TSTP Solution File: REL028+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL028+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n013.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 13:44:10 EDT 2023
% Result : Theorem 12.45s 2.00s
% Output : Proof 13.45s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL028+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n013.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 20:20:02 EDT 2023
% 0.13/0.35 % CPUTime :
% 12.45/2.00 Command-line arguments: --no-flatten-goal
% 12.45/2.00
% 12.45/2.00 % SZS status Theorem
% 12.45/2.00
% 13.10/2.06 % SZS output start Proof
% 13.10/2.06 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 13.10/2.06 Axiom 2 (composition_identity): composition(X, one) = X.
% 13.10/2.06 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 13.10/2.06 Axiom 4 (goals): join(x1, one) = one.
% 13.10/2.06 Axiom 5 (goals_1): join(x0, one) = one.
% 13.10/2.06 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 13.10/2.06 Axiom 7 (def_top): top = join(X, complement(X)).
% 13.10/2.06 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 13.10/2.06 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 13.10/2.06 Axiom 10 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 13.10/2.06 Axiom 11 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 13.10/2.06 Axiom 12 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 13.10/2.06 Axiom 13 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 13.10/2.06 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 13.10/2.06 Axiom 15 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 13.10/2.06
% 13.10/2.06 Lemma 16: complement(top) = zero.
% 13.10/2.06 Proof:
% 13.10/2.06 complement(top)
% 13.10/2.06 = { by axiom 7 (def_top) }
% 13.10/2.06 complement(join(complement(X), complement(complement(X))))
% 13.10/2.06 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 13.10/2.06 meet(X, complement(X))
% 13.10/2.06 = { by axiom 6 (def_zero) R->L }
% 13.10/2.06 zero
% 13.10/2.06
% 13.10/2.06 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 13.10/2.06 Proof:
% 13.10/2.06 join(X, join(Y, complement(X)))
% 13.10/2.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.06 join(X, join(complement(X), Y))
% 13.10/2.06 = { by axiom 11 (maddux2_join_associativity) }
% 13.10/2.06 join(join(X, complement(X)), Y)
% 13.10/2.06 = { by axiom 7 (def_top) R->L }
% 13.10/2.06 join(top, Y)
% 13.10/2.06 = { by axiom 3 (maddux1_join_commutativity) }
% 13.10/2.06 join(Y, top)
% 13.10/2.06
% 13.10/2.06 Lemma 18: composition(converse(one), X) = X.
% 13.10/2.06 Proof:
% 13.10/2.06 composition(converse(one), X)
% 13.10/2.06 = { by axiom 1 (converse_idempotence) R->L }
% 13.10/2.06 composition(converse(one), converse(converse(X)))
% 13.10/2.06 = { by axiom 8 (converse_multiplicativity) R->L }
% 13.10/2.06 converse(composition(converse(X), one))
% 13.10/2.06 = { by axiom 2 (composition_identity) }
% 13.10/2.06 converse(converse(X))
% 13.10/2.06 = { by axiom 1 (converse_idempotence) }
% 13.10/2.06 X
% 13.10/2.06
% 13.10/2.06 Lemma 19: composition(one, X) = X.
% 13.10/2.06 Proof:
% 13.10/2.06 composition(one, X)
% 13.10/2.06 = { by lemma 18 R->L }
% 13.10/2.06 composition(converse(one), composition(one, X))
% 13.10/2.06 = { by axiom 9 (composition_associativity) }
% 13.10/2.06 composition(composition(converse(one), one), X)
% 13.10/2.06 = { by axiom 2 (composition_identity) }
% 13.10/2.06 composition(converse(one), X)
% 13.10/2.06 = { by lemma 18 }
% 13.10/2.06 X
% 13.10/2.06
% 13.10/2.06 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 13.10/2.06 Proof:
% 13.10/2.06 join(complement(X), complement(X))
% 13.10/2.06 = { by lemma 18 R->L }
% 13.10/2.06 join(complement(X), composition(converse(one), complement(X)))
% 13.10/2.06 = { by lemma 19 R->L }
% 13.10/2.06 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 13.10/2.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.06 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 13.10/2.06 = { by axiom 14 (converse_cancellativity) }
% 13.10/2.06 complement(X)
% 13.10/2.06
% 13.10/2.06 Lemma 21: join(top, complement(X)) = top.
