TSTP Solution File: REL022+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL022+2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:44:03 EDT 2023
% Result : Theorem 26.31s 3.78s
% Output : Proof 28.02s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : REL022+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n025.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 22:02:07 EDT 2023
% 0.13/0.34 % CPUTime :
% 26.31/3.78 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 26.31/3.78
% 26.31/3.78 % SZS status Theorem
% 26.31/3.78
% 27.43/3.89 % SZS output start Proof
% 27.43/3.89 Axiom 1 (composition_identity): composition(X, one) = X.
% 27.43/3.89 Axiom 2 (goals): composition(x0, top) = x0.
% 27.43/3.89 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 27.43/3.89 Axiom 4 (converse_idempotence): converse(converse(X)) = X.
% 27.43/3.89 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 27.43/3.89 Axiom 6 (def_top): top = join(X, complement(X)).
% 27.43/3.89 Axiom 7 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 27.43/3.89 Axiom 8 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 27.43/3.89 Axiom 9 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 27.43/3.89 Axiom 10 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 27.43/3.89 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 27.43/3.89 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 27.43/3.89 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 27.43/3.89 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 27.43/3.89 Axiom 15 (modular_law_1): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 27.43/3.89 Axiom 16 (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).
% 27.43/3.89 Axiom 17 (dedekind_law): join(meet(composition(X, Y), Z), composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z)))) = composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z))).
% 27.43/3.89
% 27.43/3.89 Lemma 18: complement(top) = zero.
% 27.43/3.89 Proof:
% 27.43/3.89 complement(top)
% 27.43/3.89 = { by axiom 6 (def_top) }
% 27.43/3.89 complement(join(complement(X), complement(complement(X))))
% 27.43/3.89 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 27.43/3.89 meet(X, complement(X))
% 27.43/3.89 = { by axiom 5 (def_zero) R->L }
% 27.43/3.89 zero
% 27.43/3.89
% 27.43/3.89 Lemma 19: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 27.43/3.89 Proof:
% 27.43/3.89 join(meet(X, Y), complement(join(complement(X), Y)))
% 27.43/3.89 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.43/3.89 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 27.43/3.89 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 27.43/3.89 X
% 27.43/3.89
% 27.43/3.89 Lemma 20: join(zero, meet(X, X)) = X.
% 27.43/3.89 Proof:
% 27.43/3.89 join(zero, meet(X, X))
% 27.43/3.89 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.43/3.89 join(zero, complement(join(complement(X), complement(X))))
% 27.43/3.89 = { by axiom 5 (def_zero) }
% 27.43/3.89 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 27.43/3.89 = { by lemma 19 }
% 27.43/3.89 X
% 27.43/3.89
% 27.43/3.89 Lemma 21: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 27.43/3.89 Proof:
% 27.43/3.89 converse(composition(converse(X), Y))
% 27.43/3.89 = { by axiom 7 (converse_multiplicativity) }
% 27.43/3.89 composition(converse(Y), converse(converse(X)))
% 27.43/3.89 = { by axiom 4 (converse_idempotence) }
% 27.43/3.89 composition(converse(Y), X)
% 27.43/3.89
% 27.43/3.89 Lemma 22: composition(converse(one), X) = X.
% 27.43/3.89 Proof:
% 27.43/3.89 composition(converse(one), X)
% 27.43/3.89 = { by lemma 21 R->L }
% 27.43/3.89 converse(composition(converse(X), one))
% 27.43/3.89 = { by axiom 1 (composition_identity) }
% 27.43/3.89 converse(converse(X))
% 27.43/3.89 = { by axiom 4 (converse_idempotence) }
% 27.43/3.89 X
% 27.43/3.89
% 27.43/3.89 Lemma 23: composition(one, X) = X.
% 27.43/3.89 Proof:
% 27.43/3.89 composition(one, X)
% 27.43/3.89 = { by lemma 22 R->L }
% 27.43/3.89 composition(converse(one), composition(one, X))
% 27.43/3.89 = { by axiom 8 (composition_associativity) }
% 27.43/3.89 composition(composition(converse(one), one), X)
% 27.43/3.89 = { by axiom 1 (composition_identity) }
% 27.43/3.89 composition(converse(one), X)
% 27.43/3.89 = { by lemma 22 }
% 27.43/3.89 X
% 27.43/3.89
% 27.43/3.89 Lemma 24: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 27.43/3.89 Proof:
% 27.43/3.89 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 27.43/3.89 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.89 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 27.43/3.89 = { by axiom 13 (converse_cancellativity) }
% 27.43/3.89 complement(X)
% 27.43/3.89
% 27.43/3.89 Lemma 25: join(complement(X), complement(X)) = complement(X).
% 27.43/3.89 Proof:
% 27.43/3.89 join(complement(X), complement(X))
% 27.43/3.89 = { by lemma 22 R->L }
% 27.43/3.89 join(complement(X), composition(converse(one), complement(X)))
% 27.43/3.89 = { by lemma 23 R->L }
% 27.43/3.89 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 27.43/3.89 = { by lemma 24 }
% 27.43/3.89 complement(X)
% 27.43/3.89
% 27.43/3.89 Lemma 26: join(zero, zero) = zero.
% 27.43/3.89 Proof:
% 27.43/3.89 join(zero, zero)
% 27.43/3.89 = { by lemma 18 R->L }
% 27.43/3.89 join(zero, complement(top))
% 27.43/3.89 = { by lemma 18 R->L }
% 27.43/3.89 join(complement(top), complement(top))
% 27.43/3.89 = { by lemma 25 }
% 27.43/3.89 complement(top)
% 27.43/3.89 = { by lemma 18 }
% 27.43/3.89 zero
% 27.43/3.89
% 27.43/3.89 Lemma 27: join(zero, join(zero, X)) = join(X, zero).
% 27.43/3.89 Proof:
% 27.43/3.89 join(zero, join(zero, X))
% 27.43/3.89 = { by axiom 10 (maddux2_join_associativity) }
% 27.43/3.89 join(join(zero, zero), X)
% 27.43/3.89 = { by lemma 26 }
% 27.43/3.89 join(zero, X)
% 27.43/3.89 = { by axiom 3 (maddux1_join_commutativity) }
% 27.43/3.89 join(X, zero)
% 27.43/3.89
% 27.43/3.89 Lemma 28: join(X, zero) = X.
% 27.43/3.89 Proof:
% 27.43/3.89 join(X, zero)
% 27.43/3.89 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.89 join(zero, X)
% 27.43/3.89 = { by lemma 20 R->L }
% 27.43/3.89 join(zero, join(zero, meet(X, X)))
% 27.43/3.89 = { by lemma 27 }
% 27.43/3.89 join(meet(X, X), zero)
% 27.43/3.89 = { by axiom 3 (maddux1_join_commutativity) }
% 27.43/3.89 join(zero, meet(X, X))
% 27.43/3.89 = { by lemma 20 }
% 27.43/3.89 X
% 27.43/3.89
% 27.43/3.89 Lemma 29: join(zero, X) = X.
% 27.43/3.89 Proof:
% 27.43/3.89 join(zero, X)
% 27.43/3.89 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.89 join(X, zero)
% 27.43/3.89 = { by lemma 28 }
% 27.43/3.89 X
% 27.43/3.89
% 27.43/3.89 Lemma 30: complement(zero) = top.
% 27.43/3.89 Proof:
% 27.43/3.89 complement(zero)
% 27.43/3.89 = { by lemma 29 R->L }
% 27.43/3.89 join(zero, complement(zero))
% 27.43/3.89 = { by axiom 6 (def_top) R->L }
% 27.43/3.89 top
% 27.43/3.89
% 27.43/3.89 Lemma 31: converse(one) = one.
% 27.43/3.89 Proof:
% 27.43/3.89 converse(one)
% 27.43/3.89 = { by axiom 1 (composition_identity) R->L }
% 27.43/3.89 composition(converse(one), one)
% 27.43/3.89 = { by lemma 22 }
% 27.43/3.89 one
% 27.43/3.89
% 27.43/3.89 Lemma 32: join(X, join(Y, complement(X))) = join(Y, top).
% 27.43/3.89 Proof:
% 27.43/3.89 join(X, join(Y, complement(X)))
% 27.43/3.89 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.89 join(X, join(complement(X), Y))
% 27.43/3.89 = { by axiom 10 (maddux2_join_associativity) }
% 27.43/3.89 join(join(X, complement(X)), Y)
% 27.43/3.89 = { by axiom 6 (def_top) R->L }
% 27.43/3.89 join(top, Y)
% 27.43/3.89 = { by axiom 3 (maddux1_join_commutativity) }
% 27.43/3.89 join(Y, top)
% 27.43/3.90
% 27.43/3.90 Lemma 33: join(top, complement(X)) = top.
% 27.43/3.90 Proof:
% 27.43/3.90 join(top, complement(X))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 join(complement(X), top)
% 27.43/3.90 = { by lemma 32 R->L }
% 27.43/3.90 join(X, join(complement(X), complement(X)))
% 27.43/3.90 = { by lemma 25 }
% 27.43/3.90 join(X, complement(X))
% 27.43/3.90 = { by axiom 6 (def_top) R->L }
% 27.43/3.90 top
% 27.43/3.90
% 27.43/3.90 Lemma 34: join(top, X) = join(Y, top).
% 27.43/3.90 Proof:
% 27.43/3.90 join(top, X)
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 join(X, top)
% 27.43/3.90 = { by lemma 33 R->L }
% 27.43/3.90 join(X, join(top, complement(X)))
% 27.43/3.90 = { by lemma 32 }
% 27.43/3.90 join(top, top)
% 27.43/3.90 = { by lemma 32 R->L }
% 27.43/3.90 join(Y, join(top, complement(Y)))
% 27.43/3.90 = { by lemma 33 }
% 27.43/3.90 join(Y, top)
% 27.43/3.90
% 27.43/3.90 Lemma 35: join(X, top) = top.
