement(Y))) = X.
% 161.00/20.96 Proof:
% 161.00/20.96 join(meet(X, Y), meet(X, complement(Y)))
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 join(meet(X, complement(Y)), meet(X, Y))
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) }
% 161.00/20.96 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 161.00/20.96 = { by lemma 25 }
% 161.00/20.96 X
% 161.00/20.96
% 161.00/20.96 Lemma 45: meet(X, top) = X.
% 161.00/20.96 Proof:
% 161.00/20.96 meet(X, top)
% 161.00/20.96 = { by lemma 41 R->L }
% 161.00/20.96 complement(join(zero, complement(X)))
% 161.00/20.96 = { by lemma 30 R->L }
% 161.00/20.96 join(zero, complement(join(zero, complement(X))))
% 161.00/20.96 = { by lemma 41 }
% 161.00/20.96 join(zero, meet(X, top))
% 161.00/20.96 = { by lemma 42 R->L }
% 161.00/20.96 join(zero, meet(X, join(complement(zero), complement(zero))))
% 161.00/20.96 = { by lemma 22 }
% 161.00/20.96 join(zero, meet(X, complement(zero)))
% 161.00/20.96 = { by lemma 43 R->L }
% 161.00/20.96 join(meet(X, zero), meet(X, complement(zero)))
% 161.00/20.96 = { by lemma 44 }
% 161.00/20.96 X
% 161.00/20.96
% 161.00/20.96 Lemma 46: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 161.00/20.96 Proof:
% 161.00/20.96 join(meet(X, Y), meet(X, Y))
% 161.00/20.96 = { by lemma 40 }
% 161.00/20.96 join(meet(Y, X), meet(X, Y))
% 161.00/20.96 = { by lemma 40 }
% 161.00/20.96 join(meet(Y, X), meet(Y, X))
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) }
% 161.00/20.96 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) }
% 161.00/20.96 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 161.00/20.96 = { by lemma 22 }
% 161.00/20.96 complement(join(complement(Y), complement(X)))
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 161.00/20.96 meet(Y, X)
% 161.00/20.96 = { by lemma 40 R->L }
% 161.00/20.96 meet(X, Y)
% 161.00/20.96
% 161.00/20.96 Lemma 47: converse(zero) = zero.
% 161.00/20.96 Proof:
% 161.00/20.96 converse(zero)
% 161.00/20.96 = { by lemma 39 R->L }
% 161.00/20.96 join(zero, converse(zero))
% 161.00/20.96 = { by lemma 34 R->L }
% 161.00/20.96 converse(join(converse(zero), zero))
% 161.00/20.96 = { by lemma 29 }
% 161.00/20.96 converse(join(converse(zero), converse(zero)))
% 161.00/20.96 = { by lemma 33 }
% 161.00/20.96 join(zero, converse(converse(zero)))
% 161.00/20.96 = { by axiom 2 (converse_idempotence) }
% 161.00/20.96 join(zero, zero)
% 161.00/20.96 = { by lemma 45 R->L }
% 161.00/20.96 join(zero, meet(zero, top))
% 161.00/20.96 = { by lemma 45 R->L }
% 161.00/20.96 join(meet(zero, top), meet(zero, top))
% 161.00/20.96 = { by lemma 46 }
% 161.00/20.96 meet(zero, top)
% 161.00/20.96 = { by lemma 45 }
% 161.00/20.96 zero
% 161.00/20.96
% 161.00/20.96 Lemma 48: join(top, X) = top.
% 161.00/20.96 Proof:
% 161.00/20.96 join(top, X)
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 join(X, top)
% 161.00/20.96 = { by lemma 24 R->L }
% 161.00/20.96 join(Y, top)
% 161.00/20.96 = { by lemma 32 }
% 161.00/20.96 top
% 161.00/20.96
% 161.00/20.96 Lemma 49: complement(complement(X)) = X.
% 161.00/20.96 Proof:
% 161.00/20.96 complement(complement(X))
% 161.00/20.96 = { by lemma 30 R->L }
% 161.00/20.96 join(zero, complement(complement(X)))
% 161.00/20.96 = { by lemma 28 }
% 161.00/20.96 X
% 161.00/20.96
% 161.00/20.96 Lemma 50: meet(zero, X) = zero.
% 161.00/20.96 Proof:
% 161.00/20.96 meet(zero, X)
% 161.00/20.96 = { by lemma 40 }
% 161.00/20.96 meet(X, zero)
% 161.00/20.96 = { by lemma 43 }
% 161.00/20.96 zero
% 161.00/20.96
% 161.00/20.96 Lemma 51: composition(join(join(composition(converse(x0), x0), one), Y), X) = join(X, composition(Y, X)).
% 161.00/20.96 Proof:
% 161.00/20.96 composition(join(join(composition(converse(x0), x0), one), Y), X)
% 161.00/20.96 = { by axiom 12 (composition_distributivity) }
% 161.00/20.96 join(composition(join(composition(converse(x0), x0), one), X), composition(Y, X))
% 161.00/20.96 = { by lemma 20 }
% 161.00/20.96 join(X, composition(Y, X))
% 161.00/20.96
% 161.00/20.96 Lemma 52: composition(top, zero) = zero.
% 161.00/20.96 Proof:
% 161.00/20.96 composition(top, zero)
% 161.00/20.96 = { by lemma 38 R->L }
% 161.00/20.96 composition(converse(top), zero)
% 161.00/20.96 = { by lemma 39 R->L }
% 161.00/20.96 join(zero, composition(converse(top), zero))
% 161.00/20.96 = { by lemma 16 R->L }
% 161.00/20.96 join(complement(top), composition(converse(top), zero))
% 161.00/20.96 = { by lemma 16 R->L }
% 161.00/20.96 join(complement(top), composition(converse(top), complement(top)))
% 161.00/20.96 = { by lemma 48 R->L }
% 161.00/20.96 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 161.00/20.96 = { by lemma 38 R->L }
% 161.00/20.96 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 161.00/20.96 = { by lemma 51 R->L }
% 161.00/20.96 join(complement(top), composition(converse(top), complement(composition(join(join(composition(converse(x0), x0), one), converse(top)), top))))
% 161.00/20.96 = { by lemma 37 }
% 161.00/20.96 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 161.00/20.96 = { by lemma 21 }
% 161.00/20.96 complement(top)
% 161.00/20.96 = { by lemma 16 }
% 161.00/20.96 zero
% 161.00/20.96
% 161.00/20.96 Lemma 53: composition(zero, X) = zero.
% 161.00/20.96 Proof:
% 161.00/20.96 composition(zero, X)
% 161.00/20.96 = { by lemma 47 R->L }
% 161.00/20.96 composition(converse(zero), X)
% 161.00/20.96 = { by lemma 18 R->L }
% 161.00/20.96 converse(composition(converse(X), zero))
% 161.00/20.96 = { by lemma 39 R->L }
% 161.00/20.96 converse(join(zero, composition(converse(X), zero)))
% 161.00/20.96 = { by lemma 52 R->L }
% 161.00/20.96 converse(join(composition(top, zero), composition(converse(X), zero)))
% 161.00/20.96 = { by axiom 12 (composition_distributivity) R->L }
% 161.00/20.96 converse(composition(join(top, converse(X)), zero))
% 161.00/20.96 = { by lemma 48 }
% 161.00/20.96 converse(composition(top, zero))
% 161.00/20.96 = { by lemma 52 }
% 161.00/20.96 converse(zero)
% 161.00/20.96 = { by lemma 47 }
% 161.00/20.96 zero
% 161.00/20.96
% 161.00/20.96 Lemma 54: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 161.00/20.96 Proof:
% 161.00/20.96 meet(X, join(complement(Y), complement(Z)))
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 meet(X, join(complement(Z), complement(Y)))
% 161.00/20.96 = { by lemma 40 }
% 161.00/20.96 meet(join(complement(Z), complement(Y)), X)
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) }
% 161.00/20.96 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 161.00/20.96 complement(join(meet(Z, Y), complement(X)))
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/20.96 complement(join(complement(X), meet(Z, Y)))
% 161.00/20.96 = { by lemma 40 R->L }
% 161.00/20.96 complement(join(complement(X), meet(Y, Z)))
% 161.00/20.96
% 161.00/20.96 Lemma 55: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 161.00/20.96 Proof:
% 161.00/20.96 complement(join(X, complement(Y)))
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 complement(join(complement(Y), X))
% 161.00/20.96 = { by lemma 45 R->L }
% 161.00/20.96 complement(join(complement(Y), meet(X, top)))
% 161.00/20.96 = { by lemma 40 R->L }
% 161.00/20.96 complement(join(complement(Y), meet(top, X)))
% 161.00/20.96 = { by lemma 54 R->L }
% 161.00/20.96 meet(Y, join(complement(top), complement(X)))
% 161.00/20.96 = { by lemma 16 }
% 161.00/20.96 meet(Y, join(zero, complement(X)))
% 161.00/20.96 = { by lemma 30 }
% 161.00/20.96 meet(Y, complement(X))
% 161.00/20.96
% 161.00/20.96 Lemma 56: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 161.00/20.96 Proof:
% 161.00/20.96 complement(join(complement(X), Y))
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 complement(join(Y, complement(X)))
% 161.00/20.96 = { by lemma 55 }
% 161.00/20.96 meet(X, complement(Y))
% 161.00/20.96
% 161.00/20.96 Lemma 57: join(complement(join(composition(converse(x0), x0), one)), composition(converse(X), complement(X))) = complement(join(composition(converse(x0), x0), one)).