% 13.10/2.06 Proof:
% 13.10/2.06 join(top, complement(X))
% 13.10/2.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.06 join(complement(X), top)
% 13.10/2.06 = { by lemma 17 R->L }
% 13.10/2.06 join(X, join(complement(X), complement(X)))
% 13.10/2.06 = { by lemma 20 }
% 13.10/2.06 join(X, complement(X))
% 13.10/2.06 = { by axiom 7 (def_top) R->L }
% 13.10/2.06 top
% 13.10/2.06
% 13.10/2.06 Lemma 22: join(Y, top) = join(X, top).
% 13.10/2.06 Proof:
% 13.10/2.06 join(Y, top)
% 13.10/2.06 = { by lemma 21 R->L }
% 13.10/2.06 join(Y, join(top, complement(Y)))
% 13.10/2.06 = { by lemma 17 }
% 13.10/2.06 join(top, top)
% 13.10/2.06 = { by lemma 17 R->L }
% 13.10/2.06 join(X, join(top, complement(X)))
% 13.10/2.06 = { by lemma 21 }
% 13.10/2.06 join(X, top)
% 13.10/2.06
% 13.10/2.06 Lemma 23: join(X, top) = top.
% 13.10/2.06 Proof:
% 13.10/2.06 join(X, top)
% 13.10/2.06 = { by lemma 22 }
% 13.10/2.06 join(join(zero, zero), top)
% 13.10/2.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.06 join(top, join(zero, zero))
% 13.10/2.06 = { by lemma 16 R->L }
% 13.10/2.06 join(top, join(zero, complement(top)))
% 13.10/2.06 = { by lemma 16 R->L }
% 13.10/2.06 join(top, join(complement(top), complement(top)))
% 13.10/2.06 = { by lemma 20 }
% 13.10/2.06 join(top, complement(top))
% 13.10/2.06 = { by axiom 7 (def_top) R->L }
% 13.10/2.06 top
% 13.10/2.06
% 13.10/2.06 Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 13.10/2.06 Proof:
% 13.10/2.06 join(meet(X, Y), complement(join(complement(X), Y)))
% 13.10/2.06 = { by axiom 12 (maddux4_definiton_of_meet) }
% 13.10/2.06 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 13.10/2.06 = { by axiom 15 (maddux3_a_kind_of_de_Morgan) R->L }
% 13.10/2.06 X
% 13.10/2.06
% 13.10/2.06 Lemma 25: join(zero, meet(X, X)) = X.
% 13.10/2.06 Proof:
% 13.10/2.06 join(zero, meet(X, X))
% 13.10/2.06 = { by axiom 12 (maddux4_definiton_of_meet) }
% 13.10/2.06 join(zero, complement(join(complement(X), complement(X))))
% 13.10/2.06 = { by axiom 6 (def_zero) }
% 13.10/2.06 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 13.10/2.06 = { by lemma 24 }
% 13.10/2.06 X
% 13.10/2.06
% 13.10/2.06 Lemma 26: complement(complement(X)) = meet(X, X).
% 13.10/2.06 Proof:
% 13.10/2.06 complement(complement(X))
% 13.10/2.06 = { by lemma 20 R->L }
% 13.10/2.06 complement(join(complement(X), complement(X)))
% 13.10/2.06 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 13.10/2.06 meet(X, X)
% 13.10/2.06
% 13.10/2.06 Lemma 27: meet(Y, X) = meet(X, Y).