% 27.43/3.90 Proof:
% 27.43/3.90 join(X, top)
% 27.43/3.90 = { by lemma 34 R->L }
% 27.43/3.90 join(top, complement(top))
% 27.43/3.90 = { by axiom 6 (def_top) R->L }
% 27.43/3.90 top
% 27.43/3.90
% 27.43/3.90 Lemma 36: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 27.43/3.90 Proof:
% 27.43/3.90 converse(join(X, converse(Y)))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 converse(join(converse(Y), X))
% 27.43/3.90 = { by axiom 9 (converse_additivity) }
% 27.43/3.90 join(converse(converse(Y)), converse(X))
% 27.43/3.90 = { by axiom 4 (converse_idempotence) }
% 27.43/3.90 join(Y, converse(X))
% 27.43/3.90
% 27.43/3.90 Lemma 37: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 27.43/3.90 Proof:
% 27.43/3.90 converse(join(converse(X), Y))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 converse(join(Y, converse(X)))
% 27.43/3.90 = { by lemma 36 }
% 27.43/3.90 join(X, converse(Y))
% 27.43/3.90
% 27.43/3.90 Lemma 38: converse(top) = top.
% 27.43/3.90 Proof:
% 27.43/3.90 converse(top)
% 27.43/3.90 = { by lemma 35 R->L }
% 27.43/3.90 converse(join(converse(top), top))
% 27.43/3.90 = { by lemma 37 }
% 27.43/3.90 join(top, converse(top))
% 27.43/3.90 = { by lemma 34 }
% 27.43/3.90 join(X, top)
% 27.43/3.90 = { by lemma 35 }
% 27.43/3.90 top
% 27.43/3.90
% 27.43/3.90 Lemma 39: converse(zero) = zero.
% 27.43/3.90 Proof:
% 27.43/3.90 converse(zero)
% 27.43/3.90 = { by lemma 28 R->L }
% 27.43/3.90 join(converse(zero), zero)
% 27.43/3.90 = { by lemma 27 R->L }
% 27.43/3.90 join(zero, join(zero, converse(zero)))
% 27.43/3.90 = { by lemma 37 R->L }
% 27.43/3.90 join(zero, converse(join(converse(zero), zero)))
% 27.43/3.90 = { by lemma 28 }
% 27.43/3.90 join(zero, converse(converse(zero)))
% 27.43/3.90 = { by axiom 4 (converse_idempotence) }
% 27.43/3.90 join(zero, zero)
% 27.43/3.90 = { by lemma 26 }
% 27.43/3.90 zero
% 27.43/3.90
% 27.43/3.90 Lemma 40: complement(complement(X)) = X.
% 27.43/3.90 Proof:
% 27.43/3.90 complement(complement(X))
% 27.43/3.90 = { by lemma 29 R->L }
% 27.43/3.90 join(zero, complement(complement(X)))
% 27.43/3.90 = { by lemma 25 R->L }
% 27.43/3.90 join(zero, complement(join(complement(X), complement(X))))
% 27.43/3.90 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 27.43/3.90 join(zero, meet(X, X))
% 27.43/3.90 = { by lemma 20 }
% 27.43/3.90 X
% 27.43/3.90
% 27.43/3.90 Lemma 41: meet(Y, X) = meet(X, Y).
% 27.43/3.90 Proof:
% 27.43/3.90 meet(Y, X)
% 27.43/3.90 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.43/3.90 complement(join(complement(Y), complement(X)))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 complement(join(complement(X), complement(Y)))
% 27.43/3.90 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 27.43/3.90 meet(X, Y)
% 27.43/3.90
% 27.43/3.90 Lemma 42: join(meet(X, Y), meet(X, complement(Y))) = X.
% 27.43/3.90 Proof:
% 27.43/3.90 join(meet(X, Y), meet(X, complement(Y)))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 join(meet(X, complement(Y)), meet(X, Y))
% 27.43/3.90 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.43/3.90 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 27.43/3.90 = { by lemma 19 }
% 27.43/3.90 X
% 27.43/3.90
% 27.43/3.90 Lemma 43: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 27.43/3.90 Proof:
% 27.43/3.90 join(meet(X, Y), meet(Y, complement(X)))
% 27.43/3.90 = { by lemma 41 }
% 27.43/3.90 join(meet(Y, X), meet(Y, complement(X)))
% 27.43/3.90 = { by lemma 42 }
% 27.43/3.90 Y
% 27.43/3.90
% 27.43/3.90 Lemma 44: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 27.43/3.90 Proof:
% 27.43/3.90 join(meet(X, Y), meet(complement(X), Y))
% 27.43/3.90 = { by lemma 41 }
% 27.43/3.90 join(meet(X, Y), meet(Y, complement(X)))
% 27.43/3.90 = { by lemma 43 }
% 27.43/3.90 Y
% 27.43/3.90
% 27.43/3.90 Lemma 45: complement(join(zero, complement(X))) = meet(X, top).
% 27.43/3.90 Proof:
% 27.43/3.90 complement(join(zero, complement(X)))
% 27.43/3.90 = { by lemma 18 R->L }
% 27.43/3.90 complement(join(complement(top), complement(X)))
% 27.43/3.90 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 27.43/3.90 meet(top, X)
% 27.43/3.90 = { by lemma 41 R->L }
% 27.43/3.90 meet(X, top)
% 27.43/3.90
% 27.43/3.90 Lemma 46: meet(X, top) = X.
% 27.43/3.90 Proof:
% 27.43/3.90 meet(X, top)
% 27.43/3.90 = { by lemma 45 R->L }
% 27.43/3.90 complement(join(zero, complement(X)))
% 27.43/3.90 = { by lemma 29 }
% 27.43/3.90 complement(complement(X))
% 27.43/3.90 = { by lemma 40 }
% 27.43/3.90 X
% 27.43/3.90
% 27.43/3.90 Lemma 47: meet(top, X) = X.
% 27.43/3.90 Proof:
% 27.43/3.90 meet(top, X)
% 27.43/3.90 = { by lemma 41 }
% 27.43/3.90 meet(X, top)
% 27.43/3.90 = { by lemma 46 }
% 27.43/3.90 X
% 27.43/3.90
% 27.43/3.90 Lemma 48: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 27.43/3.90 Proof:
% 27.43/3.90 complement(join(complement(X), meet(Y, Z)))
% 27.43/3.90 = { by lemma 41 }
% 27.43/3.90 complement(join(complement(X), meet(Z, Y)))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 complement(join(meet(Z, Y), complement(X)))
% 27.43/3.90 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.43/3.90 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 27.43/3.90 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 27.43/3.90 meet(join(complement(Z), complement(Y)), X)
% 27.43/3.90 = { by lemma 41 R->L }
% 27.43/3.90 meet(X, join(complement(Z), complement(Y)))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) }
% 27.43/3.90 meet(X, join(complement(Y), complement(Z)))
% 27.43/3.90
% 27.43/3.90 Lemma 49: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 27.43/3.90 Proof:
% 27.43/3.90 join(complement(X), complement(Y))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 join(complement(Y), complement(X))
% 27.43/3.90 = { by lemma 47 R->L }
% 27.43/3.90 meet(top, join(complement(Y), complement(X)))
% 27.43/3.90 = { by lemma 48 R->L }
% 27.43/3.90 complement(join(complement(top), meet(Y, X)))
% 27.43/3.90 = { by lemma 18 }
% 27.43/3.90 complement(join(zero, meet(Y, X)))
% 27.43/3.90 = { by lemma 29 }
% 27.43/3.90 complement(meet(Y, X))
% 27.43/3.90 = { by lemma 41 R->L }
% 27.43/3.90 complement(meet(X, Y))
% 27.43/3.90
% 27.43/3.90 Lemma 50: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 27.43/3.90 Proof:
% 27.43/3.90 complement(meet(X, complement(Y)))
% 27.43/3.90 = { by lemma 41 }
% 27.43/3.90 complement(meet(complement(Y), X))
% 27.43/3.90 = { by lemma 29 R->L }
% 27.43/3.90 complement(meet(join(zero, complement(Y)), X))
% 27.43/3.90 = { by lemma 49 R->L }
% 27.43/3.90 join(complement(join(zero, complement(Y))), complement(X))
% 27.43/3.90 = { by lemma 45 }
% 27.43/3.90 join(meet(Y, top), complement(X))
% 27.43/3.90 = { by lemma 46 }
% 27.43/3.90 join(Y, complement(X))
% 27.43/3.90
% 27.43/3.90 Lemma 51: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 27.43/3.90 Proof:
% 27.43/3.90 complement(join(X, complement(Y)))
% 27.43/3.90 = { by lemma 29 R->L }
% 27.43/3.90 complement(join(zero, join(X, complement(Y))))
% 27.43/3.90 = { by lemma 50 R->L }
% 27.43/3.90 complement(join(zero, complement(meet(Y, complement(X)))))
% 27.43/3.90 = { by lemma 45 }
% 27.43/3.90 meet(meet(Y, complement(X)), top)
% 27.43/3.90 = { by lemma 46 }
% 27.43/3.90 meet(Y, complement(X))
% 27.43/3.90
% 27.43/3.90 Lemma 52: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 27.43/3.90 Proof:
% 27.43/3.90 complement(join(complement(X), Y))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 complement(join(Y, complement(X)))
% 27.43/3.90 = { by lemma 51 }
% 27.43/3.90 meet(X, complement(Y))
% 27.43/3.90
% 27.43/3.90 Lemma 53: join(X, converse(complement(converse(X)))) = top.
