TSTP Solution File: REL044+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL044+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 : n010.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 13:44:30 EDT 2023
% Result : Theorem 92.40s 12.21s
% Output : Proof 95.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL044+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n010.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 19:34:50 EDT 2023
% 0.13/0.35 % CPUTime :
% 92.40/12.21 Command-line arguments: --flatten
% 92.40/12.21
% 92.40/12.21 % SZS status Theorem
% 92.40/12.21
% 94.94/12.56 % SZS output start Proof
% 94.94/12.56 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 94.94/12.56 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 94.94/12.56 Axiom 3 (composition_identity): composition(X, one) = X.
% 94.94/12.56 Axiom 4 (def_zero): zero = meet(X, complement(X)).
% 94.94/12.57 Axiom 5 (def_top): top = join(X, complement(X)).
% 94.94/12.57 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 94.94/12.57 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 94.94/12.57 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 94.94/12.57 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 94.94/12.57 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 94.94/12.57 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 94.94/12.57 Axiom 12 (goals): join(composition(complement(x0), x1), complement(x2)) = complement(x2).
% 94.94/12.57 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 94.94/12.57 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 94.94/12.57 Axiom 15 (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))).
% 94.94/12.57
% 94.94/12.57 Lemma 16: complement(top) = zero.
% 94.94/12.57 Proof:
% 94.94/12.57 complement(top)
% 94.94/12.57 = { by axiom 5 (def_top) }
% 94.94/12.57 complement(join(complement(X), complement(complement(X))))
% 94.94/12.57 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 94.94/12.57 meet(X, complement(X))
% 94.94/12.57 = { by axiom 4 (def_zero) R->L }
% 94.94/12.57 zero
% 94.94/12.57
% 94.94/12.57 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 94.94/12.57 Proof:
% 94.94/12.57 join(X, join(Y, complement(X)))
% 94.94/12.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.57 join(X, join(complement(X), Y))
% 94.94/12.57 = { by axiom 7 (maddux2_join_associativity) }
% 94.94/12.57 join(join(X, complement(X)), Y)
% 94.94/12.57 = { by axiom 5 (def_top) R->L }
% 94.94/12.57 join(top, Y)
% 94.94/12.57 = { by axiom 2 (maddux1_join_commutativity) }
% 94.94/12.57 join(Y, top)
% 94.94/12.57
% 94.94/12.57 Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 94.94/12.57 Proof:
% 94.94/12.57 converse(composition(converse(X), Y))
% 94.94/12.57 = { by axiom 8 (converse_multiplicativity) }
% 94.94/12.57 composition(converse(Y), converse(converse(X)))
% 94.94/12.57 = { by axiom 1 (converse_idempotence) }
% 94.94/12.57 composition(converse(Y), X)
% 94.94/12.57
% 94.94/12.57 Lemma 19: composition(converse(one), X) = X.
% 94.94/12.57 Proof:
% 94.94/12.57 composition(converse(one), X)
% 94.94/12.57 = { by lemma 18 R->L }
% 94.94/12.57 converse(composition(converse(X), one))
% 94.94/12.57 = { by axiom 3 (composition_identity) }
% 94.94/12.57 converse(converse(X))
% 94.94/12.57 = { by axiom 1 (converse_idempotence) }
% 94.94/12.57 X
% 94.94/12.57
% 94.94/12.57 Lemma 20: composition(one, X) = X.
% 94.94/12.57 Proof:
% 94.94/12.57 composition(one, X)
% 94.94/12.57 = { by lemma 19 R->L }
% 94.94/12.57 composition(converse(one), composition(one, X))
% 94.94/12.57 = { by axiom 9 (composition_associativity) }
% 94.94/12.57 composition(composition(converse(one), one), X)
% 94.94/12.57 = { by axiom 3 (composition_identity) }
% 94.94/12.57 composition(converse(one), X)
% 94.94/12.57 = { by lemma 19 }
% 94.94/12.57 X
% 94.94/12.57
% 94.94/12.57 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 94.94/12.57 Proof:
% 94.94/12.57 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 94.94/12.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.57 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 94.94/12.57 = { by axiom 13 (converse_cancellativity) }
% 94.94/12.57 complement(X)
% 94.94/12.57
% 94.94/12.57 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 94.94/12.57 Proof:
% 94.94/12.57 join(complement(X), complement(X))
% 94.94/12.57 = { by lemma 19 R->L }
% 94.94/12.57 join(complement(X), composition(converse(one), complement(X)))
% 94.94/12.57 = { by lemma 20 R->L }
% 94.94/12.57 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 94.94/12.57 = { by lemma 21 }
% 94.94/12.57 complement(X)
% 94.94/12.57
% 94.94/12.57 Lemma 23: join(top, complement(X)) = top.
% 94.94/12.57 Proof:
% 94.94/12.57 join(top, complement(X))
% 94.94/12.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.57 join(complement(X), top)
% 94.94/12.57 = { by lemma 17 R->L }
% 94.94/12.57 join(X, join(complement(X), complement(X)))
% 94.94/12.57 = { by lemma 22 }
% 94.94/12.57 join(X, complement(X))
% 94.94/12.57 = { by axiom 5 (def_top) R->L }
% 94.94/12.57 top
% 94.94/12.57
% 94.94/12.57 Lemma 24: join(Y, top) = join(X, top).
% 94.94/12.57 Proof:
% 94.94/12.57 join(Y, top)
% 94.94/12.57 = { by lemma 23 R->L }
% 94.94/12.57 join(Y, join(top, complement(Y)))
% 94.94/12.57 = { by lemma 17 }
% 94.94/12.57 join(top, top)
% 94.94/12.57 = { by lemma 17 R->L }
% 94.94/12.57 join(X, join(top, complement(X)))
% 94.94/12.57 = { by lemma 23 }
% 94.94/12.57 join(X, top)
% 94.94/12.57
% 94.94/12.57 Lemma 25: join(X, top) = top.
% 94.94/12.57 Proof:
% 94.94/12.57 join(X, top)
% 94.94/12.57 = { by lemma 24 }
% 94.94/12.57 join(complement(Y), top)
% 94.94/12.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.57 join(top, complement(Y))
% 94.94/12.57 = { by lemma 23 }
% 94.94/12.57 top
% 94.94/12.57
% 94.94/12.57 Lemma 26: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 94.94/12.57 Proof:
% 94.94/12.57 converse(join(X, converse(Y)))
% 94.94/12.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.57 converse(join(converse(Y), X))
% 94.94/12.57 = { by axiom 6 (converse_additivity) }
% 94.94/12.57 join(converse(converse(Y)), converse(X))
% 94.94/12.57 = { by axiom 1 (converse_idempotence) }
% 94.94/12.57 join(Y, converse(X))
% 94.94/12.57
% 94.94/12.57 Lemma 27: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 94.94/12.57 Proof:
% 94.94/12.57 converse(join(converse(X), Y))
% 94.94/12.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.57 converse(join(Y, converse(X)))
% 94.94/12.57 = { by lemma 26 }
% 94.94/12.57 join(X, converse(Y))
% 94.94/12.57
% 94.94/12.57 Lemma 28: join(X, converse(complement(converse(X)))) = converse(top).
% 94.94/12.57 Proof:
% 94.94/12.57 join(X, converse(complement(converse(X))))
% 94.94/12.57 = { by lemma 27 R->L }
% 94.94/12.57 converse(join(converse(X), complement(converse(X))))
% 94.94/12.57 = { by axiom 5 (def_top) R->L }
% 94.94/12.57 converse(top)
% 94.94/12.57
% 94.94/12.57 Lemma 29: join(X, join(complement(X), Y)) = top.
% 94.94/12.57 Proof:
% 94.94/12.57 join(X, join(complement(X), Y))
% 94.94/12.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.57 join(X, join(Y, complement(X)))
% 94.94/12.57 = { by lemma 17 }
% 94.94/12.57 join(Y, top)
% 94.94/12.57 = { by lemma 24 R->L }
% 94.94/12.57 join(Z, top)
% 94.94/12.57 = { by lemma 25 }
% 94.94/12.57 top
% 94.94/12.57
% 94.94/12.57 Lemma 30: converse(top) = top.