% 161.00/20.96 Proof:
% 161.00/20.96 join(complement(join(composition(converse(x0), x0), one)), composition(converse(X), complement(X)))
% 161.00/20.96 = { by axiom 11 (goals) }
% 161.00/20.96 join(complement(one), composition(converse(X), complement(X)))
% 161.00/20.96 = { by axiom 3 (composition_identity) R->L }
% 161.00/20.96 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 161.00/20.96 = { by lemma 21 }
% 161.00/20.96 complement(one)
% 161.00/20.96 = { by axiom 11 (goals) R->L }
% 161.00/20.96 complement(join(composition(converse(x0), x0), one))
% 161.00/20.96
% 161.00/20.96 Lemma 58: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 161.00/20.96 Proof:
% 161.00/20.96 join(complement(X), complement(Y))
% 161.00/20.96 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.96 join(complement(Y), complement(X))
% 161.00/20.96 = { by lemma 26 R->L }
% 161.00/20.96 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 161.00/20.96 = { by lemma 54 }
% 161.00/20.96 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 161.00/20.96 = { by lemma 30 }
% 161.00/20.96 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 161.00/20.96 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 161.00/20.96 complement(join(meet(Y, X), meet(Y, X)))
% 161.00/20.96 = { by lemma 46 }
% 161.00/20.96 complement(meet(Y, X))
% 161.00/20.96 = { by lemma 40 R->L }
% 161.00/20.96 complement(meet(X, Y))
% 161.00/20.97
% 161.00/20.97 Lemma 59: join(X, complement(meet(X, Y))) = top.
% 161.00/20.97 Proof:
% 161.00/20.97 join(X, complement(meet(X, Y)))
% 161.00/20.97 = { by lemma 40 }
% 161.00/20.97 join(X, complement(meet(Y, X)))
% 161.00/20.97 = { by lemma 58 R->L }
% 161.00/20.97 join(X, join(complement(Y), complement(X)))
% 161.00/20.97 = { by lemma 17 }
% 161.00/20.97 join(complement(Y), top)
% 161.00/20.97 = { by lemma 32 }
% 161.00/20.97 top
% 161.00/20.97
% 161.00/20.97 Lemma 60: meet(X, meet(Y, complement(X))) = zero.
% 161.00/20.97 Proof:
% 161.00/20.97 meet(X, meet(Y, complement(X)))
% 161.00/20.97 = { by lemma 40 }
% 161.00/20.97 meet(X, meet(complement(X), Y))
% 161.00/20.97 = { by axiom 10 (maddux4_definiton_of_meet) }
% 161.00/20.97 complement(join(complement(X), complement(meet(complement(X), Y))))
% 161.00/20.97 = { by lemma 59 }
% 161.00/20.97 complement(top)
% 161.00/20.97 = { by lemma 16 }
% 161.00/20.97 zero
% 161.00/20.97
% 161.00/20.97 Lemma 61: meet(join(composition(converse(x0), x0), one), composition(converse(complement(X)), X)) = zero.
% 161.00/20.97 Proof:
% 161.00/20.97 meet(join(composition(converse(x0), x0), one), composition(converse(complement(X)), X))
% 161.00/20.97 = { by lemma 40 }
% 161.00/20.97 meet(composition(converse(complement(X)), X), join(composition(converse(x0), x0), one))
% 161.00/20.97 = { by lemma 49 R->L }
% 161.00/20.97 meet(composition(converse(complement(X)), X), complement(complement(join(composition(converse(x0), x0), one))))
% 161.00/20.97 = { by lemma 57 R->L }
% 161.00/20.97 meet(composition(converse(complement(X)), X), complement(join(complement(join(composition(converse(x0), x0), one)), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 161.00/20.97 = { by lemma 41 }
% 161.00/20.97 meet(composition(converse(complement(X)), X), complement(join(complement(join(composition(converse(x0), x0), one)), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 161.00/20.97 = { by lemma 30 }
% 161.00/20.97 meet(composition(converse(complement(X)), X), complement(join(complement(join(composition(converse(x0), x0), one)), composition(converse(complement(X)), meet(X, top)))))
% 161.00/20.97 = { by lemma 45 }
% 161.00/20.97 meet(composition(converse(complement(X)), X), complement(join(complement(join(composition(converse(x0), x0), one)), composition(converse(complement(X)), X))))
% 161.00/20.97 = { by lemma 56 }
% 161.00/20.97 meet(composition(converse(complement(X)), X), meet(join(composition(converse(x0), x0), one), complement(composition(converse(complement(X)), X))))
% 161.00/20.97 = { by lemma 60 }
% 161.00/20.97 zero
% 161.00/20.97
% 161.00/20.97 Lemma 62: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 161.00/20.97 Proof:
% 161.00/20.97 join(meet(X, Y), meet(Y, complement(X)))
% 161.00/20.97 = { by lemma 40 }
% 161.00/20.97 join(meet(Y, X), meet(Y, complement(X)))
% 161.00/20.97 = { by lemma 44 }
% 161.00/20.97 Y
% 161.00/20.97
% 161.00/20.97 Lemma 63: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 161.00/20.97 Proof:
% 161.00/20.97 join(meet(X, Y), meet(complement(X), Y))
% 161.00/20.97 = { by lemma 40 }
% 161.00/20.97 join(meet(X, Y), meet(Y, complement(X)))
% 161.00/20.97 = { by lemma 62 }
% 161.00/20.97 Y
% 161.00/20.97
% 161.00/20.97 Lemma 64: converse(complement(X)) = complement(converse(X)).
% 161.00/20.97 Proof:
% 161.00/20.97 converse(complement(X))
% 161.00/20.97 = { by lemma 30 R->L }
% 161.00/20.97 converse(join(zero, complement(X)))
% 161.00/20.97 = { by lemma 25 R->L }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), complement(converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 161.00/20.97 = { by lemma 56 R->L }
% 161.00/20.97 converse(join(complement(join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 161.00/20.97 = { by lemma 35 }
% 161.00/20.97 converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 161.00/20.97 = { by lemma 38 }
% 161.00/20.97 converse(join(complement(top), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 161.00/20.97 = { by lemma 16 }
% 161.00/20.97 converse(join(zero, complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 161.00/20.97 = { by lemma 30 }
% 161.00/20.97 converse(complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
% 161.00/20.97 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 161.00/20.97 converse(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))))
% 161.00/20.97 = { by lemma 31 R->L }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), zero))
% 161.00/20.97 = { by lemma 50 R->L }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(X)))))))))
% 161.00/20.97 = { by lemma 53 R->L }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 161.00/20.97 = { by lemma 61 R->L }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(join(composition(converse(x0), x0), one), composition(converse(complement(converse(complement(join(zero, complement(X)))))), converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 161.00/20.97 = { by axiom 15 (modular_law_2) R->L }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(join(composition(converse(x0), x0), one), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(join(composition(converse(x0), x0), one), composition(converse(complement(converse(complement(join(zero, complement(X)))))), converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 161.00/20.97 = { by lemma 61 }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(join(composition(converse(x0), x0), one), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 161.00/20.97 = { by lemma 20 }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 161.00/20.97 = { by lemma 53 }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(X))))))))))
% 161.00/20.97 = { by lemma 50 }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), zero)))
% 161.00/20.97 = { by lemma 31 }
% 161.00/20.97 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 161.00/20.97 = { by lemma 63 }
% 161.00/20.97 converse(converse(complement(converse(complement(join(zero, complement(X)))))))
% 161.00/20.97 = { by axiom 2 (converse_idempotence) }
% 161.00/20.97 complement(converse(complement(join(zero, complement(X)))))
% 161.00/20.97 = { by lemma 41 }
% 161.00/20.97 complement(converse(meet(X, top)))
% 161.00/20.97 = { by lemma 45 }
% 161.00/20.97 complement(converse(X))
% 161.00/20.97
% 161.00/20.97 Lemma 65: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 161.00/20.97 Proof:
% 161.00/20.97 complement(meet(X, complement(Y)))
% 161.00/20.97 = { by lemma 39 R->L }
% 161.00/20.97 complement(join(zero, meet(X, complement(Y))))
% 161.00/20.97 = { by lemma 55 R->L }
% 161.00/20.97 complement(join(zero, complement(join(Y, complement(X)))))
% 161.00/20.97 = { by lemma 41 }
% 161.00/20.97 meet(join(Y, complement(X)), top)
% 161.00/20.97 = { by lemma 45 }
% 161.00/20.97 join(Y, complement(X))
% 161.00/20.97
% 161.00/20.97 Lemma 66: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 161.00/20.97 Proof:
% 161.00/20.97 complement(meet(complement(X), Y))
% 161.00/20.97 = { by lemma 40 }
% 161.00/20.97 complement(meet(Y, complement(X)))
% 161.00/20.97 = { by lemma 65 }
% 161.00/20.97 join(X, complement(Y))
% 161.00/20.97
% 161.00/20.97 Lemma 67: meet(X, join(X, complement(Y))) = X.
% 161.00/20.97 Proof:
% 161.00/20.97 meet(X, join(X, complement(Y)))
% 161.00/20.97 = { by lemma 65 R->L }
% 161.00/20.97 meet(X, complement(meet(Y, complement(X))))
% 161.00/20.97 = { by lemma 56 R->L }
% 161.00/20.97 complement(join(complement(X), meet(Y, complement(X))))
% 161.00/20.97 = { by lemma 30 R->L }
% 161.00/20.97 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 161.00/20.97 = { by lemma 60 R->L }
% 161.00/20.97 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 161.00/20.97 = { by lemma 25 }
% 161.00/20.97 X
% 161.00/20.97
% 161.00/20.97 Lemma 68: join(X, meet(X, Y)) = X.
% 161.00/20.97 Proof:
% 161.00/20.97 join(X, meet(X, Y))
% 161.00/20.97 = { by axiom 10 (maddux4_definiton_of_meet) }
% 161.00/20.97 join(X, complement(join(complement(X), complement(Y))))
% 161.00/20.97 = { by lemma 66 R->L }
% 161.00/20.97 complement(meet(complement(X), join(complement(X), complement(Y))))
% 161.00/20.97 = { by lemma 67 }
% 161.00/20.97 complement(complement(X))
% 161.00/20.97 = { by lemma 49 }
% 161.00/20.97 X
% 161.00/20.97
% 161.00/20.97 Lemma 69: meet(X, join(X, Y)) = X.