% 27.43/3.90 Proof:
% 27.43/3.90 join(X, converse(complement(converse(X))))
% 27.43/3.90 = { by lemma 37 R->L }
% 27.43/3.90 converse(join(converse(X), complement(converse(X))))
% 27.43/3.90 = { by axiom 6 (def_top) R->L }
% 27.43/3.90 converse(top)
% 27.43/3.90 = { by lemma 38 }
% 27.43/3.90 top
% 27.43/3.90
% 27.43/3.90 Lemma 54: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 27.43/3.90 Proof:
% 27.43/3.90 join(X, composition(Y, X))
% 27.43/3.90 = { by lemma 23 R->L }
% 27.43/3.90 join(composition(one, X), composition(Y, X))
% 27.43/3.90 = { by axiom 12 (composition_distributivity) R->L }
% 27.43/3.90 composition(join(one, Y), X)
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) }
% 27.43/3.90 composition(join(Y, one), X)
% 27.43/3.90
% 27.43/3.90 Lemma 55: composition(top, zero) = zero.
% 27.43/3.90 Proof:
% 27.43/3.90 composition(top, zero)
% 27.43/3.90 = { by lemma 18 R->L }
% 27.43/3.90 composition(top, complement(top))
% 27.43/3.90 = { by lemma 35 R->L }
% 27.43/3.90 composition(join(X, top), complement(top))
% 27.43/3.90 = { by lemma 34 R->L }
% 27.43/3.90 composition(join(top, one), complement(top))
% 27.43/3.90 = { by lemma 38 R->L }
% 27.43/3.90 composition(join(converse(top), one), complement(top))
% 27.43/3.90 = { by lemma 54 R->L }
% 27.43/3.90 join(complement(top), composition(converse(top), complement(top)))
% 27.43/3.90 = { by lemma 35 R->L }
% 27.43/3.90 join(complement(top), composition(converse(top), complement(join(Y, top))))
% 27.43/3.90 = { by lemma 34 R->L }
% 27.43/3.90 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 27.43/3.90 = { by lemma 54 }
% 27.43/3.90 join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
% 27.43/3.90 = { by lemma 34 }
% 27.43/3.90 join(complement(top), composition(converse(top), complement(composition(join(Z, top), top))))
% 27.43/3.90 = { by lemma 35 }
% 27.43/3.90 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 27.43/3.90 = { by lemma 24 }
% 27.43/3.90 complement(top)
% 27.43/3.90 = { by lemma 18 }
% 27.43/3.90 zero
% 27.43/3.90
% 27.43/3.90 Lemma 56: composition(X, zero) = zero.
% 27.43/3.90 Proof:
% 27.43/3.90 composition(X, zero)
% 27.43/3.90 = { by lemma 29 R->L }
% 27.43/3.90 join(zero, composition(X, zero))
% 27.43/3.90 = { by lemma 55 R->L }
% 27.43/3.90 join(composition(top, zero), composition(X, zero))
% 27.43/3.90 = { by axiom 12 (composition_distributivity) R->L }
% 27.43/3.90 composition(join(top, X), zero)
% 27.43/3.90 = { by lemma 34 }
% 27.43/3.90 composition(join(Y, top), zero)
% 27.43/3.90 = { by lemma 35 }
% 27.43/3.90 composition(top, zero)
% 27.43/3.90 = { by lemma 55 }
% 27.43/3.90 zero
% 27.43/3.90
% 27.43/3.90 Lemma 57: composition(zero, X) = zero.
% 27.43/3.90 Proof:
% 27.43/3.90 composition(zero, X)
% 27.43/3.90 = { by lemma 39 R->L }
% 27.43/3.90 composition(converse(zero), X)
% 27.43/3.90 = { by lemma 21 R->L }
% 27.43/3.90 converse(composition(converse(X), zero))
% 27.43/3.90 = { by lemma 56 }
% 27.43/3.90 converse(zero)
% 27.43/3.90 = { by lemma 39 }
% 27.43/3.90 zero
% 27.43/3.90
% 27.43/3.90 Lemma 58: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 27.43/3.90 Proof:
% 27.43/3.90 join(complement(one), composition(converse(X), complement(X)))
% 27.43/3.90 = { by axiom 1 (composition_identity) R->L }
% 27.43/3.90 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 27.43/3.90 = { by lemma 24 }
% 27.43/3.90 complement(one)
% 27.43/3.90
% 27.43/3.90 Lemma 59: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 27.43/3.90 Proof:
% 27.43/3.90 join(Y, join(X, Z))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 join(join(X, Z), Y)
% 27.43/3.90 = { by axiom 10 (maddux2_join_associativity) R->L }
% 27.43/3.90 join(X, join(Z, Y))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) }
% 27.43/3.90 join(X, join(Y, Z))
% 27.43/3.90
% 27.43/3.90 Lemma 60: meet(X, complement(join(X, Y))) = zero.
% 27.43/3.90 Proof:
% 27.43/3.90 meet(X, complement(join(X, Y)))
% 27.43/3.90 = { by lemma 51 R->L }
% 27.43/3.90 complement(join(join(X, Y), complement(X)))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 complement(join(complement(X), join(X, Y)))
% 27.43/3.90 = { by lemma 59 }
% 27.43/3.90 complement(join(X, join(complement(X), Y)))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) }
% 27.43/3.90 complement(join(X, join(Y, complement(X))))
% 27.43/3.90 = { by lemma 32 }
% 27.43/3.90 complement(join(Y, top))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) }
% 27.43/3.90 complement(join(top, Y))
% 27.43/3.90 = { by lemma 34 }
% 27.43/3.90 complement(join(Z, top))
% 27.43/3.90 = { by lemma 35 }
% 27.43/3.90 complement(top)
% 27.43/3.90 = { by lemma 18 }
% 27.43/3.90 zero
% 27.43/3.90
% 27.43/3.90 Lemma 61: meet(X, complement(join(Y, X))) = zero.
% 27.43/3.90 Proof:
% 27.43/3.90 meet(X, complement(join(Y, X)))
% 27.43/3.90 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.43/3.90 meet(X, complement(join(X, Y)))
% 27.43/3.90 = { by lemma 60 }
% 27.43/3.91 zero
% 27.43/3.91
% 27.43/3.91 Lemma 62: meet(one, composition(converse(complement(X)), X)) = zero.
% 27.43/3.91 Proof:
% 27.43/3.91 meet(one, composition(converse(complement(X)), X))
% 27.43/3.91 = { by lemma 41 }
% 27.43/3.91 meet(composition(converse(complement(X)), X), one)
% 27.43/3.91 = { by lemma 40 R->L }
% 27.43/3.91 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 27.43/3.91 = { by lemma 58 R->L }
% 27.43/3.91 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 27.43/3.91 = { by lemma 45 }
% 27.43/3.91 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 27.43/3.91 = { by lemma 29 }
% 27.43/3.91 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), meet(X, top)))))
% 27.43/3.91 = { by lemma 46 }
% 27.43/3.91 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 27.43/3.91 = { by lemma 61 }
% 27.43/3.91 zero
% 27.43/3.91
% 27.43/3.91 Lemma 63: converse(complement(converse(X))) = complement(X).
% 27.43/3.91 Proof:
% 27.43/3.91 converse(complement(converse(X)))
% 27.43/3.91 = { by lemma 40 R->L }
% 27.43/3.91 converse(complement(converse(complement(complement(X)))))
% 27.43/3.91 = { by lemma 44 R->L }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), converse(complement(converse(complement(complement(X)))))))
% 27.43/3.91 = { by lemma 40 R->L }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), complement(complement(converse(complement(converse(complement(complement(X)))))))))
% 27.43/3.91 = { by lemma 52 R->L }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X)))))))))
% 27.43/3.91 = { by lemma 47 R->L }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(top, join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))))))
% 27.43/3.91 = { by lemma 53 R->L }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), converse(complement(converse(complement(complement(X)))))), join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))))))
% 27.43/3.91 = { by lemma 41 }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))), join(complement(complement(X)), converse(complement(converse(complement(complement(X)))))))))
% 27.43/3.91 = { by lemma 40 R->L }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))), join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))
% 27.43/3.91 = { by lemma 50 R->L }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(complement(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X))))), join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))
% 27.43/3.91 = { by lemma 41 }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))), complement(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X))))))))
% 27.43/3.91 = { by lemma 51 R->L }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X)))), complement(join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))))
% 27.43/3.91 = { by lemma 51 }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X)))), meet(complement(converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))))))
% 27.43/3.91 = { by lemma 44 }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(complement(complement(X))))))
% 27.43/3.91 = { by lemma 40 }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))
% 27.43/3.91 = { by lemma 40 }
% 27.43/3.91 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(X))
% 27.43/3.91 = { by lemma 40 }
% 27.43/3.91 join(meet(X, converse(complement(converse(X)))), complement(X))
% 27.43/3.91 = { by lemma 28 R->L }
% 27.43/3.91 join(join(meet(X, converse(complement(converse(X)))), zero), complement(X))
% 27.43/3.91 = { by lemma 57 R->L }
% 27.43/3.91 join(join(meet(X, converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 27.43/3.91 = { by lemma 23 R->L }
% 27.43/3.91 join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 27.43/3.91 = { by lemma 62 R->L }
% 27.43/3.91 join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 27.43/3.91 = { by axiom 17 (dedekind_law) }
% 27.43/3.91 join(composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))
% 27.43/3.91 = { by lemma 62 }
% 27.43/3.91 join(composition(zero, meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))
% 27.43/3.91 = { by lemma 57 }
% 27.43/3.91 join(zero, complement(X))
% 27.43/3.91 = { by lemma 29 }
% 27.43/3.91 complement(X)
% 27.43/3.91
% 27.43/3.91 Lemma 64: complement(converse(X)) = converse(complement(X)).