% 94.94/12.57 Proof:
% 94.94/12.57 converse(top)
% 94.94/12.57 = { by lemma 25 R->L }
% 94.94/12.57 converse(join(X, top))
% 94.94/12.57 = { by axiom 6 (converse_additivity) }
% 94.94/12.57 join(converse(X), converse(top))
% 94.94/12.57 = { by lemma 28 R->L }
% 94.94/12.57 join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X)))))))
% 94.94/12.57 = { by lemma 29 }
% 94.94/12.57 top
% 94.94/12.57
% 94.94/12.57 Lemma 31: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 94.94/12.57 Proof:
% 94.94/12.57 join(meet(X, Y), complement(join(complement(X), Y)))
% 94.94/12.57 = { by axiom 10 (maddux4_definiton_of_meet) }
% 94.94/12.57 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 94.94/12.57 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 94.94/12.57 X
% 94.94/12.57
% 94.94/12.57 Lemma 32: join(zero, meet(X, X)) = X.
% 94.94/12.57 Proof:
% 94.94/12.57 join(zero, meet(X, X))
% 94.94/12.57 = { by axiom 10 (maddux4_definiton_of_meet) }
% 94.94/12.57 join(zero, complement(join(complement(X), complement(X))))
% 94.94/12.57 = { by axiom 4 (def_zero) }
% 94.94/12.57 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 94.94/12.57 = { by lemma 31 }
% 94.94/12.57 X
% 94.94/12.57
% 94.94/12.57 Lemma 33: complement(complement(X)) = meet(X, X).
% 94.94/12.57 Proof:
% 94.94/12.58 complement(complement(X))
% 94.94/12.58 = { by lemma 22 R->L }
% 94.94/12.58 complement(join(complement(X), complement(X)))
% 94.94/12.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 94.94/12.58 meet(X, X)
% 94.94/12.58
% 94.94/12.58 Lemma 34: meet(Y, X) = meet(X, Y).
% 94.94/12.58 Proof:
% 94.94/12.58 meet(Y, X)
% 94.94/12.58 = { by axiom 10 (maddux4_definiton_of_meet) }
% 94.94/12.58 complement(join(complement(Y), complement(X)))
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.58 complement(join(complement(X), complement(Y)))
% 94.94/12.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 94.94/12.58 meet(X, Y)
% 94.94/12.58
% 94.94/12.58 Lemma 35: complement(join(zero, complement(X))) = meet(X, top).
% 94.94/12.58 Proof:
% 94.94/12.58 complement(join(zero, complement(X)))
% 94.94/12.58 = { by lemma 16 R->L }
% 94.94/12.58 complement(join(complement(top), complement(X)))
% 94.94/12.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 94.94/12.58 meet(top, X)
% 94.94/12.58 = { by lemma 34 R->L }
% 94.94/12.58 meet(X, top)
% 94.94/12.58
% 94.94/12.58 Lemma 36: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 94.94/12.58 Proof:
% 94.94/12.58 join(zero, join(X, meet(Y, Y)))
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.58 join(zero, join(meet(Y, Y), X))
% 94.94/12.58 = { by axiom 7 (maddux2_join_associativity) }
% 94.94/12.58 join(join(zero, meet(Y, Y)), X)
% 94.94/12.58 = { by lemma 32 }
% 94.94/12.58 join(Y, X)
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) }
% 94.94/12.58 join(X, Y)
% 94.94/12.58
% 94.94/12.58 Lemma 37: join(X, complement(zero)) = top.
% 94.94/12.58 Proof:
% 94.94/12.58 join(X, complement(zero))
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.58 join(complement(zero), X)
% 94.94/12.58 = { by lemma 36 R->L }
% 94.94/12.58 join(zero, join(complement(zero), meet(X, X)))
% 94.94/12.58 = { by lemma 29 }
% 94.94/12.58 top
% 94.94/12.58
% 94.94/12.58 Lemma 38: join(meet(X, Y), meet(X, complement(Y))) = X.
% 94.94/12.58 Proof:
% 94.94/12.58 join(meet(X, Y), meet(X, complement(Y)))
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.58 join(meet(X, complement(Y)), meet(X, Y))
% 94.94/12.58 = { by axiom 10 (maddux4_definiton_of_meet) }
% 94.94/12.58 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 94.94/12.58 = { by lemma 31 }
% 94.94/12.58 X
% 94.94/12.58
% 94.94/12.58 Lemma 39: join(zero, meet(X, top)) = X.
% 94.94/12.58 Proof:
% 94.94/12.58 join(zero, meet(X, top))
% 94.94/12.58 = { by lemma 37 R->L }
% 94.94/12.58 join(zero, meet(X, join(complement(zero), complement(zero))))
% 94.94/12.58 = { by lemma 22 }
% 94.94/12.58 join(zero, meet(X, complement(zero)))
% 94.94/12.58 = { by lemma 16 R->L }
% 94.94/12.58 join(complement(top), meet(X, complement(zero)))
% 94.94/12.58 = { by lemma 37 R->L }
% 94.94/12.58 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 94.94/12.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 94.94/12.58 join(meet(X, zero), meet(X, complement(zero)))
% 94.94/12.58 = { by lemma 38 }
% 94.94/12.58 X
% 94.94/12.58
% 94.94/12.58 Lemma 40: join(zero, complement(X)) = complement(X).
% 94.94/12.58 Proof:
% 94.94/12.58 join(zero, complement(X))
% 94.94/12.58 = { by lemma 32 R->L }
% 94.94/12.58 join(zero, complement(join(zero, meet(X, X))))
% 94.94/12.58 = { by lemma 33 R->L }
% 94.94/12.58 join(zero, complement(join(zero, complement(complement(X)))))
% 94.94/12.58 = { by lemma 35 }
% 94.94/12.58 join(zero, meet(complement(X), top))
% 94.94/12.58 = { by lemma 39 }
% 94.94/12.58 complement(X)
% 94.94/12.58
% 94.94/12.58 Lemma 41: complement(complement(X)) = X.
% 94.94/12.58 Proof:
% 94.94/12.58 complement(complement(X))
% 94.94/12.58 = { by lemma 40 R->L }
% 94.94/12.58 join(zero, complement(complement(X)))
% 94.94/12.58 = { by lemma 33 }
% 94.94/12.58 join(zero, meet(X, X))
% 94.94/12.58 = { by lemma 32 }
% 94.94/12.58 X
% 94.94/12.58
% 94.94/12.58 Lemma 42: join(X, zero) = X.
% 94.94/12.58 Proof:
% 94.94/12.58 join(X, zero)
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.58 join(zero, X)
% 94.94/12.58 = { by lemma 41 R->L }
% 94.94/12.58 join(zero, complement(complement(X)))
% 94.94/12.58 = { by lemma 33 }
% 94.94/12.58 join(zero, meet(X, X))
% 94.94/12.58 = { by lemma 32 }
% 94.94/12.58 X
% 94.94/12.58
% 94.94/12.58 Lemma 43: join(zero, X) = X.
% 94.94/12.58 Proof:
% 94.94/12.58 join(zero, X)
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.58 join(X, zero)
% 94.94/12.58 = { by lemma 42 }
% 94.94/12.58 X
% 94.94/12.58
% 94.94/12.58 Lemma 44: join(X, X) = X.
% 94.94/12.58 Proof:
% 94.94/12.58 join(X, X)
% 94.94/12.58 = { by lemma 41 R->L }
% 94.94/12.58 join(X, complement(complement(X)))
% 94.94/12.58 = { by lemma 41 R->L }
% 94.94/12.58 join(complement(complement(X)), complement(complement(X)))
% 94.94/12.58 = { by lemma 22 }
% 94.94/12.58 complement(complement(X))
% 94.94/12.58 = { by lemma 41 }
% 94.94/12.58 X
% 94.94/12.58
% 94.94/12.58 Lemma 45: converse(zero) = zero.