% 161.00/20.97 Proof:
% 161.00/20.97 meet(X, join(X, Y))
% 161.00/20.97 = { by lemma 45 R->L }
% 161.00/20.97 meet(X, join(X, meet(Y, top)))
% 161.00/20.97 = { by lemma 41 R->L }
% 161.00/20.97 meet(X, join(X, complement(join(zero, complement(Y)))))
% 161.00/20.97 = { by lemma 67 }
% 161.00/20.97 X
% 161.00/20.97
% 161.00/20.97 Lemma 70: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 161.00/20.97 Proof:
% 161.00/20.97 meet(complement(X), complement(Y))
% 161.00/20.97 = { by lemma 40 }
% 161.00/20.97 meet(complement(Y), complement(X))
% 161.00/20.97 = { by lemma 30 R->L }
% 161.00/20.97 meet(join(zero, complement(Y)), complement(X))
% 161.00/20.97 = { by lemma 55 R->L }
% 161.00/20.97 complement(join(X, complement(join(zero, complement(Y)))))
% 161.00/20.97 = { by lemma 41 }
% 161.00/20.97 complement(join(X, meet(Y, top)))
% 161.00/20.97 = { by lemma 45 }
% 161.00/20.97 complement(join(X, Y))
% 161.00/20.97
% 161.00/20.97 Lemma 71: converse(composition(Y, converse(X))) = composition(X, converse(Y)).
% 161.00/20.97 Proof:
% 161.00/20.97 converse(composition(Y, converse(X)))
% 161.00/20.97 = { by axiom 8 (converse_multiplicativity) }
% 161.00/20.97 composition(converse(converse(X)), converse(Y))
% 161.00/20.97 = { by axiom 2 (converse_idempotence) }
% 161.00/20.97 composition(X, converse(Y))
% 161.00/20.97
% 161.00/20.97 Lemma 72: join(complement(X), meet(Y, complement(Z))) = complement(meet(X, join(Z, complement(Y)))).
% 161.00/20.97 Proof:
% 161.00/20.97 join(complement(X), meet(Y, complement(Z)))
% 161.00/20.97 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.97 join(meet(Y, complement(Z)), complement(X))
% 161.00/20.97 = { by lemma 55 R->L }
% 161.00/20.97 join(complement(join(Z, complement(Y))), complement(X))
% 161.00/20.97 = { by lemma 58 }
% 161.00/20.97 complement(meet(join(Z, complement(Y)), X))
% 161.00/20.97 = { by lemma 40 R->L }
% 161.00/20.97 complement(meet(X, join(Z, complement(Y))))
% 161.00/20.97
% 161.00/20.97 Lemma 73: complement(meet(Y, join(X, complement(Y)))) = complement(meet(X, join(Y, complement(X)))).
% 161.00/20.97 Proof:
% 161.00/20.97 complement(meet(Y, join(X, complement(Y))))
% 161.00/20.97 = { by lemma 72 R->L }
% 161.00/20.97 join(complement(Y), meet(Y, complement(X)))
% 161.00/20.97 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/20.97 join(meet(Y, complement(X)), complement(Y))
% 161.00/20.97 = { by lemma 63 R->L }
% 161.00/20.97 join(meet(Y, complement(X)), join(meet(X, complement(Y)), meet(complement(X), complement(Y))))
% 161.00/20.97 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.97 join(meet(Y, complement(X)), join(meet(complement(X), complement(Y)), meet(X, complement(Y))))
% 161.00/20.97 = { by axiom 7 (maddux2_join_associativity) }
% 161.00/20.97 join(join(meet(Y, complement(X)), meet(complement(X), complement(Y))), meet(X, complement(Y)))
% 161.00/20.97 = { by lemma 62 }
% 161.00/20.97 join(complement(X), meet(X, complement(Y)))
% 161.00/20.97 = { by lemma 72 }
% 161.00/20.97 complement(meet(X, join(Y, complement(X))))
% 161.00/20.97
% 161.00/20.97 Lemma 74: join(complement(Y), meet(Y, X)) = join(X, complement(join(X, Y))).
% 161.00/20.97 Proof:
% 161.00/20.97 join(complement(Y), meet(Y, X))
% 161.00/20.97 = { by lemma 40 }
% 161.00/20.97 join(complement(Y), meet(X, Y))
% 161.00/20.97 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.97 join(meet(X, Y), complement(Y))
% 161.00/20.97 = { by lemma 65 R->L }
% 161.00/20.97 complement(meet(Y, complement(meet(X, Y))))
% 161.00/20.97 = { by lemma 58 R->L }
% 161.00/20.97 complement(meet(Y, join(complement(X), complement(Y))))
% 161.00/20.97 = { by lemma 73 R->L }
% 161.00/20.97 complement(meet(complement(X), join(Y, complement(complement(X)))))
% 161.00/20.97 = { by lemma 66 }
% 161.00/20.97 join(X, complement(join(Y, complement(complement(X)))))
% 161.00/20.97 = { by lemma 55 }
% 161.00/20.97 join(X, meet(complement(X), complement(Y)))
% 161.00/20.97 = { by lemma 70 }
% 161.00/20.97 join(X, complement(join(X, Y)))
% 161.00/20.97
% 161.00/20.97 Lemma 75: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 161.00/20.97 Proof:
% 161.00/20.97 join(X, complement(join(X, Y)))
% 161.00/20.97 = { by lemma 70 R->L }
% 161.00/20.97 join(X, meet(complement(X), complement(Y)))
% 161.00/20.97 = { by lemma 55 R->L }
% 161.00/20.97 join(X, complement(join(Y, complement(complement(X)))))
% 161.00/20.97 = { by lemma 66 R->L }
% 161.00/20.97 complement(meet(complement(X), join(Y, complement(complement(X)))))
% 161.00/20.97 = { by lemma 73 R->L }
% 161.00/20.97 complement(meet(Y, join(complement(X), complement(Y))))
% 161.00/20.97 = { by lemma 58 }
% 161.00/20.97 complement(meet(Y, complement(meet(X, Y))))
% 161.00/20.97 = { by lemma 65 }
% 161.00/20.97 join(meet(X, Y), complement(Y))
% 161.00/20.97 = { by lemma 63 R->L }
% 161.00/20.97 join(meet(X, Y), complement(join(meet(X, Y), meet(complement(X), Y))))
% 161.00/20.98 = { by lemma 74 R->L }
% 161.00/20.98 join(complement(meet(complement(X), Y)), meet(meet(complement(X), Y), meet(X, Y)))
% 161.00/20.98 = { by lemma 66 }
% 161.00/20.98 join(join(X, complement(Y)), meet(meet(complement(X), Y), meet(X, Y)))
% 161.00/20.98 = { by axiom 7 (maddux2_join_associativity) R->L }
% 161.00/20.98 join(X, join(complement(Y), meet(meet(complement(X), Y), meet(X, Y))))
% 161.00/20.98 = { by lemma 40 }
% 161.00/20.98 join(X, join(complement(Y), meet(meet(X, Y), meet(complement(X), Y))))
% 161.00/20.98 = { by lemma 40 }
% 161.00/20.98 join(X, join(complement(Y), meet(meet(X, Y), meet(Y, complement(X)))))
% 161.00/20.98 = { by lemma 45 R->L }
% 161.00/20.98 join(X, join(complement(Y), meet(meet(X, Y), meet(Y, meet(complement(X), top)))))
% 161.00/20.98 = { by lemma 41 R->L }
% 161.00/20.98 join(X, join(complement(Y), meet(meet(X, Y), meet(Y, complement(join(zero, complement(complement(X))))))))
% 161.00/20.98 = { by lemma 56 R->L }
% 161.00/20.98 join(X, join(complement(Y), meet(meet(X, Y), complement(join(complement(Y), join(zero, complement(complement(X))))))))
% 161.00/20.98 = { by lemma 56 R->L }
% 161.00/20.98 join(X, join(complement(Y), complement(join(complement(meet(X, Y)), join(complement(Y), join(zero, complement(complement(X))))))))
% 161.00/20.98 = { by axiom 7 (maddux2_join_associativity) }
% 161.00/20.98 join(X, join(complement(Y), complement(join(join(complement(meet(X, Y)), complement(Y)), join(zero, complement(complement(X)))))))
% 161.00/20.98 = { by lemma 70 R->L }
% 161.00/20.98 join(X, join(complement(Y), meet(complement(join(complement(meet(X, Y)), complement(Y))), complement(join(zero, complement(complement(X)))))))
% 161.00/20.98 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 161.00/20.98 join(X, join(complement(Y), meet(meet(meet(X, Y), Y), complement(join(zero, complement(complement(X)))))))
% 161.00/20.98 = { by lemma 40 R->L }
% 161.00/20.98 join(X, join(complement(Y), meet(complement(join(zero, complement(complement(X)))), meet(meet(X, Y), Y))))
% 161.00/20.98 = { by lemma 40 R->L }
% 161.00/20.98 join(X, join(complement(Y), meet(complement(join(zero, complement(complement(X)))), meet(Y, meet(X, Y)))))
% 161.00/20.98 = { by lemma 41 }
% 161.00/20.98 join(X, join(complement(Y), meet(meet(complement(X), top), meet(Y, meet(X, Y)))))
% 161.00/20.98 = { by lemma 45 }
% 161.00/20.98 join(X, join(complement(Y), meet(complement(X), meet(Y, meet(X, Y)))))
% 161.00/20.98 = { by lemma 40 }
% 161.00/20.98 join(X, join(complement(Y), meet(meet(Y, meet(X, Y)), complement(X))))
% 161.00/20.98 = { by lemma 72 }
% 161.00/20.98 join(X, complement(meet(Y, join(X, complement(meet(Y, meet(X, Y)))))))
% 161.00/20.98 = { by lemma 40 }
% 161.00/20.98 join(X, complement(meet(Y, join(X, complement(meet(Y, meet(Y, X)))))))
% 161.00/20.98 = { by lemma 40 }
% 161.00/20.98 join(X, complement(meet(Y, join(X, complement(meet(meet(Y, X), Y))))))
% 161.00/20.98 = { by lemma 25 R->L }
% 161.00/20.98 join(X, complement(meet(Y, join(X, complement(meet(meet(Y, X), join(meet(Y, X), complement(join(complement(Y), X)))))))))
% 161.00/20.98 = { by lemma 67 }
% 161.00/20.98 join(X, complement(meet(Y, join(X, complement(meet(Y, X))))))
% 161.00/20.98 = { by lemma 40 }
% 161.00/20.98 join(X, complement(meet(Y, join(X, complement(meet(X, Y))))))
% 161.00/20.98 = { by lemma 59 }
% 161.00/20.98 join(X, complement(meet(Y, top)))
% 161.00/20.98 = { by lemma 45 }
% 161.00/20.98 join(X, complement(Y))
% 161.00/20.98
% 161.00/20.98 Lemma 76: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 161.00/20.98 Proof:
% 161.00/20.98 meet(complement(X), join(Y, complement(Z)))
% 161.00/20.98 = { by lemma 40 }
% 161.00/20.98 meet(join(Y, complement(Z)), complement(X))
% 161.00/20.98 = { by lemma 55 R->L }
% 161.00/20.98 complement(join(X, complement(join(Y, complement(Z)))))
% 161.00/20.98 = { by lemma 55 }
% 161.00/20.98 complement(join(X, meet(Z, complement(Y))))
% 161.00/20.98
% 161.00/20.98 Lemma 77: meet(X, join(Y, complement(X))) = meet(X, Y).