% 27.43/3.91 Proof:
% 27.43/3.91 complement(converse(X))
% 27.43/3.91 = { by axiom 4 (converse_idempotence) R->L }
% 27.43/3.91 converse(converse(complement(converse(X))))
% 27.43/3.91 = { by lemma 63 }
% 27.43/3.91 converse(complement(X))
% 27.43/3.91
% 27.43/3.91 Lemma 65: meet(X, X) = X.
% 27.43/3.91 Proof:
% 27.43/3.91 meet(X, X)
% 27.43/3.91 = { by lemma 29 R->L }
% 27.43/3.91 join(zero, meet(X, X))
% 27.43/3.91 = { by lemma 20 }
% 27.43/3.91 X
% 27.43/3.91
% 27.43/3.91 Lemma 66: meet(X, zero) = zero.
% 27.43/3.91 Proof:
% 27.43/3.91 meet(X, zero)
% 27.43/3.91 = { by lemma 41 }
% 27.43/3.91 meet(zero, X)
% 27.43/3.91 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.77/3.91 complement(join(complement(zero), complement(X)))
% 27.77/3.91 = { by lemma 30 }
% 27.77/3.91 complement(join(top, complement(X)))
% 27.77/3.91 = { by lemma 33 }
% 27.77/3.91 complement(top)
% 27.77/3.91 = { by lemma 18 }
% 27.77/3.91 zero
% 27.77/3.91
% 27.77/3.91 Lemma 67: meet(zero, X) = zero.
% 27.77/3.91 Proof:
% 27.77/3.91 meet(zero, X)
% 27.77/3.91 = { by lemma 41 }
% 27.77/3.91 meet(X, zero)
% 27.77/3.91 = { by lemma 66 }
% 27.77/3.91 zero
% 27.77/3.91
% 27.77/3.91 Lemma 68: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 27.77/3.91 Proof:
% 27.77/3.91 complement(meet(complement(X), Y))
% 27.77/3.91 = { by lemma 41 }
% 27.77/3.91 complement(meet(Y, complement(X)))
% 27.77/3.91 = { by lemma 50 }
% 27.77/3.91 join(X, complement(Y))
% 27.77/3.91
% 27.77/3.91 Lemma 69: join(X, complement(meet(X, Y))) = top.
% 27.77/3.91 Proof:
% 27.77/3.91 join(X, complement(meet(X, Y)))
% 27.77/3.91 = { by lemma 41 }
% 27.77/3.91 join(X, complement(meet(Y, X)))
% 27.77/3.91 = { by lemma 49 R->L }
% 27.77/3.91 join(X, join(complement(Y), complement(X)))
% 27.77/3.91 = { by lemma 32 }
% 27.77/3.91 join(complement(Y), top)
% 27.77/3.91 = { by lemma 35 }
% 27.77/3.91 top
% 27.77/3.91
% 27.77/3.91 Lemma 70: meet(X, join(X, complement(Y))) = X.
% 27.77/3.91 Proof:
% 27.77/3.91 meet(X, join(X, complement(Y)))
% 27.77/3.91 = { by lemma 28 R->L }
% 27.77/3.91 join(meet(X, join(X, complement(Y))), zero)
% 27.77/3.91 = { by lemma 18 R->L }
% 27.77/3.91 join(meet(X, join(X, complement(Y))), complement(top))
% 27.77/3.91 = { by lemma 68 R->L }
% 27.77/3.91 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 27.77/3.91 = { by lemma 69 R->L }
% 27.77/3.91 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 27.77/3.91 = { by lemma 19 }
% 27.77/3.91 X
% 27.77/3.91
% 27.77/3.91 Lemma 71: join(X, meet(X, Y)) = X.
% 27.77/3.91 Proof:
% 27.77/3.91 join(X, meet(X, Y))
% 27.77/3.91 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.77/3.91 join(X, complement(join(complement(X), complement(Y))))
% 27.77/3.91 = { by lemma 68 R->L }
% 27.77/3.91 complement(meet(complement(X), join(complement(X), complement(Y))))
% 27.77/3.91 = { by lemma 70 }
% 27.77/3.91 complement(complement(X))
% 27.77/3.91 = { by lemma 40 }
% 27.77/3.91 X
% 27.77/3.91
% 27.77/3.91 Lemma 72: join(X, meet(Y, X)) = X.
% 27.77/3.91 Proof:
% 27.77/3.91 join(X, meet(Y, X))
% 27.77/3.91 = { by lemma 41 }
% 27.77/3.91 join(X, meet(X, Y))
% 27.77/3.91 = { by lemma 71 }
% 27.77/3.91 X
% 27.77/3.91
% 27.77/3.91 Lemma 73: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 27.77/3.91 Proof:
% 27.77/3.91 converse(composition(X, converse(Y)))
% 27.77/3.91 = { by axiom 7 (converse_multiplicativity) }
% 27.77/3.91 composition(converse(converse(Y)), converse(X))
% 27.77/3.91 = { by axiom 4 (converse_idempotence) }
% 27.77/3.91 composition(Y, converse(X))
% 27.77/3.91
% 27.77/3.91 Lemma 74: join(X, composition(X, Y)) = composition(X, join(Y, one)).
% 27.77/3.91 Proof:
% 27.77/3.91 join(X, composition(X, Y))
% 27.77/3.91 = { by axiom 4 (converse_idempotence) R->L }
% 27.77/3.91 join(X, composition(X, converse(converse(Y))))
% 27.77/3.91 = { by lemma 73 R->L }
% 27.77/3.91 join(X, converse(composition(converse(Y), converse(X))))
% 27.77/3.91 = { by lemma 37 R->L }
% 27.77/3.91 converse(join(converse(X), composition(converse(Y), converse(X))))
% 27.77/3.91 = { by lemma 54 }
% 27.77/3.91 converse(composition(join(converse(Y), one), converse(X)))
% 27.77/3.91 = { by lemma 73 }
% 27.77/3.91 composition(X, converse(join(converse(Y), one)))
% 27.77/3.91 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.91 composition(X, converse(join(one, converse(Y))))
% 27.77/3.91 = { by axiom 9 (converse_additivity) }
% 27.77/3.91 composition(X, join(converse(one), converse(converse(Y))))
% 27.77/3.91 = { by lemma 31 }
% 27.77/3.91 composition(X, join(one, converse(converse(Y))))
% 27.77/3.91 = { by axiom 4 (converse_idempotence) }
% 27.77/3.91 composition(X, join(one, Y))
% 27.77/3.91 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.91 composition(X, join(Y, one))
% 27.77/3.91
% 27.77/3.91 Lemma 75: meet(X, join(X, Y)) = X.
% 27.77/3.91 Proof:
% 27.77/3.91 meet(X, join(X, Y))
% 27.77/3.91 = { by lemma 46 R->L }
% 27.77/3.91 meet(X, join(X, meet(Y, top)))
% 27.77/3.91 = { by lemma 45 R->L }
% 27.77/3.91 meet(X, join(X, complement(join(zero, complement(Y)))))
% 27.77/3.91 = { by lemma 70 }
% 27.77/3.91 X
% 27.77/3.91
% 27.77/3.91 Lemma 76: meet(X, join(Y, X)) = X.
% 27.77/3.91 Proof:
% 27.77/3.91 meet(X, join(Y, X))
% 27.77/3.91 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.91 meet(X, join(X, Y))
% 27.77/3.91 = { by lemma 75 }
% 27.77/3.91 X
% 27.77/3.91
% 27.77/3.91 Lemma 77: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 27.77/3.91 Proof:
% 27.77/3.91 meet(Y, meet(X, Z))
% 27.77/3.91 = { by lemma 41 }
% 27.77/3.91 meet(Y, meet(Z, X))
% 27.77/3.91 = { by lemma 46 R->L }
% 27.77/3.91 meet(meet(Y, meet(Z, X)), top)
% 27.77/3.91 = { by lemma 45 R->L }
% 27.77/3.91 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 27.77/3.91 = { by lemma 41 }
% 27.77/3.91 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 27.77/3.91 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.77/3.91 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 27.77/3.91 = { by lemma 50 }
% 27.77/3.91 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 27.77/3.91 = { by axiom 10 (maddux2_join_associativity) R->L }
% 27.77/3.91 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 27.77/3.91 = { by lemma 49 }
% 27.77/3.91 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 27.77/3.91 = { by lemma 49 }
% 27.77/3.91 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 27.77/3.91 = { by lemma 41 R->L }
% 27.77/3.91 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 27.77/3.91 = { by lemma 45 }
% 27.77/3.91 meet(meet(X, meet(Y, Z)), top)
% 27.77/3.91 = { by lemma 46 }
% 27.77/3.91 meet(X, meet(Y, Z))
% 27.77/3.91
% 27.77/3.92 Lemma 78: meet(Z, meet(X, Y)) = meet(X, meet(Y, Z)).
% 27.77/3.92 Proof:
% 27.77/3.92 meet(Z, meet(X, Y))
% 27.77/3.92 = { by lemma 77 R->L }
% 27.77/3.92 meet(X, meet(Z, Y))
% 27.77/3.92 = { by lemma 41 R->L }
% 27.77/3.92 meet(X, meet(Y, Z))
% 27.77/3.92
% 27.77/3.92 Lemma 79: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 27.77/3.92 Proof:
% 27.77/3.92 meet(complement(X), complement(Y))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 meet(complement(Y), complement(X))
% 27.77/3.92 = { by lemma 29 R->L }
% 27.77/3.92 meet(join(zero, complement(Y)), complement(X))
% 27.77/3.92 = { by lemma 51 R->L }
% 27.77/3.92 complement(join(X, complement(join(zero, complement(Y)))))
% 27.77/3.92 = { by lemma 45 }
% 27.77/3.92 complement(join(X, meet(Y, top)))
% 27.77/3.92 = { by lemma 46 }
% 27.77/3.92 complement(join(X, Y))
% 27.77/3.92
% 27.77/3.92 Lemma 80: composition(converse(x0), complement(x0)) = zero.