% 94.94/12.58 Proof:
% 94.94/12.58 converse(zero)
% 94.94/12.58 = { by lemma 43 R->L }
% 94.94/12.58 join(zero, converse(zero))
% 94.94/12.58 = { by lemma 27 R->L }
% 94.94/12.58 converse(join(converse(zero), zero))
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.58 converse(join(zero, converse(zero)))
% 94.94/12.58 = { by lemma 32 R->L }
% 94.94/12.58 converse(join(zero, join(zero, meet(converse(zero), converse(zero)))))
% 94.94/12.58 = { by axiom 10 (maddux4_definiton_of_meet) }
% 94.94/12.58 converse(join(zero, join(zero, complement(join(complement(converse(zero)), complement(converse(zero)))))))
% 94.94/12.58 = { by lemma 22 R->L }
% 94.94/12.58 converse(join(zero, join(zero, join(complement(join(complement(converse(zero)), complement(converse(zero)))), complement(join(complement(converse(zero)), complement(converse(zero))))))))
% 94.94/12.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 94.94/12.58 converse(join(zero, join(zero, join(meet(converse(zero), converse(zero)), complement(join(complement(converse(zero)), complement(converse(zero))))))))
% 94.94/12.58 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 94.94/12.58 converse(join(zero, join(zero, join(meet(converse(zero), converse(zero)), meet(converse(zero), converse(zero))))))
% 94.94/12.58 = { by lemma 36 }
% 94.94/12.58 converse(join(zero, join(meet(converse(zero), converse(zero)), converse(zero))))
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) }
% 94.94/12.58 converse(join(zero, join(converse(zero), meet(converse(zero), converse(zero)))))
% 94.94/12.58 = { by lemma 36 }
% 94.94/12.58 converse(join(converse(zero), converse(zero)))
% 94.94/12.58 = { by lemma 26 }
% 94.94/12.58 join(zero, converse(converse(zero)))
% 94.94/12.58 = { by axiom 1 (converse_idempotence) }
% 94.94/12.58 join(zero, zero)
% 94.94/12.58 = { by lemma 44 }
% 94.94/12.58 zero
% 94.94/12.58
% 94.94/12.58 Lemma 46: join(top, X) = top.
% 94.94/12.58 Proof:
% 94.94/12.58 join(top, X)
% 94.94/12.58 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.58 join(X, top)
% 94.94/12.58 = { by lemma 24 R->L }
% 94.94/12.58 join(Y, top)
% 94.94/12.58 = { by lemma 25 }
% 94.94/12.58 top
% 94.94/12.58
% 94.94/12.58 Lemma 47: meet(X, X) = X.
% 94.94/12.58 Proof:
% 94.94/12.58 meet(X, X)
% 94.94/12.58 = { by lemma 33 R->L }
% 94.94/12.58 complement(complement(X))
% 94.94/12.58 = { by lemma 41 }
% 94.94/12.58 X
% 94.94/12.58
% 94.94/12.58 Lemma 48: meet(X, top) = X.
% 94.94/12.58 Proof:
% 94.94/12.58 meet(X, top)
% 94.94/12.58 = { by lemma 35 R->L }
% 94.94/12.58 complement(join(zero, complement(X)))
% 94.94/12.58 = { by lemma 40 R->L }
% 94.94/12.58 join(zero, complement(join(zero, complement(X))))
% 94.94/12.58 = { by lemma 35 }
% 94.94/12.58 join(zero, meet(X, top))
% 94.94/12.58 = { by lemma 39 }
% 94.94/12.58 X
% 94.94/12.58
% 94.94/12.58 Lemma 49: meet(top, X) = X.
% 94.94/12.58 Proof:
% 94.94/12.58 meet(top, X)
% 94.94/12.58 = { by lemma 34 }
% 94.94/12.58 meet(X, top)
% 94.94/12.58 = { by lemma 48 }
% 94.94/12.58 X
% 94.94/12.58
% 94.94/12.58 Lemma 50: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 94.94/12.58 Proof:
% 94.94/12.58 converse(composition(X, converse(Y)))
% 94.94/12.58 = { by axiom 8 (converse_multiplicativity) }
% 94.94/12.58 composition(converse(converse(Y)), converse(X))
% 94.94/12.59 = { by axiom 1 (converse_idempotence) }
% 94.94/12.59 composition(Y, converse(X))
% 94.94/12.59
% 94.94/12.59 Lemma 51: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 94.94/12.59 Proof:
% 94.94/12.59 composition(join(X, one), Y)
% 94.94/12.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.59 composition(join(one, X), Y)
% 94.94/12.59 = { by axiom 11 (composition_distributivity) }
% 94.94/12.59 join(composition(one, Y), composition(X, Y))
% 94.94/12.59 = { by lemma 20 }
% 94.94/12.59 join(Y, composition(X, Y))
% 94.94/12.59
% 94.94/12.59 Lemma 52: composition(zero, X) = zero.
% 94.94/12.59 Proof:
% 94.94/12.59 composition(zero, X)
% 94.94/12.59 = { by lemma 43 R->L }
% 94.94/12.59 join(zero, composition(zero, X))
% 94.94/12.59 = { by axiom 1 (converse_idempotence) R->L }
% 94.94/12.59 join(zero, composition(zero, converse(converse(X))))
% 94.94/12.59 = { by lemma 50 R->L }
% 94.94/12.59 join(zero, converse(composition(converse(X), converse(zero))))
% 94.94/12.59 = { by lemma 27 R->L }
% 94.94/12.59 converse(join(converse(zero), composition(converse(X), converse(zero))))
% 94.94/12.59 = { by lemma 51 R->L }
% 94.94/12.59 converse(composition(join(converse(X), one), converse(zero)))
% 94.94/12.59 = { by lemma 50 }
% 94.94/12.59 composition(zero, converse(join(converse(X), one)))
% 94.94/12.59 = { by lemma 27 }
% 94.94/12.59 composition(zero, join(X, converse(one)))
% 94.94/12.59 = { by axiom 3 (composition_identity) R->L }
% 94.94/12.59 composition(zero, join(X, composition(converse(one), one)))
% 94.94/12.59 = { by lemma 19 }
% 94.94/12.59 composition(zero, join(X, one))
% 94.94/12.59 = { by lemma 45 R->L }
% 94.94/12.59 composition(converse(zero), join(X, one))
% 94.94/12.59 = { by lemma 16 R->L }
% 94.94/12.59 composition(converse(complement(top)), join(X, one))
% 94.94/12.59 = { by lemma 46 R->L }
% 94.94/12.59 composition(converse(complement(join(top, composition(X, top)))), join(X, one))
% 94.94/12.59 = { by lemma 51 R->L }
% 94.94/12.59 composition(converse(complement(composition(join(X, one), top))), join(X, one))
% 94.94/12.59 = { by lemma 18 R->L }
% 94.94/12.59 converse(composition(converse(join(X, one)), complement(composition(join(X, one), top))))
% 94.94/12.59 = { by lemma 43 R->L }
% 94.94/12.59 converse(join(zero, composition(converse(join(X, one)), complement(composition(join(X, one), top)))))
% 94.94/12.59 = { by lemma 16 R->L }
% 94.94/12.59 converse(join(complement(top), composition(converse(join(X, one)), complement(composition(join(X, one), top)))))
% 94.94/12.59 = { by lemma 21 }
% 94.94/12.59 converse(complement(top))
% 94.94/12.59 = { by lemma 16 }
% 94.94/12.59 converse(zero)
% 94.94/12.59 = { by lemma 45 }
% 94.94/12.59 zero
% 94.94/12.59
% 94.94/12.59 Lemma 53: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 94.94/12.59 Proof:
% 94.94/12.59 complement(join(complement(X), meet(Y, Z)))
% 94.94/12.59 = { by lemma 34 }
% 94.94/12.59 complement(join(complement(X), meet(Z, Y)))
% 94.94/12.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.59 complement(join(meet(Z, Y), complement(X)))
% 94.94/12.59 = { by axiom 10 (maddux4_definiton_of_meet) }
% 94.94/12.59 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 94.94/12.59 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 94.94/12.59 meet(join(complement(Z), complement(Y)), X)
% 94.94/12.59 = { by lemma 34 R->L }
% 94.94/12.59 meet(X, join(complement(Z), complement(Y)))
% 94.94/12.59 = { by axiom 2 (maddux1_join_commutativity) }
% 94.94/12.59 meet(X, join(complement(Y), complement(Z)))
% 94.94/12.59
% 94.94/12.59 Lemma 54: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 94.94/12.59 Proof:
% 94.94/12.59 join(complement(X), complement(Y))
% 94.94/12.59 = { by lemma 49 R->L }
% 94.94/12.59 meet(top, join(complement(X), complement(Y)))
% 94.94/12.59 = { by lemma 53 R->L }
% 94.94/12.59 complement(join(complement(top), meet(X, Y)))
% 94.94/12.59 = { by lemma 16 }
% 94.94/12.59 complement(join(zero, meet(X, Y)))
% 94.94/12.59 = { by lemma 34 R->L }
% 94.94/12.59 complement(join(zero, meet(Y, X)))
% 94.94/12.59 = { by axiom 2 (maddux1_join_commutativity) }
% 94.94/12.59 complement(join(meet(Y, X), zero))
% 94.94/12.59 = { by lemma 42 }
% 94.94/12.59 complement(meet(Y, X))
% 94.94/12.59 = { by lemma 34 R->L }
% 94.94/12.59 complement(meet(X, Y))
% 94.94/12.59
% 94.94/12.59 Lemma 55: complement(join(complement(X), complement(Y))) = meet(Y, X).