% 161.00/20.98 Proof:
% 161.00/20.98 meet(X, join(Y, complement(X)))
% 161.00/20.98 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.98 meet(X, join(complement(X), Y))
% 161.00/20.98 = { by lemma 49 R->L }
% 161.00/20.98 meet(X, complement(complement(join(complement(X), Y))))
% 161.00/20.98 = { by lemma 56 R->L }
% 161.00/20.98 complement(join(complement(X), complement(join(complement(X), Y))))
% 161.00/20.98 = { by lemma 75 }
% 161.00/20.98 complement(join(complement(X), complement(Y)))
% 161.00/20.98 = { by lemma 56 }
% 161.00/20.98 meet(X, complement(complement(Y)))
% 161.00/20.98 = { by lemma 49 }
% 161.00/20.98 meet(X, Y)
% 161.00/20.98
% 161.00/20.98 Lemma 78: join(meet(Y, X), complement(Y)) = join(X, complement(Y)).
% 161.00/20.98 Proof:
% 161.00/20.98 join(meet(Y, X), complement(Y))
% 161.00/20.98 = { by lemma 68 R->L }
% 161.00/20.98 join(meet(Y, X), complement(join(Y, meet(Y, complement(X)))))
% 161.00/20.98 = { by lemma 76 R->L }
% 161.00/20.98 join(meet(Y, X), meet(complement(Y), join(X, complement(Y))))
% 161.00/20.98 = { by lemma 40 R->L }
% 161.00/20.98 join(meet(Y, X), meet(join(X, complement(Y)), complement(Y)))
% 161.00/20.98 = { by lemma 77 R->L }
% 161.00/20.98 join(meet(Y, join(X, complement(Y))), meet(join(X, complement(Y)), complement(Y)))
% 161.00/20.98 = { by lemma 62 }
% 161.00/20.98 join(X, complement(Y))
% 161.00/20.98
% 161.00/20.98 Lemma 79: meet(X, join(complement(X), Y)) = meet(X, Y).
% 161.00/20.98 Proof:
% 161.00/20.98 meet(X, join(complement(X), Y))
% 161.00/20.98 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.98 meet(X, join(Y, complement(X)))
% 161.00/20.98 = { by lemma 77 }
% 161.00/20.98 meet(X, Y)
% 161.00/20.98
% 161.00/20.98 Lemma 80: meet(complement(X), join(X, Y)) = meet(Y, complement(meet(Y, X))).
% 161.00/20.98 Proof:
% 161.00/20.98 meet(complement(X), join(X, Y))
% 161.00/20.98 = { by lemma 40 }
% 161.00/20.98 meet(join(X, Y), complement(X))
% 161.00/20.98 = { by lemma 55 R->L }
% 161.00/20.98 complement(join(X, complement(join(X, Y))))
% 161.00/20.98 = { by lemma 74 R->L }
% 161.00/20.98 complement(join(complement(Y), meet(Y, X)))
% 161.00/20.98 = { by lemma 56 }
% 161.00/20.98 meet(Y, complement(meet(Y, X)))
% 161.00/20.98
% 161.00/20.98 Lemma 81: composition(converse(X), complement(composition(X, top))) = zero.
% 161.00/20.98 Proof:
% 161.00/20.98 composition(converse(X), complement(composition(X, top)))
% 161.00/20.98 = { by lemma 39 R->L }
% 161.00/20.98 join(zero, composition(converse(X), complement(composition(X, top))))
% 161.00/20.98 = { by lemma 16 R->L }
% 161.00/20.98 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 161.00/20.98 = { by lemma 21 }
% 161.00/20.98 complement(top)
% 161.00/20.98 = { by lemma 16 }
% 161.00/20.98 zero
% 161.00/20.98
% 161.00/20.98 Lemma 82: composition(join(X, converse(Y)), converse(Z)) = converse(composition(Z, join(Y, converse(X)))).
% 161.00/20.98 Proof:
% 161.00/20.98 composition(join(X, converse(Y)), converse(Z))
% 161.00/20.98 = { by lemma 33 R->L }
% 161.00/20.98 composition(converse(join(Y, converse(X))), converse(Z))
% 161.00/20.98 = { by axiom 8 (converse_multiplicativity) R->L }
% 161.00/20.98 converse(composition(Z, join(Y, converse(X))))
% 161.00/20.98
% 161.00/20.98 Lemma 83: join(converse(composition(X, Y)), composition(Z, converse(X))) = converse(composition(X, join(Y, converse(Z)))).
% 161.00/20.98 Proof:
% 161.00/20.98 join(converse(composition(X, Y)), composition(Z, converse(X)))
% 161.00/20.98 = { by axiom 8 (converse_multiplicativity) }
% 161.00/20.98 join(composition(converse(Y), converse(X)), composition(Z, converse(X)))
% 161.00/20.98 = { by axiom 12 (composition_distributivity) R->L }
% 161.00/20.98 composition(join(converse(Y), Z), converse(X))
% 161.00/20.98 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.98 composition(join(Z, converse(Y)), converse(X))
% 161.00/20.98 = { by lemma 82 }
% 161.00/20.98 converse(composition(X, join(Y, converse(Z))))
% 161.00/20.98
% 161.00/20.98 Lemma 84: composition(top, composition(converse(X), X)) = composition(top, X).
% 161.00/20.98 Proof:
% 161.00/20.98 composition(top, composition(converse(X), X))
% 161.00/20.98 = { by axiom 9 (composition_associativity) }
% 161.00/20.98 composition(composition(top, converse(X)), X)
% 161.00/20.98 = { by axiom 2 (converse_idempotence) R->L }
% 161.00/20.98 composition(composition(top, converse(X)), converse(converse(X)))
% 161.00/20.98 = { by lemma 39 R->L }
% 161.00/20.98 join(zero, composition(composition(top, converse(X)), converse(converse(X))))
% 161.00/20.98 = { by lemma 47 R->L }
% 161.00/20.98 join(converse(zero), composition(composition(top, converse(X)), converse(converse(X))))
% 161.00/20.98 = { by lemma 81 R->L }
% 161.00/20.98 join(converse(composition(converse(X), complement(composition(X, top)))), composition(composition(top, converse(X)), converse(converse(X))))
% 161.00/20.98 = { by lemma 83 }
% 161.00/20.98 converse(composition(converse(X), join(complement(composition(X, top)), converse(composition(top, converse(X))))))
% 161.00/20.98 = { by lemma 18 }
% 161.00/20.98 composition(converse(join(complement(composition(X, top)), converse(composition(top, converse(X))))), X)
% 161.00/20.98 = { by lemma 33 }
% 161.00/20.98 composition(join(composition(top, converse(X)), converse(complement(composition(X, top)))), X)
% 161.00/20.98 = { by lemma 64 }
% 161.00/20.98 composition(join(composition(top, converse(X)), complement(converse(composition(X, top)))), X)
% 161.00/20.98 = { by axiom 8 (converse_multiplicativity) }
% 161.00/20.98 composition(join(composition(top, converse(X)), complement(composition(converse(top), converse(X)))), X)
% 161.00/20.98 = { by lemma 38 }
% 161.00/20.98 composition(join(composition(top, converse(X)), complement(composition(top, converse(X)))), X)
% 161.00/20.98 = { by axiom 4 (def_top) R->L }
% 161.00/20.98 composition(top, X)
% 161.00/20.98
% 161.00/20.98 Lemma 85: converse(join(X, composition(converse(Y), Z))) = join(converse(X), composition(converse(Z), Y)).