% 27.77/3.92 Proof:
% 27.77/3.92 composition(converse(x0), complement(x0))
% 27.77/3.92 = { by lemma 29 R->L }
% 27.77/3.92 join(zero, composition(converse(x0), complement(x0)))
% 27.77/3.92 = { by lemma 18 R->L }
% 27.77/3.92 join(complement(top), composition(converse(x0), complement(x0)))
% 27.77/3.92 = { by axiom 2 (goals) R->L }
% 27.77/3.92 join(complement(top), composition(converse(x0), complement(composition(x0, top))))
% 27.77/3.92 = { by lemma 24 }
% 27.77/3.92 complement(top)
% 27.77/3.92 = { by lemma 18 }
% 27.77/3.92 zero
% 27.77/3.92
% 27.77/3.92 Lemma 81: composition(converse(complement(x0)), x0) = zero.
% 27.77/3.92 Proof:
% 27.77/3.92 composition(converse(complement(x0)), x0)
% 27.77/3.92 = { by lemma 21 R->L }
% 27.77/3.92 converse(composition(converse(x0), complement(x0)))
% 27.77/3.92 = { by lemma 80 }
% 27.77/3.92 converse(zero)
% 27.77/3.92 = { by lemma 39 }
% 27.77/3.92 zero
% 27.77/3.92
% 27.77/3.92 Lemma 82: converse(complement(join(X, converse(Y)))) = complement(join(Y, converse(X))).
% 27.77/3.92 Proof:
% 27.77/3.92 converse(complement(join(X, converse(Y))))
% 27.77/3.92 = { by lemma 36 R->L }
% 27.77/3.92 converse(complement(converse(join(Y, converse(X)))))
% 27.77/3.92 = { by lemma 63 }
% 27.77/3.92 complement(join(Y, converse(X)))
% 27.77/3.92
% 27.77/3.92 Lemma 83: converse(meet(X, converse(Y))) = meet(Y, converse(X)).
% 27.77/3.92 Proof:
% 27.77/3.92 converse(meet(X, converse(Y)))
% 27.77/3.92 = { by lemma 40 R->L }
% 27.77/3.92 converse(complement(complement(meet(X, converse(Y)))))
% 27.77/3.92 = { by lemma 49 R->L }
% 27.77/3.92 converse(complement(join(complement(X), complement(converse(Y)))))
% 27.77/3.92 = { by lemma 64 }
% 27.77/3.92 converse(complement(join(complement(X), converse(complement(Y)))))
% 27.77/3.92 = { by lemma 82 }
% 27.77/3.92 complement(join(complement(Y), converse(complement(X))))
% 27.77/3.92 = { by lemma 52 }
% 27.77/3.92 meet(Y, complement(converse(complement(X))))
% 27.77/3.92 = { by lemma 64 }
% 27.77/3.92 meet(Y, converse(complement(complement(X))))
% 27.77/3.92 = { by lemma 40 }
% 27.77/3.92 meet(Y, converse(X))
% 27.77/3.92
% 27.77/3.92 Lemma 84: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 27.77/3.92 Proof:
% 27.77/3.92 join(complement(X), meet(X, Y))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 join(complement(X), meet(Y, X))
% 27.77/3.92 = { by lemma 71 R->L }
% 27.77/3.92 join(join(complement(X), meet(complement(X), Y)), meet(Y, X))
% 27.77/3.92 = { by axiom 10 (maddux2_join_associativity) R->L }
% 27.77/3.92 join(complement(X), join(meet(complement(X), Y), meet(Y, X)))
% 27.77/3.92 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.92 join(complement(X), join(meet(Y, X), meet(complement(X), Y)))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 join(complement(X), join(meet(Y, X), meet(Y, complement(X))))
% 27.77/3.92 = { by lemma 42 }
% 27.77/3.92 join(complement(X), Y)
% 27.77/3.92 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.92 join(Y, complement(X))
% 27.77/3.92
% 27.77/3.92 Lemma 85: join(complement(X), meet(Y, X)) = join(Y, complement(X)).
% 27.77/3.92 Proof:
% 27.77/3.92 join(complement(X), meet(Y, X))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 join(complement(X), meet(X, Y))
% 27.77/3.92 = { by lemma 84 }
% 27.77/3.92 join(Y, complement(X))
% 27.77/3.92
% 27.77/3.92 Lemma 86: join(complement(one), converse(complement(one))) = complement(one).
% 27.77/3.92 Proof:
% 27.77/3.92 join(complement(one), converse(complement(one)))
% 27.77/3.92 = { by lemma 46 R->L }
% 27.77/3.92 join(complement(one), converse(meet(complement(one), top)))
% 27.77/3.92 = { by lemma 53 R->L }
% 27.77/3.92 join(complement(one), converse(meet(complement(one), join(one, converse(complement(converse(one)))))))
% 27.77/3.92 = { by lemma 31 }
% 27.77/3.92 join(complement(one), converse(meet(complement(one), join(one, converse(complement(one))))))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 join(complement(one), converse(meet(join(one, converse(complement(one))), complement(one))))
% 27.77/3.92 = { by lemma 51 R->L }
% 27.77/3.92 join(complement(one), converse(complement(join(one, complement(join(one, converse(complement(one))))))))
% 27.77/3.92 = { by lemma 79 R->L }
% 27.77/3.92 join(complement(one), converse(meet(complement(one), complement(complement(join(one, converse(complement(one))))))))
% 27.77/3.92 = { by lemma 79 R->L }
% 27.77/3.92 join(complement(one), converse(meet(complement(one), complement(meet(complement(one), complement(converse(complement(one))))))))
% 27.77/3.92 = { by lemma 49 R->L }
% 27.77/3.92 join(complement(one), converse(meet(complement(one), join(complement(complement(one)), complement(complement(converse(complement(one))))))))
% 27.77/3.92 = { by lemma 48 R->L }
% 27.77/3.92 join(complement(one), converse(complement(join(complement(complement(one)), meet(complement(one), complement(converse(complement(one))))))))
% 27.77/3.92 = { by lemma 84 }
% 27.77/3.92 join(complement(one), converse(complement(join(complement(converse(complement(one))), complement(complement(one))))))
% 27.77/3.92 = { by lemma 51 }
% 27.77/3.92 join(complement(one), converse(meet(complement(one), complement(complement(converse(complement(one)))))))
% 27.77/3.92 = { by lemma 79 }
% 27.77/3.92 join(complement(one), converse(complement(join(one, complement(converse(complement(one)))))))
% 27.77/3.92 = { by lemma 51 }
% 27.77/3.92 join(complement(one), converse(meet(converse(complement(one)), complement(one))))
% 27.77/3.92 = { by lemma 37 R->L }
% 27.77/3.92 converse(join(converse(complement(one)), meet(converse(complement(one)), complement(one))))
% 27.77/3.92 = { by lemma 71 }
% 27.77/3.92 converse(converse(complement(one)))
% 27.77/3.92 = { by axiom 4 (converse_idempotence) }
% 27.77/3.92 complement(one)
% 27.77/3.92
% 27.77/3.92 Lemma 87: join(meet(X, Y), complement(X)) = join(Y, complement(X)).
% 27.77/3.92 Proof:
% 27.77/3.92 join(meet(X, Y), complement(X))
% 27.77/3.92 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.92 join(complement(X), meet(X, Y))
% 27.77/3.92 = { by lemma 84 }
% 27.77/3.92 join(Y, complement(X))
% 27.77/3.92
% 27.77/3.92 Lemma 88: meet(meet(X, one), converse(Y)) = meet(X, converse(meet(Y, one))).
% 27.77/3.92 Proof:
% 27.77/3.92 meet(meet(X, one), converse(Y))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 meet(converse(Y), meet(X, one))
% 27.77/3.92 = { by lemma 78 }
% 27.77/3.92 meet(X, meet(one, converse(Y)))
% 27.77/3.92 = { by lemma 40 R->L }
% 27.77/3.92 meet(X, meet(one, converse(complement(complement(Y)))))
% 27.77/3.92 = { by lemma 64 R->L }
% 27.77/3.92 meet(X, meet(one, complement(converse(complement(Y)))))
% 27.77/3.92 = { by lemma 52 R->L }
% 27.77/3.92 meet(X, complement(join(complement(one), converse(complement(Y)))))
% 27.77/3.92 = { by lemma 82 R->L }
% 27.77/3.92 meet(X, converse(complement(join(complement(Y), converse(complement(one))))))
% 27.77/3.92 = { by lemma 86 R->L }
% 27.77/3.92 meet(X, converse(complement(join(complement(Y), converse(join(complement(one), converse(complement(one))))))))
% 27.77/3.92 = { by lemma 36 }
% 27.77/3.92 meet(X, converse(complement(join(complement(Y), join(complement(one), converse(complement(one)))))))
% 27.77/3.92 = { by lemma 86 }
% 27.77/3.92 meet(X, converse(complement(join(complement(Y), complement(one)))))
% 27.77/3.92 = { by lemma 51 }
% 27.77/3.92 meet(X, converse(meet(one, complement(complement(Y)))))
% 27.77/3.92 = { by lemma 40 }
% 27.77/3.92 meet(X, converse(meet(one, Y)))
% 27.77/3.92 = { by lemma 41 R->L }
% 27.77/3.92 meet(X, converse(meet(Y, one)))
% 27.77/3.92
% 27.77/3.92 Lemma 89: meet(meet(x0, X), composition(complement(x0), Y)) = zero.