% 94.94/12.59 Proof:
% 94.94/12.59 complement(join(complement(X), complement(Y)))
% 94.94/12.59 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 94.94/12.59 meet(X, Y)
% 94.94/12.59 = { by lemma 34 R->L }
% 94.94/12.59 meet(Y, X)
% 94.94/12.59
% 94.94/12.59 Lemma 56: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 94.94/12.59 Proof:
% 94.94/12.59 complement(join(X, complement(Y)))
% 94.94/12.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.59 complement(join(complement(Y), X))
% 94.94/12.59 = { by lemma 48 R->L }
% 94.94/12.59 complement(join(complement(Y), meet(X, top)))
% 94.94/12.59 = { by lemma 35 R->L }
% 94.94/12.59 complement(join(complement(Y), complement(join(zero, complement(X)))))
% 94.94/12.59 = { by lemma 55 }
% 94.94/12.59 meet(join(zero, complement(X)), Y)
% 94.94/12.59 = { by lemma 40 }
% 94.94/12.59 meet(complement(X), Y)
% 94.94/12.59 = { by lemma 34 R->L }
% 94.94/12.59 meet(Y, complement(X))
% 94.94/12.59
% 94.94/12.59 Lemma 57: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 94.94/12.59 Proof:
% 94.94/12.59 complement(join(complement(X), Y))
% 94.94/12.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.59 complement(join(Y, complement(X)))
% 94.94/12.59 = { by lemma 56 }
% 94.94/12.59 meet(X, complement(Y))
% 94.94/12.59
% 94.94/12.59 Lemma 58: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 94.94/12.59 Proof:
% 94.94/12.59 complement(meet(X, complement(Y)))
% 94.94/12.59 = { by lemma 34 }
% 94.94/12.59 complement(meet(complement(Y), X))
% 94.94/12.59 = { by lemma 40 R->L }
% 94.94/12.59 complement(meet(join(zero, complement(Y)), X))
% 94.94/12.59 = { by lemma 54 R->L }
% 94.94/12.59 join(complement(join(zero, complement(Y))), complement(X))
% 94.94/12.59 = { by lemma 35 }
% 94.94/12.59 join(meet(Y, top), complement(X))
% 94.94/12.59 = { by lemma 48 }
% 94.94/12.59 join(Y, complement(X))
% 94.94/12.59
% 94.94/12.59 Lemma 59: meet(X, meet(Y, complement(X))) = zero.
% 94.94/12.59 Proof:
% 94.94/12.59 meet(X, meet(Y, complement(X)))
% 94.94/12.59 = { by lemma 34 }
% 94.94/12.59 meet(X, meet(complement(X), Y))
% 94.94/12.59 = { by axiom 10 (maddux4_definiton_of_meet) }
% 94.94/12.59 complement(join(complement(X), complement(meet(complement(X), Y))))
% 94.94/12.59 = { by lemma 34 }
% 94.94/12.59 complement(join(complement(X), complement(meet(Y, complement(X)))))
% 94.94/12.59 = { by lemma 54 R->L }
% 94.94/12.59 complement(join(complement(X), join(complement(Y), complement(complement(X)))))
% 94.94/12.59 = { by lemma 17 }
% 94.94/12.59 complement(join(complement(Y), top))
% 94.94/12.59 = { by lemma 25 }
% 94.94/12.59 complement(top)
% 94.94/12.59 = { by lemma 16 }
% 94.94/12.59 zero
% 94.94/12.59
% 94.94/12.59 Lemma 60: meet(X, join(Y, X)) = X.
% 94.94/12.59 Proof:
% 94.94/12.59 meet(X, join(Y, X))
% 94.94/12.59 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.59 meet(X, join(X, Y))
% 94.94/12.59 = { by lemma 47 R->L }
% 94.94/12.59 meet(X, join(X, meet(Y, Y)))
% 94.94/12.59 = { by lemma 33 R->L }
% 94.94/12.59 meet(X, join(X, complement(complement(Y))))
% 94.94/12.59 = { by lemma 58 R->L }
% 94.94/12.59 meet(X, complement(meet(complement(Y), complement(X))))
% 94.94/12.59 = { by lemma 54 R->L }
% 94.94/12.59 meet(X, join(complement(complement(Y)), complement(complement(X))))
% 94.94/12.59 = { by lemma 53 R->L }
% 94.94/12.59 complement(join(complement(X), meet(complement(Y), complement(X))))
% 94.94/12.59 = { by lemma 40 R->L }
% 94.94/12.59 join(zero, complement(join(complement(X), meet(complement(Y), complement(X)))))
% 94.94/12.59 = { by lemma 59 R->L }
% 94.94/12.59 join(meet(X, meet(complement(Y), complement(X))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 94.94/12.59 = { by lemma 31 }
% 94.94/12.59 X
% 94.94/12.59
% 94.94/12.59 Lemma 61: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 94.94/12.60 Proof:
% 94.94/12.60 meet(Y, meet(X, Z))
% 94.94/12.60 = { by lemma 34 }
% 94.94/12.60 meet(Y, meet(Z, X))
% 94.94/12.60 = { by lemma 41 R->L }
% 94.94/12.60 complement(complement(meet(Y, meet(Z, X))))
% 94.94/12.60 = { by lemma 34 }
% 94.94/12.60 complement(complement(meet(Y, meet(X, Z))))
% 94.94/12.60 = { by axiom 10 (maddux4_definiton_of_meet) }
% 94.94/12.60 complement(complement(meet(Y, complement(join(complement(X), complement(Z))))))
% 94.94/12.60 = { by lemma 58 }
% 94.94/12.60 complement(join(join(complement(X), complement(Z)), complement(Y)))
% 94.94/12.60 = { by axiom 7 (maddux2_join_associativity) R->L }
% 94.94/12.60 complement(join(complement(X), join(complement(Z), complement(Y))))
% 94.94/12.60 = { by lemma 54 }
% 94.94/12.60 complement(join(complement(X), complement(meet(Z, Y))))
% 94.94/12.60 = { by lemma 54 }
% 94.94/12.60 complement(complement(meet(X, meet(Z, Y))))
% 94.94/12.60 = { by lemma 34 R->L }
% 94.94/12.60 complement(complement(meet(X, meet(Y, Z))))
% 94.94/12.60 = { by lemma 33 }
% 94.94/12.60 meet(meet(X, meet(Y, Z)), meet(X, meet(Y, Z)))
% 94.94/12.60 = { by lemma 47 }
% 94.94/12.60 meet(X, meet(Y, Z))
% 94.94/12.60
% 94.94/12.60 Lemma 62: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 94.94/12.60 Proof:
% 94.94/12.60 meet(complement(X), complement(Y))
% 94.94/12.60 = { by lemma 34 }
% 94.94/12.60 meet(complement(Y), complement(X))
% 94.94/12.60 = { by lemma 56 R->L }
% 94.94/12.60 complement(join(X, complement(complement(Y))))
% 94.94/12.60 = { by lemma 33 }
% 94.94/12.60 complement(join(X, meet(Y, Y)))
% 94.94/12.60 = { by lemma 47 }
% 94.94/12.60 complement(join(X, Y))
% 94.94/12.60
% 94.94/12.60 Lemma 63: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 94.94/12.60 Proof:
% 94.94/12.60 meet(meet(X, Y), Z)
% 94.94/12.60 = { by lemma 34 }
% 94.94/12.60 meet(Z, meet(X, Y))
% 94.94/12.60 = { by lemma 61 R->L }
% 94.94/12.60 meet(X, meet(Z, Y))
% 94.94/12.60
% 94.94/12.60 Lemma 64: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 94.94/12.60 Proof:
% 94.94/12.60 join(complement(one), composition(converse(X), complement(X)))
% 94.94/12.60 = { by axiom 3 (composition_identity) R->L }
% 94.94/12.60 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 94.94/12.60 = { by lemma 21 }
% 94.94/12.60 complement(one)
% 94.94/12.60
% 94.94/12.60 Lemma 65: join(complement(one), composition(converse(complement(X)), X)) = complement(one).