% 161.00/20.98 Proof:
% 161.00/20.98 converse(join(X, composition(converse(Y), Z)))
% 161.00/20.98 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.98 converse(join(composition(converse(Y), Z), X))
% 161.00/20.98 = { by axiom 6 (converse_additivity) }
% 161.00/20.98 join(converse(composition(converse(Y), Z)), converse(X))
% 161.00/20.98 = { by lemma 18 }
% 161.00/20.98 join(composition(converse(Z), Y), converse(X))
% 161.00/20.98 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/20.98 join(converse(X), composition(converse(Z), Y))
% 161.00/20.98
% 161.00/20.98 Lemma 86: join(converse(composition(X, Y)), composition(Z, converse(X))) = composition(join(Z, converse(Y)), converse(X)).
% 161.00/20.98 Proof:
% 161.00/20.98 join(converse(composition(X, Y)), composition(Z, converse(X)))
% 161.00/20.98 = { by axiom 8 (converse_multiplicativity) }
% 161.00/20.98 join(composition(converse(Y), converse(X)), composition(Z, converse(X)))
% 161.00/20.98 = { by axiom 12 (composition_distributivity) R->L }
% 161.00/20.98 composition(join(converse(Y), Z), converse(X))
% 161.00/20.98 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/20.98 composition(join(Z, converse(Y)), converse(X))
% 161.00/20.98
% 161.00/20.98 Lemma 87: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 161.00/20.98 Proof:
% 161.00/20.98 join(composition(X, Y), composition(X, Z))
% 161.00/20.98 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.98 join(composition(X, Z), composition(X, Y))
% 161.00/20.98 = { by axiom 2 (converse_idempotence) R->L }
% 161.00/20.98 join(composition(X, Z), composition(converse(converse(X)), Y))
% 161.00/20.98 = { by axiom 2 (converse_idempotence) R->L }
% 161.00/20.98 join(converse(converse(composition(X, Z))), composition(converse(converse(X)), Y))
% 161.00/20.98 = { by lemma 85 R->L }
% 161.00/20.98 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 161.00/20.98 = { by lemma 86 }
% 161.00/20.98 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 161.00/20.98 = { by lemma 71 }
% 161.00/20.98 composition(X, converse(join(converse(Y), converse(Z))))
% 161.00/20.98 = { by lemma 33 }
% 161.00/20.98 composition(X, join(Z, converse(converse(Y))))
% 161.00/20.98 = { by axiom 2 (converse_idempotence) }
% 161.00/20.98 composition(X, join(Z, Y))
% 161.00/20.98 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/20.98 composition(X, join(Y, Z))
% 161.00/20.98
% 161.00/20.98 Lemma 88: composition(converse(X), composition(converse(Y), Z)) = composition(converse(composition(Y, X)), Z).
% 161.00/20.98 Proof:
% 161.00/20.98 composition(converse(X), composition(converse(Y), Z))
% 161.00/20.98 = { by axiom 9 (composition_associativity) }
% 161.00/20.98 composition(composition(converse(X), converse(Y)), Z)
% 161.00/20.98 = { by axiom 8 (converse_multiplicativity) R->L }
% 161.00/20.98 composition(converse(composition(Y, X)), Z)
% 161.00/20.98
% 161.00/20.98 Lemma 89: composition(converse(X), composition(converse(Y), Z)) = converse(composition(composition(converse(Z), Y), X)).
% 161.00/20.98 Proof:
% 161.00/20.98 composition(converse(X), composition(converse(Y), Z))
% 161.00/20.98 = { by lemma 18 R->L }
% 161.00/20.98 composition(converse(X), converse(composition(converse(Z), Y)))
% 161.00/20.98 = { by axiom 8 (converse_multiplicativity) R->L }
% 161.00/20.98 converse(composition(composition(converse(Z), Y), X))
% 161.00/20.98
% 161.00/20.98 Lemma 90: join(join(composition(converse(x0), x0), one), X) = join(X, one).
% 161.00/20.98 Proof:
% 161.00/20.98 join(join(composition(converse(x0), x0), one), X)
% 161.00/20.98 = { by axiom 11 (goals) }
% 161.00/20.98 join(one, X)
% 161.00/20.98 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.98 join(X, one)
% 161.00/20.98
% 161.00/20.98 Lemma 91: composition(join(X, complement(composition(top, composition(converse(Y), Y)))), converse(Y)) = composition(X, converse(Y)).
% 161.00/20.98 Proof:
% 161.00/20.98 composition(join(X, complement(composition(top, composition(converse(Y), Y)))), converse(Y))
% 161.00/20.98 = { by lemma 84 }
% 161.00/20.98 composition(join(X, complement(composition(top, Y))), converse(Y))
% 161.00/20.98 = { by lemma 38 R->L }
% 161.00/20.98 composition(join(X, complement(composition(converse(top), Y))), converse(Y))
% 161.00/20.98 = { by lemma 18 R->L }
% 161.00/20.98 composition(join(X, complement(converse(composition(converse(Y), top)))), converse(Y))
% 161.00/20.98 = { by lemma 64 R->L }
% 161.00/20.98 composition(join(X, converse(complement(composition(converse(Y), top)))), converse(Y))
% 161.00/20.98 = { by lemma 86 R->L }
% 161.00/20.98 join(converse(composition(Y, complement(composition(converse(Y), top)))), composition(X, converse(Y)))
% 161.00/20.98 = { by axiom 2 (converse_idempotence) R->L }
% 161.00/20.98 join(converse(composition(converse(converse(Y)), complement(composition(converse(Y), top)))), composition(X, converse(Y)))
% 161.00/20.98 = { by lemma 81 }
% 161.00/20.98 join(converse(zero), composition(X, converse(Y)))
% 161.00/20.98 = { by lemma 47 }
% 161.00/20.98 join(zero, composition(X, converse(Y)))
% 161.00/20.98 = { by lemma 39 }
% 161.00/20.98 composition(X, converse(Y))
% 161.00/20.98
% 161.00/20.98 Lemma 92: composition(composition(converse(x0), x0), converse(x0)) = converse(x0).
% 161.00/20.98 Proof:
% 161.00/20.98 composition(composition(converse(x0), x0), converse(x0))
% 161.00/20.98 = { by lemma 25 R->L }
% 161.00/20.98 composition(join(meet(composition(converse(x0), x0), complement(join(composition(converse(x0), x0), one))), complement(join(complement(composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one))))), converse(x0))
% 161.00/20.98 = { by lemma 55 R->L }
% 161.00/20.98 composition(join(complement(join(join(composition(converse(x0), x0), one), complement(composition(converse(x0), x0)))), complement(join(complement(composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one))))), converse(x0))
% 161.00/20.98 = { by lemma 90 R->L }
% 161.00/20.98 composition(join(complement(join(join(join(composition(converse(x0), x0), one), composition(converse(x0), x0)), complement(composition(converse(x0), x0)))), complement(join(complement(composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one))))), converse(x0))
% 161.00/20.98 = { by axiom 7 (maddux2_join_associativity) R->L }
% 161.00/20.98 composition(join(complement(join(join(composition(converse(x0), x0), one), join(composition(converse(x0), x0), complement(composition(converse(x0), x0))))), complement(join(complement(composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one))))), converse(x0))
% 161.00/20.98 = { by axiom 4 (def_top) R->L }
% 161.00/20.98 composition(join(complement(join(join(composition(converse(x0), x0), one), top)), complement(join(complement(composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one))))), converse(x0))
% 161.00/20.98 = { by lemma 32 }
% 161.00/20.98 composition(join(complement(top), complement(join(complement(composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one))))), converse(x0))
% 161.00/20.98 = { by lemma 16 }
% 161.00/20.98 composition(join(zero, complement(join(complement(composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one))))), converse(x0))
% 161.00/20.98 = { by lemma 30 }
% 161.00/20.98 composition(complement(join(complement(composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one)))), converse(x0))
% 161.00/20.98 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 161.00/20.98 composition(meet(composition(converse(x0), x0), join(composition(converse(x0), x0), one)), converse(x0))
% 161.00/20.98 = { by lemma 40 R->L }
% 161.00/20.98 composition(meet(join(composition(converse(x0), x0), one), composition(converse(x0), x0)), converse(x0))
% 161.00/20.98 = { by lemma 77 R->L }
% 161.00/20.98 composition(meet(join(composition(converse(x0), x0), one), join(composition(converse(x0), x0), complement(join(composition(converse(x0), x0), one)))), converse(x0))
% 161.00/20.98 = { by lemma 65 R->L }
% 161.00/20.98 composition(meet(join(composition(converse(x0), x0), one), complement(meet(join(composition(converse(x0), x0), one), complement(composition(converse(x0), x0))))), converse(x0))
% 161.00/20.98 = { by lemma 56 R->L }
% 161.00/20.98 composition(complement(join(complement(join(composition(converse(x0), x0), one)), meet(join(composition(converse(x0), x0), one), complement(composition(converse(x0), x0))))), converse(x0))
% 161.00/20.98 = { by lemma 76 R->L }
% 161.00/20.98 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(converse(x0), x0), complement(join(composition(converse(x0), x0), one)))), converse(x0))
% 161.00/20.99 = { by lemma 57 R->L }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(converse(x0), x0), join(complement(join(composition(converse(x0), x0), one)), composition(converse(join(zero, complement(x0))), complement(join(zero, complement(x0))))))), converse(x0))
% 161.00/20.99 = { by lemma 41 }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(converse(x0), x0), join(complement(join(composition(converse(x0), x0), one)), composition(converse(join(zero, complement(x0))), meet(x0, top))))), converse(x0))
% 161.00/20.99 = { by lemma 30 }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(converse(x0), x0), join(complement(join(composition(converse(x0), x0), one)), composition(converse(complement(x0)), meet(x0, top))))), converse(x0))
% 161.00/20.99 = { by lemma 45 }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(converse(x0), x0), join(complement(join(composition(converse(x0), x0), one)), composition(converse(complement(x0)), x0)))), converse(x0))