% 27.77/3.92 Proof:
% 27.77/3.92 meet(meet(x0, X), composition(complement(x0), Y))
% 27.77/3.92 = { by lemma 43 R->L }
% 27.77/3.92 meet(meet(x0, X), join(meet(x0, composition(complement(x0), Y)), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 meet(meet(x0, X), join(meet(composition(complement(x0), Y), x0), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 28 R->L }
% 27.77/3.92 meet(meet(x0, X), join(join(meet(composition(complement(x0), Y), x0), zero), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 67 R->L }
% 27.77/3.92 meet(meet(x0, X), join(join(meet(composition(complement(x0), Y), x0), meet(zero, x0)), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 56 R->L }
% 27.77/3.92 meet(meet(x0, X), join(join(meet(composition(complement(x0), Y), x0), meet(composition(complement(x0), zero), x0)), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 66 R->L }
% 27.77/3.92 meet(meet(x0, X), join(join(meet(composition(complement(x0), Y), x0), meet(composition(complement(x0), meet(Y, zero)), x0)), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 81 R->L }
% 27.77/3.92 meet(meet(x0, X), join(join(meet(composition(complement(x0), Y), x0), meet(composition(complement(x0), meet(Y, composition(converse(complement(x0)), x0))), x0)), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by axiom 15 (modular_law_1) }
% 27.77/3.92 meet(meet(x0, X), join(meet(composition(complement(x0), meet(Y, composition(converse(complement(x0)), x0))), x0), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 81 }
% 27.77/3.92 meet(meet(x0, X), join(meet(composition(complement(x0), meet(Y, zero)), x0), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 66 }
% 27.77/3.92 meet(meet(x0, X), join(meet(composition(complement(x0), zero), x0), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 56 }
% 27.77/3.92 meet(meet(x0, X), join(meet(zero, x0), meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 67 }
% 27.77/3.92 meet(meet(x0, X), join(zero, meet(composition(complement(x0), Y), complement(x0))))
% 27.77/3.92 = { by lemma 29 }
% 27.77/3.92 meet(meet(x0, X), meet(composition(complement(x0), Y), complement(x0)))
% 27.77/3.92 = { by lemma 77 }
% 27.77/3.92 meet(composition(complement(x0), Y), meet(meet(x0, X), complement(x0)))
% 27.77/3.92 = { by lemma 51 R->L }
% 27.77/3.92 meet(composition(complement(x0), Y), complement(join(x0, complement(meet(x0, X)))))
% 27.77/3.92 = { by lemma 69 }
% 27.77/3.92 meet(composition(complement(x0), Y), complement(top))
% 27.77/3.92 = { by lemma 18 }
% 27.77/3.92 meet(composition(complement(x0), Y), zero)
% 27.77/3.92 = { by lemma 66 }
% 27.77/3.92 zero
% 27.77/3.92
% 27.77/3.92 Lemma 90: meet(complement(x0), composition(meet(x0, X), Y)) = zero.
% 27.77/3.92 Proof:
% 27.77/3.92 meet(complement(x0), composition(meet(x0, X), Y))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 meet(composition(meet(x0, X), Y), complement(x0))
% 27.77/3.92 = { by lemma 28 R->L }
% 27.77/3.92 join(meet(composition(meet(x0, X), Y), complement(x0)), zero)
% 27.77/3.92 = { by lemma 67 R->L }
% 27.77/3.92 join(meet(composition(meet(x0, X), Y), complement(x0)), meet(zero, complement(x0)))
% 27.77/3.92 = { by lemma 57 R->L }
% 27.77/3.92 join(meet(composition(meet(x0, X), Y), complement(x0)), meet(composition(zero, Y), complement(x0)))
% 27.77/3.92 = { by lemma 89 R->L }
% 27.77/3.92 join(meet(composition(meet(x0, X), Y), complement(x0)), meet(composition(meet(meet(x0, X), composition(complement(x0), converse(Y))), Y), complement(x0)))
% 27.77/3.92 = { by axiom 16 (modular_law_2) }
% 27.77/3.92 meet(composition(meet(meet(x0, X), composition(complement(x0), converse(Y))), Y), complement(x0))
% 27.77/3.92 = { by lemma 89 }
% 27.77/3.92 meet(composition(zero, Y), complement(x0))
% 27.77/3.92 = { by lemma 57 }
% 27.77/3.92 meet(zero, complement(x0))
% 27.77/3.92 = { by lemma 67 }
% 27.77/3.92 zero
% 27.77/3.92
% 27.77/3.92 Lemma 91: meet(meet(X, one), composition(Y, converse(complement(Y)))) = zero.
% 27.77/3.92 Proof:
% 27.77/3.92 meet(meet(X, one), composition(Y, converse(complement(Y))))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 meet(composition(Y, converse(complement(Y))), meet(X, one))
% 27.77/3.92 = { by lemma 64 R->L }
% 27.77/3.92 meet(composition(Y, complement(converse(Y))), meet(X, one))
% 27.77/3.92 = { by lemma 78 }
% 27.77/3.92 meet(X, meet(one, composition(Y, complement(converse(Y)))))
% 27.77/3.92 = { by lemma 41 }
% 27.77/3.92 meet(X, meet(composition(Y, complement(converse(Y))), one))
% 27.77/3.92 = { by lemma 40 R->L }
% 27.77/3.92 meet(X, meet(composition(Y, complement(converse(Y))), complement(complement(one))))
% 27.77/3.92 = { by lemma 58 R->L }
% 27.77/3.92 meet(X, meet(composition(Y, complement(converse(Y))), complement(join(complement(one), composition(converse(converse(Y)), complement(converse(Y)))))))
% 27.77/3.92 = { by axiom 4 (converse_idempotence) }
% 27.77/3.92 meet(X, meet(composition(Y, complement(converse(Y))), complement(join(complement(one), composition(Y, complement(converse(Y)))))))
% 27.77/3.92 = { by lemma 61 }
% 27.77/3.92 meet(X, zero)
% 27.77/3.92 = { by lemma 66 }
% 27.77/3.92 zero
% 27.77/3.92
% 27.77/3.92 Lemma 92: composition(meet(x0, one), complement(x0)) = zero.
% 27.77/3.92 Proof:
% 27.77/3.92 composition(meet(x0, one), complement(x0))
% 27.77/3.92 = { by lemma 43 R->L }
% 27.77/3.92 join(meet(complement(x0), composition(meet(x0, one), complement(x0))), meet(composition(meet(x0, one), complement(x0)), complement(complement(x0))))
% 27.77/3.92 = { by lemma 90 }
% 27.77/3.92 join(zero, meet(composition(meet(x0, one), complement(x0)), complement(complement(x0))))
% 27.77/3.92 = { by lemma 29 }
% 27.77/3.92 meet(composition(meet(x0, one), complement(x0)), complement(complement(x0)))
% 27.77/3.93 = { by lemma 40 }
% 27.77/3.93 meet(composition(meet(x0, one), complement(x0)), x0)
% 27.77/3.93 = { by lemma 28 R->L }
% 27.77/3.93 join(meet(composition(meet(x0, one), complement(x0)), x0), zero)
% 27.77/3.93 = { by lemma 57 R->L }
% 27.77/3.93 join(meet(composition(meet(x0, one), complement(x0)), x0), composition(zero, meet(complement(x0), composition(converse(meet(x0, one)), x0))))
% 27.77/3.93 = { by lemma 91 R->L }
% 27.77/3.93 join(meet(composition(meet(x0, one), complement(x0)), x0), composition(meet(meet(x0, one), composition(x0, converse(complement(x0)))), meet(complement(x0), composition(converse(meet(x0, one)), x0))))
% 27.77/3.93 = { by axiom 17 (dedekind_law) }
% 27.77/3.93 composition(meet(meet(x0, one), composition(x0, converse(complement(x0)))), meet(complement(x0), composition(converse(meet(x0, one)), x0)))
% 27.77/3.93 = { by lemma 91 }
% 27.77/3.93 composition(zero, meet(complement(x0), composition(converse(meet(x0, one)), x0)))
% 27.77/3.93 = { by lemma 57 }
% 27.77/3.93 zero
% 27.77/3.93
% 27.77/3.93 Lemma 93: composition(join(x0, converse(meet(x0, one))), top) = x0.
% 27.77/3.93 Proof:
% 27.77/3.93 composition(join(x0, converse(meet(x0, one))), top)
% 27.77/3.93 = { by axiom 12 (composition_distributivity) }
% 27.77/3.93 join(composition(x0, top), composition(converse(meet(x0, one)), top))
% 27.77/3.93 = { by axiom 2 (goals) }
% 27.77/3.93 join(x0, composition(converse(meet(x0, one)), top))
% 27.77/3.93 = { by lemma 30 R->L }
% 27.77/3.93 join(x0, composition(converse(meet(x0, one)), complement(zero)))
% 27.77/3.93 = { by lemma 40 R->L }
% 27.77/3.93 join(complement(complement(x0)), composition(converse(meet(x0, one)), complement(zero)))
% 27.77/3.93 = { by lemma 92 R->L }
% 27.77/3.93 join(complement(complement(x0)), composition(converse(meet(x0, one)), complement(composition(meet(x0, one), complement(x0)))))
% 27.77/3.93 = { by lemma 24 }
% 27.77/3.93 complement(complement(x0))
% 27.77/3.93 = { by lemma 40 }
% 27.77/3.93 x0
% 27.77/3.93
% 27.77/3.93 Lemma 94: meet(converse(X), converse(join(X, Y))) = converse(X).