% 94.94/12.60 Proof:
% 94.94/12.60 join(complement(one), composition(converse(complement(X)), X))
% 94.94/12.60 = { by lemma 48 R->L }
% 94.94/12.60 join(complement(one), composition(converse(complement(X)), meet(X, top)))
% 94.94/12.60 = { by lemma 40 R->L }
% 94.94/12.60 join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))
% 94.94/12.60 = { by lemma 35 R->L }
% 94.94/12.60 join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))
% 94.94/12.60 = { by lemma 64 }
% 94.94/12.60 complement(one)
% 94.94/12.60
% 94.94/12.60 Lemma 66: join(complement(one), converse(complement(one))) = complement(one).
% 94.94/12.60 Proof:
% 94.94/12.60 join(complement(one), converse(complement(one)))
% 94.94/12.60 = { by axiom 3 (composition_identity) R->L }
% 94.94/12.60 join(complement(one), composition(converse(complement(one)), one))
% 94.94/12.60 = { by lemma 65 }
% 94.94/12.60 complement(one)
% 94.94/12.60
% 94.94/12.60 Lemma 67: join(meet(X, complement(Y)), meet(X, Y)) = X.
% 94.94/12.60 Proof:
% 94.94/12.60 join(meet(X, complement(Y)), meet(X, Y))
% 94.94/12.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.60 join(meet(X, Y), meet(X, complement(Y)))
% 94.94/12.60 = { by lemma 38 }
% 94.94/12.60 X
% 94.94/12.60
% 94.94/12.60 Lemma 68: meet(X, meet(complement(Y), join(Y, complement(X)))) = zero.
% 94.94/12.60 Proof:
% 94.94/12.60 meet(X, meet(complement(Y), join(Y, complement(X))))
% 94.94/12.60 = { by lemma 34 }
% 94.94/12.60 meet(X, meet(join(Y, complement(X)), complement(Y)))
% 94.94/12.60 = { by lemma 61 }
% 94.94/12.60 meet(join(Y, complement(X)), meet(X, complement(Y)))
% 94.94/12.60 = { by lemma 56 R->L }
% 94.94/12.60 meet(join(Y, complement(X)), complement(join(Y, complement(X))))
% 94.94/12.60 = { by axiom 4 (def_zero) R->L }
% 94.94/12.60 zero
% 94.94/12.60
% 94.94/12.60 Lemma 69: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 94.94/12.60 Proof:
% 94.94/12.60 meet(X, complement(meet(X, Y)))
% 94.94/12.60 = { by lemma 34 }
% 94.94/12.60 meet(X, complement(meet(Y, X)))
% 94.94/12.60 = { by lemma 54 R->L }
% 94.94/12.60 meet(X, join(complement(Y), complement(X)))
% 94.94/12.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.60 meet(X, join(complement(X), complement(Y)))
% 94.94/12.60 = { by lemma 34 }
% 94.94/12.60 meet(join(complement(X), complement(Y)), X)
% 94.94/12.60 = { by lemma 41 R->L }
% 94.94/12.60 meet(join(complement(X), complement(Y)), complement(complement(X)))
% 94.94/12.60 = { by lemma 38 R->L }
% 94.94/12.60 join(meet(meet(join(complement(X), complement(Y)), complement(complement(X))), X), meet(meet(join(complement(X), complement(Y)), complement(complement(X))), complement(X)))
% 94.94/12.60 = { by lemma 34 R->L }
% 94.94/12.60 join(meet(meet(join(complement(X), complement(Y)), complement(complement(X))), X), meet(complement(X), meet(join(complement(X), complement(Y)), complement(complement(X)))))
% 94.94/12.60 = { by lemma 59 }
% 94.94/12.60 join(meet(meet(join(complement(X), complement(Y)), complement(complement(X))), X), zero)
% 94.94/12.60 = { by lemma 42 }
% 94.94/12.60 meet(meet(join(complement(X), complement(Y)), complement(complement(X))), X)
% 94.94/12.60 = { by lemma 41 }
% 94.94/12.60 meet(meet(join(complement(X), complement(Y)), X), X)
% 94.94/12.60 = { by lemma 34 R->L }
% 94.94/12.60 meet(X, meet(join(complement(X), complement(Y)), X))
% 94.94/12.60 = { by lemma 63 R->L }
% 94.94/12.60 meet(meet(X, X), join(complement(X), complement(Y)))
% 94.94/12.60 = { by lemma 33 R->L }
% 94.94/12.60 meet(complement(complement(X)), join(complement(X), complement(Y)))
% 94.94/12.60 = { by lemma 67 R->L }
% 94.94/12.60 join(meet(meet(complement(complement(X)), join(complement(X), complement(Y))), complement(Y)), meet(meet(complement(complement(X)), join(complement(X), complement(Y))), Y))
% 94.94/12.60 = { by lemma 34 R->L }
% 94.94/12.60 join(meet(meet(complement(complement(X)), join(complement(X), complement(Y))), complement(Y)), meet(Y, meet(complement(complement(X)), join(complement(X), complement(Y)))))
% 94.94/12.60 = { by lemma 68 }
% 94.94/12.60 join(meet(meet(complement(complement(X)), join(complement(X), complement(Y))), complement(Y)), zero)
% 94.94/12.60 = { by lemma 42 }
% 94.94/12.60 meet(meet(complement(complement(X)), join(complement(X), complement(Y))), complement(Y))
% 94.94/12.60 = { by lemma 63 }
% 94.94/12.60 meet(complement(complement(X)), meet(complement(Y), join(complement(X), complement(Y))))
% 94.94/12.60 = { by lemma 60 }
% 94.94/12.60 meet(complement(complement(X)), complement(Y))
% 94.94/12.60 = { by lemma 34 R->L }
% 94.94/12.60 meet(complement(Y), complement(complement(X)))
% 94.94/12.60 = { by lemma 41 }
% 94.94/12.60 meet(complement(Y), X)
% 94.94/12.60 = { by lemma 34 R->L }
% 94.94/12.60 meet(X, complement(Y))
% 94.94/12.60
% 94.94/12.60 Lemma 70: meet(X, join(complement(X), Y)) = meet(X, Y).
% 94.94/12.60 Proof:
% 94.94/12.60 meet(X, join(complement(X), Y))
% 94.94/12.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 94.94/12.60 meet(X, join(Y, complement(X)))
% 94.94/12.60 = { by lemma 58 R->L }
% 94.94/12.60 meet(X, complement(meet(X, complement(Y))))
% 94.94/12.60 = { by lemma 69 }
% 94.94/12.60 meet(X, complement(complement(Y)))
% 94.94/12.60 = { by lemma 41 }
% 94.94/12.60 meet(X, Y)
% 94.94/12.60
% 94.94/12.60 Lemma 71: meet(one, composition(converse(complement(X)), X)) = zero.
% 94.94/12.60 Proof:
% 94.94/12.60 meet(one, composition(converse(complement(X)), X))
% 94.94/12.60 = { by lemma 49 R->L }
% 94.94/12.60 meet(one, meet(top, composition(converse(complement(X)), X)))
% 94.94/12.60 = { by lemma 63 R->L }
% 94.94/12.60 meet(meet(one, composition(converse(complement(X)), X)), top)
% 94.94/12.60 = { by lemma 29 R->L }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(converse(composition(converse(X), complement(X))), join(complement(converse(composition(converse(X), complement(X)))), converse(complement(one)))))
% 95.56/12.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(converse(composition(converse(X), complement(X))), join(converse(complement(one)), complement(converse(composition(converse(X), complement(X)))))))
% 95.56/12.61 = { by axiom 7 (maddux2_join_associativity) }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(join(converse(composition(converse(X), complement(X))), converse(complement(one))), complement(converse(composition(converse(X), complement(X))))))
% 95.56/12.61 = { by axiom 6 (converse_additivity) R->L }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(converse(join(composition(converse(X), complement(X)), complement(one))), complement(converse(composition(converse(X), complement(X))))))
% 95.56/12.61 = { by axiom 2 (maddux1_join_commutativity) }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(complement(converse(composition(converse(X), complement(X)))), converse(join(composition(converse(X), complement(X)), complement(one)))))
% 95.56/12.61 = { by axiom 2 (maddux1_join_commutativity) }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(complement(converse(composition(converse(X), complement(X)))), converse(join(complement(one), composition(converse(X), complement(X))))))
% 95.56/12.61 = { by lemma 64 }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(complement(converse(composition(converse(X), complement(X)))), converse(complement(one))))
% 95.56/12.61 = { by lemma 18 }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(complement(composition(converse(complement(X)), X)), converse(complement(one))))
% 95.56/12.61 = { by lemma 66 R->L }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(complement(composition(converse(complement(X)), X)), converse(join(complement(one), converse(complement(one))))))
% 95.56/12.61 = { by lemma 26 }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(complement(composition(converse(complement(X)), X)), join(complement(one), converse(complement(one)))))
% 95.56/12.61 = { by lemma 66 }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), join(complement(composition(converse(complement(X)), X)), complement(one)))
% 95.56/12.61 = { by lemma 54 }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), complement(meet(composition(converse(complement(X)), X), one)))
% 95.56/12.61 = { by lemma 34 R->L }
% 95.56/12.61 meet(meet(one, composition(converse(complement(X)), X)), complement(meet(one, composition(converse(complement(X)), X))))
% 95.56/12.61 = { by axiom 4 (def_zero) R->L }
% 95.56/12.61 zero
% 95.56/12.61
% 95.56/12.61 Lemma 72: meet(X, converse(complement(converse(X)))) = zero.