% 161.00/20.99 = { by lemma 64 }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(converse(x0), x0), join(complement(join(composition(converse(x0), x0), one)), composition(complement(converse(x0)), x0)))), converse(x0))
% 161.00/20.99 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(converse(x0), x0), join(composition(complement(converse(x0)), x0), complement(join(composition(converse(x0), x0), one))))), converse(x0))
% 161.00/20.99 = { by axiom 7 (maddux2_join_associativity) }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(join(composition(converse(x0), x0), composition(complement(converse(x0)), x0)), complement(join(composition(converse(x0), x0), one)))), converse(x0))
% 161.00/20.99 = { by axiom 12 (composition_distributivity) R->L }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(join(converse(x0), complement(converse(x0))), x0), complement(join(composition(converse(x0), x0), one)))), converse(x0))
% 161.00/20.99 = { by axiom 4 (def_top) R->L }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(top, x0), complement(join(composition(converse(x0), x0), one)))), converse(x0))
% 161.00/20.99 = { by lemma 84 R->L }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(composition(top, composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one)))), converse(x0))
% 161.00/20.99 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/20.99 composition(meet(complement(complement(join(composition(converse(x0), x0), one))), join(complement(join(composition(converse(x0), x0), one)), composition(top, composition(converse(x0), x0)))), converse(x0))
% 161.00/20.99 = { by lemma 80 }
% 161.00/20.99 composition(meet(composition(top, composition(converse(x0), x0)), complement(meet(composition(top, composition(converse(x0), x0)), complement(join(composition(converse(x0), x0), one))))), converse(x0))
% 161.00/20.99 = { by lemma 65 }
% 161.00/20.99 composition(meet(composition(top, composition(converse(x0), x0)), join(join(composition(converse(x0), x0), one), complement(composition(top, composition(converse(x0), x0))))), converse(x0))
% 161.00/20.99 = { by lemma 77 }
% 161.00/20.99 composition(meet(composition(top, composition(converse(x0), x0)), join(composition(converse(x0), x0), one)), converse(x0))
% 161.00/20.99 = { by lemma 91 R->L }
% 161.00/20.99 composition(join(meet(composition(top, composition(converse(x0), x0)), join(composition(converse(x0), x0), one)), complement(composition(top, composition(converse(x0), x0)))), converse(x0))
% 161.00/20.99 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/20.99 composition(join(complement(composition(top, composition(converse(x0), x0))), meet(composition(top, composition(converse(x0), x0)), join(composition(converse(x0), x0), one))), converse(x0))
% 161.00/20.99 = { by lemma 74 }
% 161.00/20.99 composition(join(join(composition(converse(x0), x0), one), complement(join(join(composition(converse(x0), x0), one), composition(top, composition(converse(x0), x0))))), converse(x0))
% 161.00/20.99 = { by lemma 75 }
% 161.00/20.99 composition(join(join(composition(converse(x0), x0), one), complement(composition(top, composition(converse(x0), x0)))), converse(x0))
% 161.00/20.99 = { by lemma 91 }
% 161.00/20.99 composition(join(composition(converse(x0), x0), one), converse(x0))
% 161.00/20.99 = { by lemma 20 }
% 161.00/20.99 converse(x0)
% 161.00/20.99
% 161.00/20.99 Lemma 93: converse(composition(composition(converse(X), Y), top)) = composition(top, composition(converse(Y), X)).
% 161.00/20.99 Proof:
% 161.00/20.99 converse(composition(composition(converse(X), Y), top))
% 161.00/20.99 = { by lemma 89 R->L }
% 161.00/20.99 composition(converse(top), composition(converse(Y), X))
% 161.00/20.99 = { by lemma 38 }
% 161.00/20.99 composition(top, composition(converse(Y), X))
% 161.00/20.99
% 161.00/20.99 Lemma 94: join(X, composition(composition(converse(x0), x0), X)) = X.
% 161.00/20.99 Proof:
% 161.00/20.99 join(X, composition(composition(converse(x0), x0), X))
% 161.00/20.99 = { by lemma 51 R->L }
% 161.00/20.99 composition(join(join(composition(converse(x0), x0), one), composition(converse(x0), x0)), X)
% 161.00/20.99 = { by lemma 90 }
% 161.00/20.99 composition(join(composition(converse(x0), x0), one), X)
% 161.00/20.99 = { by lemma 20 }
% 161.00/20.99 X
% 161.00/20.99
% 161.00/20.99 Lemma 95: meet(X, composition(X, composition(converse(x0), x0))) = composition(X, composition(converse(x0), x0)).
% 161.00/20.99 Proof:
% 161.00/20.99 meet(X, composition(X, composition(converse(x0), x0)))
% 161.00/20.99 = { by lemma 40 }
% 161.00/20.99 meet(composition(X, composition(converse(x0), x0)), X)
% 161.00/20.99 = { by axiom 2 (converse_idempotence) R->L }
% 161.00/20.99 meet(composition(X, composition(converse(x0), x0)), converse(converse(X)))
% 161.00/20.99 = { by lemma 94 R->L }
% 161.00/20.99 meet(composition(X, composition(converse(x0), x0)), converse(join(converse(X), composition(composition(converse(x0), x0), converse(X)))))
% 161.00/20.99 = { by lemma 34 }
% 161.00/20.99 meet(composition(X, composition(converse(x0), x0)), join(X, converse(composition(composition(converse(x0), x0), converse(X)))))
% 161.00/20.99 = { by lemma 71 }
% 161.00/20.99 meet(composition(X, composition(converse(x0), x0)), join(X, composition(X, converse(composition(converse(x0), x0)))))
% 161.00/20.99 = { by lemma 18 }
% 161.00/20.99 meet(composition(X, composition(converse(x0), x0)), join(X, composition(X, composition(converse(x0), x0))))
% 161.00/20.99 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.99 meet(composition(X, composition(converse(x0), x0)), join(composition(X, composition(converse(x0), x0)), X))
% 161.00/20.99 = { by lemma 69 }
% 161.00/20.99 composition(X, composition(converse(x0), x0))
% 161.00/20.99
% 161.00/20.99 Lemma 96: meet(composition(top, composition(converse(x0), x0)), X) = composition(X, composition(converse(x0), x0)).
% 161.00/20.99 Proof:
% 161.00/20.99 meet(composition(top, composition(converse(x0), x0)), X)
% 161.00/20.99 = { by lemma 40 }
% 161.00/20.99 meet(X, composition(top, composition(converse(x0), x0)))
% 161.00/20.99 = { by lemma 79 R->L }
% 161.00/20.99 meet(X, join(complement(X), composition(top, composition(converse(x0), x0))))
% 161.00/20.99 = { by lemma 93 R->L }
% 161.00/20.99 meet(X, join(complement(X), converse(composition(composition(converse(x0), x0), top))))
% 161.00/20.99 = { by lemma 34 R->L }
% 161.00/20.99 meet(X, converse(join(converse(complement(X)), composition(composition(converse(x0), x0), top))))
% 161.00/20.99 = { by axiom 4 (def_top) }
% 161.00/20.99 meet(X, converse(join(converse(complement(X)), composition(composition(converse(x0), x0), join(converse(complement(X)), complement(converse(complement(X))))))))
% 161.00/20.99 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.99 meet(X, converse(join(converse(complement(X)), composition(composition(converse(x0), x0), join(complement(converse(complement(X))), converse(complement(X)))))))
% 161.00/20.99 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.99 meet(X, converse(join(composition(composition(converse(x0), x0), join(complement(converse(complement(X))), converse(complement(X)))), converse(complement(X)))))
% 161.00/20.99 = { by lemma 87 R->L }
% 161.00/20.99 meet(X, converse(join(join(composition(composition(converse(x0), x0), complement(converse(complement(X)))), composition(composition(converse(x0), x0), converse(complement(X)))), converse(complement(X)))))
% 161.00/20.99 = { by axiom 7 (maddux2_join_associativity) R->L }
% 161.00/20.99 meet(X, converse(join(composition(composition(converse(x0), x0), complement(converse(complement(X)))), join(composition(composition(converse(x0), x0), converse(complement(X))), converse(complement(X))))))
% 161.00/20.99 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/20.99 meet(X, converse(join(composition(composition(converse(x0), x0), complement(converse(complement(X)))), join(converse(complement(X)), composition(composition(converse(x0), x0), converse(complement(X)))))))
% 161.00/20.99 = { by lemma 94 }
% 161.00/20.99 meet(X, converse(join(composition(composition(converse(x0), x0), complement(converse(complement(X)))), converse(complement(X)))))
% 161.00/20.99 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/20.99 meet(X, converse(join(converse(complement(X)), composition(composition(converse(x0), x0), complement(converse(complement(X)))))))
% 161.00/20.99 = { by lemma 34 }
% 161.00/20.99 meet(X, join(complement(X), converse(composition(composition(converse(x0), x0), complement(converse(complement(X)))))))
% 161.00/20.99 = { by lemma 64 R->L }
% 161.00/20.99 meet(X, join(complement(X), converse(composition(composition(converse(x0), x0), converse(complement(complement(X)))))))
% 161.00/20.99 = { by lemma 18 R->L }
% 161.00/20.99 meet(X, join(complement(X), converse(composition(converse(composition(converse(x0), x0)), converse(complement(complement(X)))))))
% 161.00/20.99 = { by axiom 8 (converse_multiplicativity) R->L }
% 161.00/20.99 meet(X, join(complement(X), converse(converse(composition(complement(complement(X)), composition(converse(x0), x0))))))
% 161.00/20.99 = { by axiom 2 (converse_idempotence) }
% 161.00/20.99 meet(X, join(complement(X), composition(complement(complement(X)), composition(converse(x0), x0))))
% 161.00/20.99 = { by lemma 79 }
% 161.00/20.99 meet(X, composition(complement(complement(X)), composition(converse(x0), x0)))
% 161.00/20.99 = { by lemma 49 }
% 161.00/20.99 meet(X, composition(X, composition(converse(x0), x0)))
% 161.00/20.99 = { by lemma 95 }
% 161.00/20.99 composition(X, composition(converse(x0), x0))
% 161.00/20.99
% 161.00/20.99 Lemma 97: composition(complement(converse(X)), composition(converse(Y), Z)) = converse(composition(composition(converse(Z), Y), complement(X))).