% 27.77/3.93 Proof:
% 27.77/3.93 meet(converse(X), converse(join(X, Y)))
% 27.77/3.93 = { by axiom 9 (converse_additivity) }
% 27.77/3.93 meet(converse(X), join(converse(X), converse(Y)))
% 27.77/3.93 = { by lemma 75 }
% 27.77/3.93 converse(X)
% 27.77/3.93
% 27.77/3.93 Lemma 95: meet(x0, converse(meet(x0, one))) = meet(x0, one).
% 27.77/3.93 Proof:
% 27.77/3.93 meet(x0, converse(meet(x0, one)))
% 27.77/3.93 = { by lemma 88 R->L }
% 27.77/3.93 meet(meet(x0, one), converse(x0))
% 27.77/3.93 = { by lemma 83 R->L }
% 27.77/3.93 converse(meet(x0, converse(meet(x0, one))))
% 27.77/3.93 = { by lemma 41 R->L }
% 27.77/3.93 converse(meet(converse(meet(x0, one)), x0))
% 27.77/3.93 = { by axiom 4 (converse_idempotence) R->L }
% 27.77/3.93 converse(meet(converse(meet(x0, one)), converse(converse(x0))))
% 27.77/3.93 = { by lemma 83 }
% 27.77/3.93 meet(converse(x0), converse(converse(meet(x0, one))))
% 27.77/3.93 = { by lemma 41 R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(x0))
% 27.77/3.93 = { by lemma 93 R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(composition(join(x0, converse(meet(x0, one))), top)))
% 27.77/3.93 = { by lemma 35 R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(composition(join(x0, converse(meet(x0, one))), join(one, top))))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(composition(join(x0, converse(meet(x0, one))), join(top, one))))
% 27.77/3.93 = { by lemma 74 R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(join(x0, converse(meet(x0, one))), composition(join(x0, converse(meet(x0, one))), top))))
% 27.77/3.93 = { by lemma 93 }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(join(x0, converse(meet(x0, one))), x0)))
% 27.77/3.93 = { by axiom 10 (maddux2_join_associativity) R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(x0, join(converse(meet(x0, one)), x0))))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(x0, join(x0, converse(meet(x0, one))))))
% 27.77/3.93 = { by axiom 10 (maddux2_join_associativity) }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(join(x0, x0), converse(meet(x0, one)))))
% 27.77/3.93 = { by lemma 65 R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(join(x0, meet(x0, x0)), converse(meet(x0, one)))))
% 27.77/3.93 = { by lemma 65 R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(join(meet(x0, x0), meet(x0, x0)), converse(meet(x0, one)))))
% 27.77/3.93 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(join(meet(x0, x0), complement(join(complement(x0), complement(x0)))), converse(meet(x0, one)))))
% 27.77/3.93 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(join(complement(join(complement(x0), complement(x0))), complement(join(complement(x0), complement(x0)))), converse(meet(x0, one)))))
% 27.77/3.93 = { by lemma 25 }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(complement(join(complement(x0), complement(x0))), converse(meet(x0, one)))))
% 27.77/3.93 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(meet(x0, x0), converse(meet(x0, one)))))
% 27.77/3.93 = { by lemma 65 }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(x0, converse(meet(x0, one)))))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.93 meet(converse(converse(meet(x0, one))), converse(join(converse(meet(x0, one)), x0)))
% 27.77/3.93 = { by lemma 94 }
% 27.77/3.93 converse(converse(meet(x0, one)))
% 27.77/3.93 = { by axiom 4 (converse_idempotence) }
% 27.77/3.93 meet(x0, one)
% 27.77/3.93
% 27.77/3.93 Lemma 96: join(X, composition(meet(Y, one), X)) = X.
% 27.77/3.93 Proof:
% 27.77/3.93 join(X, composition(meet(Y, one), X))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.93 join(composition(meet(Y, one), X), X)
% 27.77/3.93 = { by lemma 22 R->L }
% 27.77/3.93 join(composition(meet(Y, one), X), composition(converse(one), X))
% 27.77/3.93 = { by axiom 12 (composition_distributivity) R->L }
% 27.77/3.93 composition(join(meet(Y, one), converse(one)), X)
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.93 composition(join(converse(one), meet(Y, one)), X)
% 27.77/3.93 = { by lemma 31 }
% 27.77/3.93 composition(join(one, meet(Y, one)), X)
% 27.77/3.93 = { by lemma 72 }
% 27.77/3.93 composition(one, X)
% 27.77/3.93 = { by lemma 23 }
% 27.77/3.93 X
% 27.77/3.93
% 27.77/3.93 Lemma 97: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 27.77/3.93 Proof:
% 27.77/3.93 join(composition(X, Y), composition(X, Z))
% 27.77/3.93 = { by axiom 4 (converse_idempotence) R->L }
% 27.77/3.93 join(composition(X, Y), composition(converse(converse(X)), Z))
% 27.77/3.93 = { by axiom 4 (converse_idempotence) R->L }
% 27.77/3.93 join(converse(converse(composition(X, Y))), composition(converse(converse(X)), Z))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.93 join(composition(converse(converse(X)), Z), converse(converse(composition(X, Y))))
% 27.77/3.93 = { by lemma 21 R->L }
% 27.77/3.93 join(converse(composition(converse(Z), converse(X))), converse(converse(composition(X, Y))))
% 27.77/3.93 = { by axiom 9 (converse_additivity) R->L }
% 27.77/3.93 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.93 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 27.77/3.93 = { by axiom 7 (converse_multiplicativity) }
% 27.77/3.93 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 27.77/3.93 = { by axiom 12 (composition_distributivity) R->L }
% 27.77/3.93 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 27.77/3.93 = { by axiom 7 (converse_multiplicativity) }
% 27.77/3.93 composition(converse(converse(X)), converse(join(converse(Y), converse(Z))))
% 27.77/3.93 = { by lemma 36 }
% 27.77/3.93 composition(converse(converse(X)), join(Z, converse(converse(Y))))
% 27.77/3.93 = { by axiom 4 (converse_idempotence) }
% 27.77/3.93 composition(X, join(Z, converse(converse(Y))))
% 27.77/3.93 = { by axiom 4 (converse_idempotence) }
% 27.77/3.93 composition(X, join(Z, Y))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.93 composition(X, join(Y, Z))
% 27.77/3.93
% 27.77/3.93 Lemma 98: meet(converse(X), converse(meet(X, Y))) = converse(meet(X, Y)).
% 27.77/3.93 Proof:
% 27.77/3.93 meet(converse(X), converse(meet(X, Y)))
% 27.77/3.93 = { by lemma 41 }
% 27.77/3.93 meet(converse(meet(X, Y)), converse(X))
% 27.77/3.93 = { by lemma 19 R->L }
% 27.77/3.93 meet(converse(meet(X, Y)), converse(join(meet(X, Y), complement(join(complement(X), Y)))))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.93 meet(converse(meet(X, Y)), converse(join(meet(X, Y), complement(join(Y, complement(X))))))
% 27.77/3.93 = { by lemma 94 }
% 27.77/3.93 converse(meet(X, Y))
% 27.77/3.93
% 27.77/3.93 Lemma 99: composition(meet(x0, one), join(X, complement(x0))) = composition(meet(x0, one), X).
% 27.77/3.93 Proof:
% 27.77/3.93 composition(meet(x0, one), join(X, complement(x0)))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.93 composition(meet(x0, one), join(complement(x0), X))
% 27.77/3.93 = { by lemma 97 R->L }
% 27.77/3.93 join(composition(meet(x0, one), complement(x0)), composition(meet(x0, one), X))
% 27.77/3.93 = { by lemma 92 }
% 27.77/3.93 join(zero, composition(meet(x0, one), X))
% 27.77/3.93 = { by lemma 29 }
% 27.77/3.93 composition(meet(x0, one), X)
% 27.77/3.93
% 27.77/3.93 Lemma 100: meet(X, join(meet(X, Y), composition(meet(Z, one), Y))) = meet(X, Y).
% 27.77/3.93 Proof:
% 27.77/3.93 meet(X, join(meet(X, Y), composition(meet(Z, one), Y)))
% 27.77/3.93 = { by lemma 76 R->L }
% 27.77/3.93 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), join(Y, join(meet(X, Y), composition(meet(Z, one), Y)))))
% 27.77/3.93 = { by lemma 59 }
% 27.77/3.93 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), join(meet(X, Y), join(Y, composition(meet(Z, one), Y)))))
% 27.77/3.93 = { by lemma 96 }
% 27.77/3.93 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), join(meet(X, Y), Y)))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.93 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), join(Y, meet(X, Y))))
% 27.77/3.93 = { by lemma 72 }
% 27.77/3.93 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), Y))
% 27.77/3.93 = { by lemma 41 R->L }
% 27.77/3.93 meet(X, meet(Y, join(meet(X, Y), composition(meet(Z, one), Y))))
% 27.77/3.93 = { by lemma 78 R->L }
% 27.77/3.93 meet(join(meet(X, Y), composition(meet(Z, one), Y)), meet(X, Y))
% 27.77/3.93 = { by lemma 41 }
% 27.77/3.93 meet(meet(X, Y), join(meet(X, Y), composition(meet(Z, one), Y)))
% 27.77/3.93 = { by axiom 11 (maddux4_definiton_of_meet) }
% 27.77/3.93 complement(join(complement(meet(X, Y)), complement(join(meet(X, Y), composition(meet(Z, one), Y)))))
% 27.77/3.93 = { by lemma 29 R->L }
% 27.77/3.93 join(zero, complement(join(complement(meet(X, Y)), complement(join(meet(X, Y), composition(meet(Z, one), Y))))))
% 27.77/3.93 = { by lemma 60 R->L }
% 27.77/3.93 join(meet(meet(X, Y), complement(join(meet(X, Y), composition(meet(Z, one), Y)))), complement(join(complement(meet(X, Y)), complement(join(meet(X, Y), composition(meet(Z, one), Y))))))
% 27.77/3.93 = { by lemma 19 }
% 27.77/3.93 meet(X, Y)
% 27.77/3.93
% 27.77/3.93 Lemma 101: composition(converse(join(meet(x0, X), composition(meet(x0, one), X))), complement(x0)) = zero.