% 95.56/12.61 Proof:
% 95.56/12.61 meet(X, converse(complement(converse(X))))
% 95.56/12.61 = { by lemma 42 R->L }
% 95.56/12.61 join(meet(X, converse(complement(converse(X)))), zero)
% 95.56/12.61 = { by lemma 52 R->L }
% 95.56/12.61 join(meet(X, converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X)))))))
% 95.56/12.61 = { by lemma 20 R->L }
% 95.56/12.61 join(meet(composition(one, X), converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X)))))))
% 95.56/12.61 = { by lemma 71 R->L }
% 95.56/12.61 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)))))))
% 95.56/12.61 = { by axiom 15 (dedekind_law) }
% 95.56/12.61 composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X))))))
% 95.56/12.61 = { by lemma 71 }
% 95.56/12.61 composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))
% 95.56/12.61 = { by lemma 52 }
% 95.56/12.61 zero
% 95.56/12.61
% 95.56/12.61 Lemma 73: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 95.56/12.61 Proof:
% 95.56/12.61 join(meet(X, Y), meet(complement(X), Y))
% 95.56/12.61 = { by lemma 34 }
% 95.56/12.61 join(meet(X, Y), meet(Y, complement(X)))
% 95.56/12.61 = { by lemma 34 }
% 95.56/12.61 join(meet(Y, X), meet(Y, complement(X)))
% 95.56/12.61 = { by lemma 38 }
% 95.56/12.61 Y
% 95.56/12.61
% 95.56/12.61 Lemma 74: meet(X, join(Y, meet(complement(Y), X))) = X.
% 95.56/12.61 Proof:
% 95.56/12.61 meet(X, join(Y, meet(complement(Y), X)))
% 95.56/12.61 = { by lemma 34 }
% 95.56/12.61 meet(X, join(Y, meet(X, complement(Y))))
% 95.56/12.61 = { by lemma 40 R->L }
% 95.56/12.61 meet(X, join(Y, meet(X, join(zero, complement(Y)))))
% 95.56/12.61 = { by lemma 48 R->L }
% 95.56/12.61 meet(X, join(meet(Y, top), meet(X, join(zero, complement(Y)))))
% 95.56/12.61 = { by lemma 35 R->L }
% 95.56/12.61 meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y)))))
% 95.56/12.61 = { by lemma 42 R->L }
% 95.56/12.61 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), zero)
% 95.56/12.61 = { by lemma 16 R->L }
% 95.56/12.61 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), complement(top))
% 95.56/12.61 = { by axiom 5 (def_top) }
% 95.56/12.61 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), complement(join(join(complement(X), complement(join(zero, complement(Y)))), complement(join(complement(X), complement(join(zero, complement(Y))))))))
% 95.56/12.61 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 95.56/12.61 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), complement(join(join(complement(X), complement(join(zero, complement(Y)))), meet(X, join(zero, complement(Y))))))
% 95.56/12.61 = { by axiom 7 (maddux2_join_associativity) R->L }
% 95.56/12.61 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), complement(join(complement(X), join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y)))))))
% 95.56/12.61 = { by lemma 31 }
% 95.56/12.61 X
% 95.56/12.61
% 95.56/12.61 Lemma 75: complement(join(join(composition(complement(x0), x1), complement(x2)), complement(X))) = meet(X, x2).
% 95.56/12.61 Proof:
% 95.56/12.61 complement(join(join(composition(complement(x0), x1), complement(x2)), complement(X)))
% 95.56/12.61 = { by axiom 12 (goals) }
% 95.56/12.61 complement(join(complement(x2), complement(X)))
% 95.56/12.61 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 95.56/12.61 meet(x2, X)
% 95.56/12.61 = { by lemma 34 R->L }
% 95.56/12.61 meet(X, x2)
% 95.56/12.61
% 95.56/12.61 Lemma 76: meet(one, composition(composition(x2, converse(x1)), converse(complement(x0)))) = zero.
% 95.56/12.61 Proof:
% 95.56/12.61 meet(one, composition(composition(x2, converse(x1)), converse(complement(x0))))
% 95.56/12.61 = { by lemma 70 R->L }
% 95.56/12.61 meet(one, join(complement(one), composition(composition(x2, converse(x1)), converse(complement(x0)))))
% 95.56/12.62 = { by lemma 64 R->L }
% 95.56/12.62 meet(one, join(join(complement(one), composition(converse(converse(x2)), complement(converse(x2)))), composition(composition(x2, converse(x1)), converse(complement(x0)))))
% 95.56/12.62 = { by axiom 1 (converse_idempotence) }
% 95.56/12.62 meet(one, join(join(complement(one), composition(x2, complement(converse(x2)))), composition(composition(x2, converse(x1)), converse(complement(x0)))))
% 95.56/12.62 = { by axiom 7 (maddux2_join_associativity) R->L }
% 95.56/12.62 meet(one, join(complement(one), join(composition(x2, complement(converse(x2))), composition(composition(x2, converse(x1)), converse(complement(x0))))))
% 95.56/12.62 = { by axiom 2 (maddux1_join_commutativity) }
% 95.56/12.62 meet(one, join(complement(one), join(composition(composition(x2, converse(x1)), converse(complement(x0))), composition(x2, complement(converse(x2))))))
% 95.56/12.62 = { by axiom 9 (composition_associativity) R->L }
% 95.56/12.62 meet(one, join(complement(one), join(composition(x2, composition(converse(x1), converse(complement(x0)))), composition(x2, complement(converse(x2))))))
% 95.56/12.62 = { by axiom 8 (converse_multiplicativity) R->L }
% 95.56/12.62 meet(one, join(complement(one), join(composition(x2, converse(composition(complement(x0), x1))), composition(x2, complement(converse(x2))))))
% 95.56/12.62 = { by axiom 1 (converse_idempotence) R->L }
% 95.56/12.62 meet(one, join(complement(one), join(composition(x2, converse(composition(complement(x0), x1))), composition(x2, converse(converse(complement(converse(x2))))))))
% 95.56/12.62 = { by axiom 1 (converse_idempotence) R->L }
% 95.56/12.62 meet(one, join(complement(one), converse(converse(join(composition(x2, converse(composition(complement(x0), x1))), composition(x2, converse(converse(complement(converse(x2))))))))))
% 95.56/12.62 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.62 meet(one, join(complement(one), converse(converse(join(composition(x2, converse(converse(complement(converse(x2))))), composition(x2, converse(composition(complement(x0), x1))))))))
% 95.56/12.62 = { by axiom 6 (converse_additivity) }
% 95.56/12.62 meet(one, join(complement(one), converse(join(converse(composition(x2, converse(converse(complement(converse(x2)))))), converse(composition(x2, converse(composition(complement(x0), x1))))))))
% 95.56/12.62 = { by lemma 50 }
% 95.56/12.62 meet(one, join(complement(one), converse(join(composition(converse(complement(converse(x2))), converse(x2)), converse(composition(x2, converse(composition(complement(x0), x1))))))))
% 95.56/12.62 = { by axiom 2 (maddux1_join_commutativity) }
% 95.56/12.62 meet(one, join(complement(one), converse(join(converse(composition(x2, converse(composition(complement(x0), x1)))), composition(converse(complement(converse(x2))), converse(x2))))))
% 95.56/12.62 = { by axiom 8 (converse_multiplicativity) }
% 95.56/12.62 meet(one, join(complement(one), converse(join(composition(converse(converse(composition(complement(x0), x1))), converse(x2)), composition(converse(complement(converse(x2))), converse(x2))))))
% 95.56/12.62 = { by axiom 11 (composition_distributivity) R->L }
% 95.56/12.62 meet(one, join(complement(one), converse(composition(join(converse(converse(composition(complement(x0), x1))), converse(complement(converse(x2)))), converse(x2)))))
% 95.56/12.62 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.62 meet(one, join(complement(one), converse(composition(join(converse(complement(converse(x2))), converse(converse(composition(complement(x0), x1)))), converse(x2)))))
% 95.56/12.62 = { by lemma 26 R->L }
% 95.56/12.62 meet(one, join(complement(one), converse(composition(converse(join(converse(composition(complement(x0), x1)), converse(converse(complement(converse(x2)))))), converse(x2)))))
% 95.56/12.62 = { by axiom 8 (converse_multiplicativity) R->L }
% 95.56/12.62 meet(one, join(complement(one), converse(converse(composition(x2, join(converse(composition(complement(x0), x1)), converse(converse(complement(converse(x2))))))))))
% 95.56/12.62 = { by axiom 1 (converse_idempotence) }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(converse(composition(complement(x0), x1)), converse(converse(complement(converse(x2))))))))
% 95.56/12.62 = { by axiom 1 (converse_idempotence) }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(converse(composition(complement(x0), x1)), complement(converse(x2))))))
% 95.56/12.62 = { by axiom 2 (maddux1_join_commutativity) }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(x2)), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 31 R->L }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(join(meet(x2, complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))), complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 34 }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(join(meet(complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2)))))), x2), complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 40 R->L }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(join(meet(join(zero, complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))), x2), complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 75 R->L }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(join(complement(join(join(composition(complement(x0), x1), complement(x2)), complement(join(zero, complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))))), complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 35 }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(join(complement(join(join(composition(complement(x0), x1), complement(x2)), meet(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), top))), complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 48 }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(join(complement(join(join(composition(complement(x0), x1), complement(x2)), converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))), complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 28 }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(join(complement(converse(top)), complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 30 }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(join(complement(top), complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 16 }