% 161.00/20.99 Proof:
% 161.00/20.99 composition(complement(converse(X)), composition(converse(Y), Z))
% 161.00/20.99 = { by lemma 64 R->L }
% 161.00/20.99 composition(converse(complement(X)), composition(converse(Y), Z))
% 161.00/20.99 = { by lemma 89 }
% 161.00/20.99 converse(composition(composition(converse(Z), Y), complement(X)))
% 161.00/20.99
% 161.00/20.99 Lemma 98: join(complement(composition(x0, X)), join(Y, composition(x0, complement(X)))) = join(Y, complement(composition(x0, X))).
% 161.00/20.99 Proof:
% 161.00/20.99 join(complement(composition(x0, X)), join(Y, composition(x0, complement(X))))
% 161.00/20.99 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/20.99 join(complement(composition(x0, X)), join(composition(x0, complement(X)), Y))
% 161.00/20.99 = { by axiom 7 (maddux2_join_associativity) }
% 161.00/20.99 join(join(complement(composition(x0, X)), composition(x0, complement(X))), Y)
% 161.00/20.99 = { by axiom 2 (converse_idempotence) R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(converse(composition(x0, complement(X))))), Y)
% 161.00/20.99 = { by axiom 8 (converse_multiplicativity) }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(converse(complement(X)), converse(x0)))), Y)
% 161.00/20.99 = { by lemma 64 }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(complement(converse(X)), converse(x0)))), Y)
% 161.00/20.99 = { by lemma 92 R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(complement(converse(X)), composition(composition(converse(x0), x0), converse(x0))))), Y)
% 161.00/20.99 = { by lemma 64 R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(converse(complement(X)), composition(composition(converse(x0), x0), converse(x0))))), Y)
% 161.00/20.99 = { by lemma 18 R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(converse(complement(X)), composition(converse(composition(converse(x0), x0)), converse(x0))))), Y)
% 161.00/20.99 = { by lemma 88 }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(converse(composition(composition(converse(x0), x0), complement(X))), converse(x0)))), Y)
% 161.00/20.99 = { by lemma 97 R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(composition(complement(converse(X)), composition(converse(x0), x0)), converse(x0)))), Y)
% 161.00/20.99 = { by lemma 96 R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(meet(composition(top, composition(converse(x0), x0)), complement(converse(X))), converse(x0)))), Y)
% 161.00/20.99 = { by lemma 55 R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), converse(x0)))), Y)
% 161.00/20.99 = { by axiom 2 (converse_idempotence) R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(complement(converse(converse(join(converse(X), complement(composition(top, composition(converse(x0), x0))))))), converse(x0)))), Y)
% 161.00/20.99 = { by lemma 64 R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(converse(complement(converse(join(converse(X), complement(composition(top, composition(converse(x0), x0))))))), converse(x0)))), Y)
% 161.00/20.99 = { by lemma 21 R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(converse(join(complement(converse(join(converse(X), complement(composition(top, composition(converse(x0), x0)))))), composition(converse(converse(converse(converse(composition(converse(x0), x0))))), complement(composition(converse(converse(converse(composition(converse(x0), x0)))), converse(join(converse(X), complement(composition(top, composition(converse(x0), x0)))))))))), converse(x0)))), Y)
% 161.00/20.99 = { by axiom 8 (converse_multiplicativity) R->L }
% 161.00/20.99 join(join(complement(composition(x0, X)), converse(composition(converse(join(complement(converse(join(converse(X), complement(composition(top, composition(converse(x0), x0)))))), composition(converse(converse(converse(converse(composition(converse(x0), x0))))), complement(converse(composition(join(converse(X), complement(composition(top, composition(converse(x0), x0)))), converse(converse(composition(converse(x0), x0))))))))), converse(x0)))), Y)
% 161.00/21.00 = { by axiom 2 (converse_idempotence) }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(converse(join(complement(converse(join(converse(X), complement(composition(top, composition(converse(x0), x0)))))), composition(converse(converse(composition(converse(x0), x0))), complement(converse(composition(join(converse(X), complement(composition(top, composition(converse(x0), x0)))), converse(converse(composition(converse(x0), x0))))))))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 85 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(converse(complement(converse(join(converse(X), complement(composition(top, composition(converse(x0), x0))))))), composition(converse(complement(converse(composition(join(converse(X), complement(composition(top, composition(converse(x0), x0)))), converse(converse(composition(converse(x0), x0))))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 64 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(converse(converse(join(converse(X), complement(composition(top, composition(converse(x0), x0))))))), composition(converse(complement(converse(composition(join(converse(X), complement(composition(top, composition(converse(x0), x0)))), converse(converse(composition(converse(x0), x0))))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by axiom 2 (converse_idempotence) }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(converse(complement(converse(composition(join(converse(X), complement(composition(top, composition(converse(x0), x0)))), converse(converse(composition(converse(x0), x0))))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 64 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(converse(converse(composition(join(converse(X), complement(composition(top, composition(converse(x0), x0)))), converse(converse(composition(converse(x0), x0))))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by axiom 2 (converse_idempotence) }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(composition(join(converse(X), complement(composition(top, composition(converse(x0), x0)))), converse(converse(composition(converse(x0), x0))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by axiom 2 (converse_idempotence) }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(composition(join(converse(X), complement(composition(top, composition(converse(x0), x0)))), composition(converse(x0), x0))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 93 R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(composition(join(converse(X), complement(converse(composition(composition(converse(x0), x0), top)))), composition(converse(x0), x0))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 64 R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(composition(join(converse(X), converse(complement(composition(composition(converse(x0), x0), top)))), composition(converse(x0), x0))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by axiom 9 (composition_associativity) }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(composition(composition(join(converse(X), converse(complement(composition(composition(converse(x0), x0), top)))), converse(x0)), x0)), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 82 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(composition(converse(composition(x0, join(complement(composition(composition(converse(x0), x0), top)), converse(converse(X))))), x0)), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 18 R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(converse(composition(converse(x0), composition(x0, join(complement(composition(composition(converse(x0), x0), top)), converse(converse(X))))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by axiom 9 (composition_associativity) }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(converse(composition(composition(converse(x0), x0), join(complement(composition(composition(converse(x0), x0), top)), converse(converse(X)))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 83 R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(join(converse(composition(composition(converse(x0), x0), complement(composition(composition(converse(x0), x0), top)))), composition(converse(X), converse(composition(converse(x0), x0))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 18 R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(join(converse(composition(converse(composition(converse(x0), x0)), complement(composition(composition(converse(x0), x0), top)))), composition(converse(X), converse(composition(converse(x0), x0))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 81 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(join(converse(zero), composition(converse(X), converse(composition(converse(x0), x0))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 47 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(join(zero, composition(converse(X), converse(composition(converse(x0), x0))))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 39 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(composition(converse(X), converse(composition(converse(x0), x0)))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 18 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(complement(join(converse(X), complement(composition(top, composition(converse(x0), x0))))), composition(complement(composition(converse(X), composition(converse(x0), x0))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 55 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(meet(composition(top, composition(converse(x0), x0)), complement(converse(X))), composition(complement(composition(converse(X), composition(converse(x0), x0))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 96 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(composition(complement(converse(X)), composition(converse(x0), x0)), composition(complement(composition(converse(X), composition(converse(x0), x0))), converse(composition(converse(x0), x0)))), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 18 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(join(composition(complement(converse(X)), composition(converse(x0), x0)), composition(complement(composition(converse(X), composition(converse(x0), x0))), composition(converse(x0), x0))), converse(x0)))), Y)
% 161.00/21.00 = { by axiom 12 (composition_distributivity) R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(composition(join(complement(converse(X)), complement(composition(converse(X), composition(converse(x0), x0)))), composition(converse(x0), x0)), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 58 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(composition(complement(meet(converse(X), composition(converse(X), composition(converse(x0), x0)))), composition(converse(x0), x0)), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 95 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(composition(complement(composition(converse(X), composition(converse(x0), x0))), composition(converse(x0), x0)), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 89 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(composition(complement(converse(composition(composition(converse(x0), x0), X))), composition(converse(x0), x0)), converse(x0)))), Y)
% 161.00/21.00 = { by lemma 97 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(composition(converse(composition(composition(converse(x0), x0), complement(composition(composition(converse(x0), x0), X)))), converse(x0)))), Y)
% 161.00/21.00 = { by axiom 8 (converse_multiplicativity) R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(converse(composition(x0, composition(composition(converse(x0), x0), complement(composition(composition(converse(x0), x0), X))))))), Y)
% 161.00/21.00 = { by lemma 18 R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(converse(composition(x0, composition(converse(composition(converse(x0), x0)), complement(composition(composition(converse(x0), x0), X))))))), Y)
% 161.00/21.00 = { by axiom 2 (converse_idempotence) R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(converse(composition(converse(converse(x0)), composition(converse(composition(converse(x0), x0)), complement(composition(composition(converse(x0), x0), X))))))), Y)
% 161.00/21.00 = { by lemma 88 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(converse(composition(converse(composition(composition(converse(x0), x0), converse(x0))), complement(composition(composition(converse(x0), x0), X)))))), Y)
% 161.00/21.00 = { by lemma 92 }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(converse(composition(converse(converse(x0)), complement(composition(composition(converse(x0), x0), X)))))), Y)
% 161.00/21.00 = { by axiom 2 (converse_idempotence) }
% 161.00/21.00 join(join(complement(composition(x0, X)), converse(converse(composition(x0, complement(composition(composition(converse(x0), x0), X)))))), Y)
% 161.00/21.00 = { by axiom 2 (converse_idempotence) }
% 161.00/21.00 join(join(complement(composition(x0, X)), composition(x0, complement(composition(composition(converse(x0), x0), X)))), Y)
% 161.00/21.00 = { by axiom 2 (converse_idempotence) R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), composition(converse(converse(x0)), complement(composition(composition(converse(x0), x0), X)))), Y)
% 161.00/21.00 = { by axiom 9 (composition_associativity) R->L }
% 161.00/21.00 join(join(complement(composition(x0, X)), composition(converse(converse(x0)), complement(composition(converse(x0), composition(x0, X))))), Y)
% 161.00/21.00 = { by lemma 21 }
% 161.00/21.00 join(complement(composition(x0, X)), Y)
% 161.00/21.00 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/21.00 join(Y, complement(composition(x0, X)))
% 161.00/21.00
% 161.00/21.00 Goal 1 (goals_1): composition(x0, meet(x1, x2)) = meet(composition(x0, x1), composition(x0, x2)).