% 27.77/3.93 Proof:
% 27.77/3.93 composition(converse(join(meet(x0, X), composition(meet(x0, one), X))), complement(x0))
% 27.77/3.93 = { by lemma 43 R->L }
% 27.77/3.93 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), meet(join(meet(x0, X), composition(meet(x0, one), X)), complement(x0)))), complement(x0))
% 27.77/3.93 = { by lemma 51 R->L }
% 27.77/3.93 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(x0, complement(join(meet(x0, X), composition(meet(x0, one), X))))))), complement(x0))
% 27.77/3.93 = { by lemma 85 R->L }
% 27.77/3.93 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(join(meet(x0, X), composition(meet(x0, one), X))), meet(x0, join(meet(x0, X), composition(meet(x0, one), X))))))), complement(x0))
% 27.77/3.93 = { by lemma 100 }
% 27.77/3.93 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(join(meet(x0, X), composition(meet(x0, one), X))), meet(x0, X))))), complement(x0))
% 27.77/3.93 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.93 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(meet(x0, X), complement(join(meet(x0, X), composition(meet(x0, one), X))))))), complement(x0))
% 27.77/3.93 = { by lemma 68 R->L }
% 27.77/3.93 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(complement(meet(complement(meet(x0, X)), join(meet(x0, X), composition(meet(x0, one), X))))))), complement(x0))
% 27.77/3.93 = { by lemma 49 R->L }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(complement(meet(x0, X))), complement(join(meet(x0, X), composition(meet(x0, one), X))))))), complement(x0))
% 27.77/3.94 = { by lemma 79 R->L }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(complement(meet(x0, X))), meet(complement(meet(x0, X)), complement(composition(meet(x0, one), X))))))), complement(x0))
% 27.77/3.94 = { by lemma 84 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(composition(meet(x0, one), X)), complement(complement(meet(x0, X))))))), complement(x0))
% 27.77/3.94 = { by lemma 49 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(complement(meet(composition(meet(x0, one), X), complement(meet(x0, X))))))), complement(x0))
% 27.77/3.94 = { by lemma 50 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(meet(x0, X), complement(composition(meet(x0, one), X)))))), complement(x0))
% 27.77/3.94 = { by lemma 85 R->L }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(composition(meet(x0, one), X)), meet(meet(x0, X), composition(meet(x0, one), X)))))), complement(x0))
% 27.77/3.94 = { by lemma 41 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(composition(meet(x0, one), X)), meet(composition(meet(x0, one), X), meet(x0, X)))))), complement(x0))
% 27.77/3.94 = { by lemma 78 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(composition(meet(x0, one), X)), meet(x0, meet(X, composition(meet(x0, one), X))))))), complement(x0))
% 27.77/3.94 = { by lemma 41 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(composition(meet(x0, one), X)), meet(x0, meet(composition(meet(x0, one), X), X)))))), complement(x0))
% 27.77/3.94 = { by lemma 96 R->L }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(composition(meet(x0, one), X)), meet(x0, meet(composition(meet(x0, one), X), join(X, composition(meet(x0, one), X)))))))), complement(x0))
% 27.77/3.94 = { by lemma 76 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(complement(composition(meet(x0, one), X)), meet(x0, composition(meet(x0, one), X)))))), complement(x0))
% 27.77/3.94 = { by lemma 85 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), complement(join(x0, complement(composition(meet(x0, one), X)))))), complement(x0))
% 27.77/3.94 = { by lemma 51 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), meet(composition(meet(x0, one), X), complement(x0)))), complement(x0))
% 27.77/3.94 = { by lemma 41 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), meet(complement(x0), composition(meet(x0, one), X)))), complement(x0))
% 27.77/3.94 = { by lemma 90 }
% 27.77/3.94 composition(converse(join(meet(x0, join(meet(x0, X), composition(meet(x0, one), X))), zero)), complement(x0))
% 27.77/3.94 = { by lemma 28 }
% 27.77/3.94 composition(converse(meet(x0, join(meet(x0, X), composition(meet(x0, one), X)))), complement(x0))
% 27.77/3.94 = { by lemma 100 }
% 27.77/3.94 composition(converse(meet(x0, X)), complement(x0))
% 27.77/3.94 = { by lemma 29 R->L }
% 27.77/3.94 join(zero, composition(converse(meet(x0, X)), complement(x0)))
% 27.77/3.94 = { by lemma 80 R->L }
% 27.77/3.94 join(composition(converse(x0), complement(x0)), composition(converse(meet(x0, X)), complement(x0)))
% 27.77/3.94 = { by axiom 12 (composition_distributivity) R->L }
% 27.77/3.94 composition(join(converse(x0), converse(meet(x0, X))), complement(x0))
% 27.77/3.94 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.94 composition(join(converse(meet(x0, X)), converse(x0)), complement(x0))
% 27.77/3.94 = { by axiom 9 (converse_additivity) R->L }
% 27.77/3.94 composition(converse(join(meet(x0, X), x0)), complement(x0))
% 27.77/3.94 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.94 composition(converse(join(x0, meet(x0, X))), complement(x0))
% 27.77/3.94 = { by lemma 71 }
% 27.77/3.94 composition(converse(x0), complement(x0))
% 27.77/3.94 = { by lemma 80 }
% 27.77/3.94 zero
% 27.77/3.94
% 27.77/3.94 Goal 1 (goals_1): join(meet(x0, x1), composition(meet(x0, one), x1)) = composition(meet(x0, one), x1).
% 27.77/3.94 Proof:
% 27.77/3.94 join(meet(x0, x1), composition(meet(x0, one), x1))
% 27.77/3.94 = { by axiom 4 (converse_idempotence) R->L }
% 27.77/3.94 converse(converse(join(meet(x0, x1), composition(meet(x0, one), x1))))
% 27.77/3.94 = { by lemma 28 R->L }
% 27.77/3.94 converse(join(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), zero))
% 27.77/3.94 = { by lemma 101 R->L }
% 27.77/3.94 converse(join(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), complement(x0))))
% 27.77/3.94 = { by lemma 74 }
% 27.77/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), join(complement(x0), one)))
% 27.77/3.94 = { by axiom 3 (maddux1_join_commutativity) }
% 27.77/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), join(one, complement(x0))))
% 27.77/3.94 = { by lemma 87 R->L }
% 27.77/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), join(meet(x0, one), complement(x0))))
% 27.77/3.94 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 27.77/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), join(complement(x0), meet(x0, one))))
% 27.77/3.94 = { by lemma 97 R->L }
% 27.77/3.94 converse(join(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), complement(x0)), composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), meet(x0, one))))
% 27.77/3.94 = { by lemma 101 }
% 27.77/3.94 converse(join(zero, composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), meet(x0, one))))
% 27.77/3.94 = { by lemma 29 }
% 27.77/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), meet(x0, one)))
% 27.77/3.94 = { by lemma 95 R->L }
% 27.77/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), meet(x0, converse(meet(x0, one)))))
% 27.77/3.94 = { by lemma 88 R->L }
% 27.77/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), meet(meet(x0, one), converse(x0))))
% 27.77/3.94 = { by lemma 83 R->L }
% 27.77/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), converse(meet(x0, converse(meet(x0, one))))))
% 27.77/3.94 = { by lemma 98 R->L }
% 27.77/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), meet(converse(x0), converse(meet(x0, converse(meet(x0, one)))))))
% 28.02/3.94 = { by lemma 95 }
% 28.02/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), meet(converse(x0), converse(meet(x0, one)))))
% 28.02/3.94 = { by lemma 98 }
% 28.02/3.94 converse(composition(converse(join(meet(x0, x1), composition(meet(x0, one), x1))), converse(meet(x0, one))))
% 28.02/3.94 = { by axiom 7 (converse_multiplicativity) R->L }
% 28.02/3.94 converse(converse(composition(meet(x0, one), join(meet(x0, x1), composition(meet(x0, one), x1)))))
% 28.02/3.94 = { by lemma 99 R->L }
% 28.02/3.94 converse(converse(composition(meet(x0, one), join(join(meet(x0, x1), composition(meet(x0, one), x1)), complement(x0)))))
% 28.02/3.94 = { by lemma 84 R->L }
% 28.02/3.94 converse(converse(composition(meet(x0, one), join(complement(x0), meet(x0, join(meet(x0, x1), composition(meet(x0, one), x1)))))))
% 28.02/3.94 = { by lemma 100 }
% 28.02/3.94 converse(converse(composition(meet(x0, one), join(complement(x0), meet(x0, x1)))))
% 28.02/3.94 = { by axiom 3 (maddux1_join_commutativity) }
% 28.02/3.94 converse(converse(composition(meet(x0, one), join(meet(x0, x1), complement(x0)))))
% 28.02/3.94 = { by lemma 87 }
% 28.02/3.94 converse(converse(composition(meet(x0, one), join(x1, complement(x0)))))
% 28.02/3.94 = { by lemma 99 }
% 28.02/3.94 converse(converse(composition(meet(x0, one), x1)))
% 28.02/3.94 = { by axiom 4 (converse_idempotence) }
% 28.02/3.94 composition(meet(x0, one), x1)
% 28.02/3.94 % SZS output end Proof
% 28.02/3.94
% 28.02/3.94 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------