% 95.56/12.62 meet(one, join(complement(one), composition(x2, join(complement(converse(join(zero, complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.62 = { by lemma 40 }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(converse(complement(join(complement(x2), complement(converse(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(converse(meet(x2, converse(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by lemma 43 R->L }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(converse(join(zero, meet(x2, converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by lemma 72 R->L }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(converse(join(meet(join(composition(complement(x0), x1), complement(x2)), converse(complement(converse(join(composition(complement(x0), x1), complement(x2)))))), meet(x2, converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by lemma 34 }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(converse(join(meet(join(composition(complement(x0), x1), complement(x2)), converse(complement(converse(join(composition(complement(x0), x1), complement(x2)))))), meet(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), x2)))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by lemma 34 }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(converse(join(meet(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), join(composition(complement(x0), x1), complement(x2))), meet(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), x2)))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(converse(join(meet(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), x2), meet(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), join(composition(complement(x0), x1), complement(x2)))))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by axiom 12 (goals) }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(converse(join(meet(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), x2), meet(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), complement(x2))))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by lemma 38 }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(converse(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by axiom 1 (converse_idempotence) }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by lemma 41 }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, join(converse(join(composition(complement(x0), x1), complement(x2))), converse(composition(complement(x0), x1))))))
% 95.56/12.63 = { by axiom 6 (converse_additivity) R->L }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, converse(join(join(composition(complement(x0), x1), complement(x2)), composition(complement(x0), x1))))))
% 95.56/12.63 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, converse(join(composition(complement(x0), x1), join(composition(complement(x0), x1), complement(x2)))))))
% 95.56/12.63 = { by axiom 12 (goals) }
% 95.56/12.63 meet(one, join(complement(one), composition(x2, converse(join(composition(complement(x0), x1), complement(x2))))))
% 95.56/12.63 = { by lemma 70 }
% 95.56/12.63 meet(one, composition(x2, converse(join(composition(complement(x0), x1), complement(x2)))))
% 95.56/12.63 = { by lemma 32 R->L }
% 95.56/12.63 meet(one, composition(join(zero, meet(x2, x2)), converse(join(composition(complement(x0), x1), complement(x2)))))
% 95.56/12.63 = { by lemma 75 R->L }
% 95.56/12.63 meet(one, composition(join(zero, complement(join(join(composition(complement(x0), x1), complement(x2)), complement(x2)))), converse(join(composition(complement(x0), x1), complement(x2)))))
% 95.56/12.63 = { by axiom 12 (goals) R->L }
% 95.56/12.63 meet(one, composition(join(zero, complement(join(join(composition(complement(x0), x1), complement(x2)), join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))))
% 95.56/12.63 = { by lemma 40 }
% 95.56/12.63 meet(one, composition(complement(join(join(composition(complement(x0), x1), complement(x2)), join(composition(complement(x0), x1), complement(x2)))), converse(join(composition(complement(x0), x1), complement(x2)))))
% 95.56/12.63 = { by lemma 44 }
% 95.56/12.63 meet(one, composition(complement(join(composition(complement(x0), x1), complement(x2))), converse(join(composition(complement(x0), x1), complement(x2)))))
% 95.56/12.63 = { by axiom 1 (converse_idempotence) R->L }
% 95.56/12.63 meet(one, composition(complement(converse(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))))
% 95.56/12.63 = { by lemma 34 }
% 95.56/12.63 meet(composition(complement(converse(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))), one)
% 95.56/12.63 = { by lemma 41 R->L }
% 95.56/12.63 meet(composition(complement(converse(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.63 = { by axiom 1 (converse_idempotence) R->L }
% 95.56/12.63 meet(composition(converse(converse(complement(converse(converse(join(composition(complement(x0), x1), complement(x2))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.63 = { by lemma 41 R->L }
% 95.56/12.63 meet(composition(converse(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.63 = { by lemma 73 R->L }
% 95.56/12.63 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), meet(complement(converse(join(composition(complement(x0), x1), complement(x2)))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by lemma 41 R->L }
% 95.56/12.64 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), meet(complement(converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by lemma 57 R->L }
% 95.56/12.64 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by lemma 48 R->L }
% 95.56/12.64 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), meet(complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), top))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by lemma 30 R->L }
% 95.56/12.64 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), meet(complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), converse(top)))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by axiom 5 (def_top) }
% 95.56/12.64 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), meet(complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), converse(join(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))), complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by lemma 27 }
% 95.56/12.64 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), meet(complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.64 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), meet(complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), join(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by lemma 34 }
% 95.56/12.64 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), meet(join(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))), complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by lemma 56 R->L }
% 95.56/12.64 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), complement(join(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), complement(join(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.64 = { by lemma 62 R->L }
% 95.56/12.65 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(join(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))), complement(join(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), meet(complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.65 = { by lemma 58 R->L }
% 95.56/12.65 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(complement(meet(join(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), meet(complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), complement(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.65 = { by lemma 34 R->L }
% 95.56/12.65 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(complement(meet(complement(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))), join(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), meet(complement(converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.65 = { by lemma 74 }
% 95.56/12.65 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(complement(complement(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.65 = { by lemma 41 }
% 95.56/12.65 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(complement(complement(converse(join(composition(complement(x0), x1), complement(x2)))))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.65 = { by lemma 41 }
% 95.56/12.65 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(complement(complement(converse(join(composition(complement(x0), x1), complement(x2))))))))), complement(converse(join(composition(complement(x0), x1), complement(x2)))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.65 = { by lemma 41 }
% 95.56/12.65 meet(composition(converse(join(meet(converse(join(composition(complement(x0), x1), complement(x2))), converse(complement(converse(converse(join(composition(complement(x0), x1), complement(x2))))))), complement(converse(join(composition(complement(x0), x1), complement(x2)))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.65 = { by lemma 72 }
% 95.56/12.65 meet(composition(converse(join(zero, complement(converse(join(composition(complement(x0), x1), complement(x2)))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.65 = { by lemma 43 }
% 95.56/12.65 meet(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))), complement(complement(one)))
% 95.56/12.65 = { by lemma 34 }
% 95.56/12.65 meet(complement(complement(one)), composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))))
% 95.56/12.65 = { by lemma 60 R->L }
% 95.56/12.65 meet(complement(complement(one)), meet(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))), join(complement(one), composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))))))
% 95.56/12.65 = { by lemma 41 R->L }
% 95.56/12.65 meet(complement(complement(one)), meet(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))), join(complement(one), complement(complement(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))))))))
% 95.56/12.65 = { by lemma 42 R->L }
% 95.56/12.65 meet(complement(complement(one)), meet(join(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))), zero), join(complement(one), complement(complement(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))))))))
% 95.56/12.65 = { by lemma 63 R->L }
% 95.56/12.65 meet(meet(complement(complement(one)), join(complement(one), complement(complement(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))))))), join(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))), zero))
% 95.56/12.65 = { by lemma 68 R->L }
% 95.56/12.65 meet(meet(complement(complement(one)), join(complement(one), complement(complement(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))))))), join(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))), meet(complement(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2))))), meet(complement(complement(one)), join(complement(one), complement(complement(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2)))))))))))
% 95.56/12.65 = { by lemma 74 }
% 95.56/12.65 meet(complement(complement(one)), join(complement(one), complement(complement(composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2))))))))
% 95.56/12.65 = { by lemma 41 }
% 95.56/12.65 meet(complement(complement(one)), join(complement(one), composition(converse(complement(converse(join(composition(complement(x0), x1), complement(x2))))), converse(join(composition(complement(x0), x1), complement(x2))))))
% 95.56/12.65 = { by lemma 65 }
% 95.56/12.65 meet(complement(complement(one)), complement(one))
% 95.56/12.65 = { by lemma 62 }
% 95.56/12.65 complement(join(complement(one), one))
% 95.56/12.65 = { by lemma 57 }
% 95.56/12.65 meet(one, complement(one))
% 95.56/12.65 = { by axiom 4 (def_zero) R->L }
% 95.56/12.65 zero
% 95.56/12.65
% 95.56/12.65 Goal 1 (goals_1): join(composition(x2, converse(x1)), x0) = x0.