% 161.00/21.00 Proof:
% 161.00/21.00 composition(x0, meet(x1, x2))
% 161.00/21.00 = { by lemma 69 R->L }
% 161.00/21.00 meet(composition(x0, meet(x1, x2)), join(composition(x0, meet(x1, x2)), composition(x0, x1)))
% 161.00/21.00 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 161.00/21.00 meet(composition(x0, meet(x1, x2)), join(composition(x0, x1), composition(x0, meet(x1, x2))))
% 161.00/21.00 = { by lemma 87 }
% 161.00/21.00 meet(composition(x0, meet(x1, x2)), composition(x0, join(x1, meet(x1, x2))))
% 161.00/21.00 = { by lemma 68 }
% 161.00/21.00 meet(composition(x0, meet(x1, x2)), composition(x0, x1))
% 161.00/21.00 = { by lemma 49 R->L }
% 161.00/21.00 meet(composition(x0, meet(x1, x2)), complement(complement(composition(x0, x1))))
% 161.00/21.00 = { by lemma 55 R->L }
% 161.00/21.00 complement(join(complement(composition(x0, x1)), complement(composition(x0, meet(x1, x2)))))
% 161.00/21.00 = { by lemma 49 R->L }
% 161.00/21.00 complement(join(complement(composition(x0, x1)), complement(complement(complement(composition(x0, meet(x1, x2)))))))
% 161.00/21.00 = { by lemma 75 R->L }
% 161.00/21.00 complement(join(complement(composition(x0, x1)), complement(join(complement(composition(x0, x1)), complement(complement(composition(x0, meet(x1, x2))))))))
% 161.00/21.00 = { by lemma 55 }
% 161.00/21.00 complement(join(complement(composition(x0, x1)), meet(complement(composition(x0, meet(x1, x2))), complement(complement(composition(x0, x1))))))
% 161.00/21.00 = { by lemma 49 R->L }
% 161.00/21.00 complement(join(complement(composition(x0, x1)), meet(complement(composition(x0, meet(x1, x2))), complement(complement(complement(complement(composition(x0, x1))))))))
% 161.00/21.00 = { by lemma 49 R->L }
% 161.00/21.00 complement(complement(complement(join(complement(composition(x0, x1)), meet(complement(composition(x0, meet(x1, x2))), complement(complement(complement(complement(composition(x0, x1))))))))))
% 161.00/21.00 = { by lemma 76 R->L }
% 161.00/21.00 complement(complement(meet(complement(complement(composition(x0, x1))), join(complement(complement(complement(composition(x0, x1)))), complement(complement(composition(x0, meet(x1, x2))))))))
% 161.00/21.00 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/21.00 complement(complement(meet(complement(complement(composition(x0, x1))), join(complement(complement(composition(x0, meet(x1, x2)))), complement(complement(complement(composition(x0, x1))))))))
% 161.00/21.00 = { by lemma 73 R->L }
% 161.00/21.00 complement(complement(meet(complement(complement(composition(x0, meet(x1, x2)))), join(complement(complement(composition(x0, x1))), complement(complement(complement(composition(x0, meet(x1, x2)))))))))
% 161.00/21.00 = { by lemma 66 }
% 161.00/21.00 complement(join(complement(composition(x0, meet(x1, x2))), complement(join(complement(complement(composition(x0, x1))), complement(complement(complement(composition(x0, meet(x1, x2)))))))))
% 161.00/21.00 = { by lemma 56 }
% 161.00/21.01 complement(join(complement(composition(x0, meet(x1, x2))), meet(complement(composition(x0, x1)), complement(complement(complement(complement(composition(x0, meet(x1, x2)))))))))
% 161.00/21.01 = { by lemma 49 }
% 161.00/21.01 complement(join(complement(composition(x0, meet(x1, x2))), meet(complement(composition(x0, x1)), complement(complement(composition(x0, meet(x1, x2)))))))
% 161.00/21.01 = { by lemma 56 }
% 161.00/21.01 meet(composition(x0, meet(x1, x2)), complement(meet(complement(composition(x0, x1)), complement(complement(composition(x0, meet(x1, x2)))))))
% 161.00/21.01 = { by lemma 65 }
% 161.00/21.01 meet(composition(x0, meet(x1, x2)), join(complement(composition(x0, meet(x1, x2))), complement(complement(composition(x0, x1)))))
% 161.00/21.01 = { by lemma 58 }
% 161.00/21.01 meet(composition(x0, meet(x1, x2)), complement(meet(composition(x0, meet(x1, x2)), complement(composition(x0, x1)))))
% 161.00/21.01 = { by lemma 80 R->L }
% 161.00/21.01 meet(complement(complement(composition(x0, x1))), join(complement(composition(x0, x1)), composition(x0, meet(x1, x2))))
% 161.00/21.01 = { by axiom 1 (maddux1_join_commutativity) }
% 161.00/21.01 meet(complement(complement(composition(x0, x1))), join(composition(x0, meet(x1, x2)), complement(composition(x0, x1))))
% 161.00/21.01 = { by lemma 98 R->L }
% 161.00/21.01 meet(complement(complement(composition(x0, x1))), join(complement(composition(x0, x1)), join(composition(x0, meet(x1, x2)), composition(x0, complement(x1)))))
% 161.00/21.01 = { by lemma 87 }
% 161.00/21.01 meet(complement(complement(composition(x0, x1))), join(complement(composition(x0, x1)), composition(x0, join(meet(x1, x2), complement(x1)))))
% 161.00/21.01 = { by lemma 78 }
% 161.00/21.01 meet(complement(complement(composition(x0, x1))), join(complement(composition(x0, x1)), composition(x0, join(x2, complement(x1)))))
% 161.00/21.01 = { by lemma 87 R->L }
% 161.00/21.01 meet(complement(complement(composition(x0, x1))), join(complement(composition(x0, x1)), join(composition(x0, x2), composition(x0, complement(x1)))))
% 161.00/21.01 = { by lemma 98 }
% 161.00/21.01 meet(complement(complement(composition(x0, x1))), join(composition(x0, x2), complement(composition(x0, x1))))
% 161.00/21.01 = { by lemma 78 R->L }
% 161.00/21.01 meet(complement(complement(composition(x0, x1))), join(meet(composition(x0, x1), composition(x0, x2)), complement(composition(x0, x1))))
% 161.00/21.01 = { by lemma 76 }
% 161.00/21.01 complement(join(complement(composition(x0, x1)), meet(composition(x0, x1), complement(meet(composition(x0, x1), composition(x0, x2))))))
% 161.00/21.01 = { by lemma 56 }
% 161.00/21.01 meet(composition(x0, x1), complement(meet(composition(x0, x1), complement(meet(composition(x0, x1), composition(x0, x2))))))
% 161.00/21.01 = { by lemma 65 }
% 161.00/21.01 meet(composition(x0, x1), join(meet(composition(x0, x1), composition(x0, x2)), complement(composition(x0, x1))))
% 161.00/21.01 = { by lemma 77 }
% 161.00/21.01 meet(composition(x0, x1), meet(composition(x0, x1), composition(x0, x2)))
% 161.00/21.01 = { by lemma 40 R->L }
% 161.00/21.01 meet(meet(composition(x0, x1), composition(x0, x2)), composition(x0, x1))
% 161.00/21.01 = { by axiom 10 (maddux4_definiton_of_meet) }
% 161.00/21.01 complement(join(complement(meet(composition(x0, x1), composition(x0, x2))), complement(composition(x0, x1))))
% 161.00/21.01 = { by lemma 30 R->L }
% 161.00/21.01 join(zero, complement(join(complement(meet(composition(x0, x1), composition(x0, x2))), complement(composition(x0, x1)))))
% 161.00/21.01 = { by lemma 16 R->L }
% 161.00/21.01 join(complement(top), complement(join(complement(meet(composition(x0, x1), composition(x0, x2))), complement(composition(x0, x1)))))
% 161.00/21.01 = { by lemma 59 R->L }
% 161.00/21.01 join(complement(join(composition(x0, x1), complement(meet(composition(x0, x1), composition(x0, x2))))), complement(join(complement(meet(composition(x0, x1), composition(x0, x2))), complement(composition(x0, x1)))))
% 161.00/21.01 = { by lemma 55 }
% 161.00/21.01 join(meet(meet(composition(x0, x1), composition(x0, x2)), complement(composition(x0, x1))), complement(join(complement(meet(composition(x0, x1), composition(x0, x2))), complement(composition(x0, x1)))))
% 161.00/21.01 = { by lemma 25 }
% 161.00/21.01 meet(composition(x0, x1), composition(x0, x2))
% 161.00/21.01 % SZS output end Proof
% 161.00/21.01
% 161.00/21.01 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------