% 95.56/12.65 Proof:
% 95.56/12.65 join(composition(x2, converse(x1)), x0)
% 95.56/12.65 = { by lemma 67 R->L }
% 95.56/12.65 join(meet(join(composition(x2, converse(x1)), x0), complement(x0)), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.65 = { by lemma 42 R->L }
% 95.56/12.65 join(join(meet(join(composition(x2, converse(x1)), x0), complement(x0)), zero), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.65 = { by lemma 52 R->L }
% 95.56/12.65 join(join(meet(join(composition(x2, converse(x1)), x0), complement(x0)), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.65 = { by lemma 41 R->L }
% 95.56/12.65 join(join(complement(complement(meet(join(composition(x2, converse(x1)), x0), complement(x0)))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 58 }
% 95.56/12.66 join(join(complement(join(x0, complement(join(composition(x2, converse(x1)), x0)))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 49 R->L }
% 95.56/12.66 join(join(complement(join(x0, complement(join(composition(x2, converse(x1)), meet(top, x0))))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 55 R->L }
% 95.56/12.66 join(join(complement(join(x0, complement(join(composition(x2, converse(x1)), complement(join(complement(x0), complement(top))))))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 16 }
% 95.56/12.66 join(join(complement(join(x0, complement(join(composition(x2, converse(x1)), complement(join(complement(x0), zero)))))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by axiom 2 (maddux1_join_commutativity) }
% 95.56/12.66 join(join(complement(join(x0, complement(join(composition(x2, converse(x1)), complement(join(zero, complement(x0))))))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 56 }
% 95.56/12.66 join(join(complement(join(x0, meet(join(zero, complement(x0)), complement(composition(x2, converse(x1)))))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 40 }
% 95.56/12.66 join(join(complement(join(x0, meet(complement(x0), complement(composition(x2, converse(x1)))))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 34 }
% 95.56/12.66 join(join(complement(join(x0, meet(complement(composition(x2, converse(x1))), complement(x0)))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.66 join(join(complement(join(meet(complement(composition(x2, converse(x1))), complement(x0)), x0)), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 73 R->L }
% 95.56/12.66 join(join(complement(join(meet(complement(composition(x2, converse(x1))), complement(x0)), join(meet(composition(x2, converse(x1)), x0), meet(complement(composition(x2, converse(x1))), x0)))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.66 join(join(complement(join(meet(complement(composition(x2, converse(x1))), complement(x0)), join(meet(complement(composition(x2, converse(x1))), x0), meet(composition(x2, converse(x1)), x0)))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by axiom 7 (maddux2_join_associativity) }
% 95.56/12.66 join(join(complement(join(join(meet(complement(composition(x2, converse(x1))), complement(x0)), meet(complement(composition(x2, converse(x1))), x0)), meet(composition(x2, converse(x1)), x0))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 67 }
% 95.56/12.66 join(join(complement(join(complement(composition(x2, converse(x1))), meet(composition(x2, converse(x1)), x0))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.66 join(join(complement(join(meet(composition(x2, converse(x1)), x0), complement(composition(x2, converse(x1))))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 58 R->L }
% 95.56/12.66 join(join(complement(complement(meet(composition(x2, converse(x1)), complement(meet(composition(x2, converse(x1)), x0))))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 69 }
% 95.56/12.66 join(join(complement(complement(meet(composition(x2, converse(x1)), complement(x0)))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 34 R->L }
% 95.56/12.66 join(join(complement(complement(meet(complement(x0), composition(x2, converse(x1))))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 41 }
% 95.56/12.66 join(join(meet(complement(x0), composition(x2, converse(x1))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 20 R->L }
% 95.56/12.66 join(join(meet(composition(one, complement(x0)), composition(x2, converse(x1))), composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 76 R->L }
% 95.56/12.66 join(join(meet(composition(one, complement(x0)), composition(x2, converse(x1))), composition(meet(one, composition(composition(x2, converse(x1)), converse(complement(x0)))), meet(complement(x0), composition(converse(one), composition(x2, converse(x1)))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by axiom 15 (dedekind_law) }
% 95.56/12.66 join(composition(meet(one, composition(composition(x2, converse(x1)), converse(complement(x0)))), meet(complement(x0), composition(converse(one), composition(x2, converse(x1))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 76 }
% 95.56/12.66 join(composition(zero, meet(complement(x0), composition(converse(one), composition(x2, converse(x1))))), meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 52 }
% 95.56/12.66 join(zero, meet(join(composition(x2, converse(x1)), x0), x0))
% 95.56/12.66 = { by lemma 43 }
% 95.56/12.66 meet(join(composition(x2, converse(x1)), x0), x0)
% 95.56/12.66 = { by lemma 42 R->L }
% 95.56/12.66 join(meet(join(composition(x2, converse(x1)), x0), x0), zero)
% 95.56/12.66 = { by lemma 16 R->L }
% 95.56/12.66 join(meet(join(composition(x2, converse(x1)), x0), x0), complement(top))
% 95.56/12.66 = { by lemma 46 R->L }
% 95.56/12.66 join(meet(join(composition(x2, converse(x1)), x0), x0), complement(join(top, composition(x2, converse(x1)))))
% 95.56/12.66 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.66 join(meet(join(composition(x2, converse(x1)), x0), x0), complement(join(composition(x2, converse(x1)), top)))
% 95.56/12.66 = { by axiom 5 (def_top) }
% 95.56/12.66 join(meet(join(composition(x2, converse(x1)), x0), x0), complement(join(composition(x2, converse(x1)), join(x0, complement(x0)))))
% 95.56/12.66 = { by axiom 7 (maddux2_join_associativity) }
% 95.56/12.66 join(meet(join(composition(x2, converse(x1)), x0), x0), complement(join(join(composition(x2, converse(x1)), x0), complement(x0))))
% 95.56/12.66 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 95.56/12.66 join(meet(join(composition(x2, converse(x1)), x0), x0), complement(join(complement(x0), join(composition(x2, converse(x1)), x0))))
% 95.56/12.66 = { by lemma 34 }
% 95.56/12.66 join(meet(x0, join(composition(x2, converse(x1)), x0)), complement(join(complement(x0), join(composition(x2, converse(x1)), x0))))
% 95.56/12.66 = { by lemma 31 }
% 95.56/12.66 x0
% 95.56/12.66 % SZS output end Proof
% 95.56/12.66
% 95.56/12.66 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------