% 13.10/2.06 Proof:
% 13.10/2.06 meet(Y, X)
% 13.10/2.06 = { by axiom 12 (maddux4_definiton_of_meet) }
% 13.10/2.06 complement(join(complement(Y), complement(X)))
% 13.10/2.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.06 complement(join(complement(X), complement(Y)))
% 13.10/2.06 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 13.10/2.06 meet(X, Y)
% 13.10/2.06
% 13.10/2.06 Lemma 28: complement(join(zero, complement(X))) = meet(X, top).
% 13.10/2.06 Proof:
% 13.10/2.06 complement(join(zero, complement(X)))
% 13.10/2.06 = { by lemma 16 R->L }
% 13.10/2.06 complement(join(complement(top), complement(X)))
% 13.10/2.06 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 13.10/2.06 meet(top, X)
% 13.10/2.06 = { by lemma 27 R->L }
% 13.10/2.06 meet(X, top)
% 13.10/2.06
% 13.10/2.06 Lemma 29: join(X, join(complement(X), Y)) = top.
% 13.10/2.06 Proof:
% 13.10/2.06 join(X, join(complement(X), Y))
% 13.10/2.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.06 join(X, join(Y, complement(X)))
% 13.10/2.06 = { by lemma 17 }
% 13.10/2.06 join(Y, top)
% 13.10/2.06 = { by lemma 22 R->L }
% 13.10/2.06 join(Z, top)
% 13.10/2.06 = { by lemma 23 }
% 13.10/2.06 top
% 13.10/2.06
% 13.10/2.06 Lemma 30: join(X, complement(zero)) = top.
% 13.10/2.06 Proof:
% 13.10/2.06 join(X, complement(zero))
% 13.10/2.06 = { by lemma 25 R->L }
% 13.10/2.06 join(join(zero, meet(X, X)), complement(zero))
% 13.10/2.06 = { by axiom 11 (maddux2_join_associativity) R->L }
% 13.10/2.06 join(zero, join(meet(X, X), complement(zero)))
% 13.10/2.06 = { by axiom 3 (maddux1_join_commutativity) }
% 13.10/2.06 join(zero, join(complement(zero), meet(X, X)))
% 13.10/2.06 = { by lemma 29 }
% 13.10/2.06 top
% 13.10/2.06
% 13.10/2.06 Lemma 31: join(meet(X, Y), meet(X, complement(Y))) = X.
% 13.10/2.06 Proof:
% 13.10/2.06 join(meet(X, Y), meet(X, complement(Y)))
% 13.10/2.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.06 join(meet(X, complement(Y)), meet(X, Y))
% 13.10/2.06 = { by axiom 12 (maddux4_definiton_of_meet) }
% 13.10/2.06 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 13.10/2.06 = { by lemma 24 }
% 13.10/2.06 X
% 13.10/2.06
% 13.10/2.06 Lemma 32: join(zero, meet(X, top)) = X.
% 13.10/2.06 Proof:
% 13.10/2.06 join(zero, meet(X, top))
% 13.10/2.06 = { by lemma 30 R->L }
% 13.10/2.06 join(zero, meet(X, join(complement(zero), complement(zero))))
% 13.10/2.06 = { by lemma 20 }
% 13.10/2.06 join(zero, meet(X, complement(zero)))
% 13.10/2.06 = { by lemma 16 R->L }
% 13.10/2.06 join(complement(top), meet(X, complement(zero)))
% 13.10/2.06 = { by lemma 30 R->L }
% 13.10/2.06 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 13.10/2.06 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 13.10/2.06 join(meet(X, zero), meet(X, complement(zero)))
% 13.10/2.06 = { by lemma 31 }
% 13.10/2.06 X
% 13.10/2.06
% 13.10/2.06 Lemma 33: join(zero, complement(X)) = complement(X).
% 13.10/2.06 Proof:
% 13.10/2.06 join(zero, complement(X))
% 13.10/2.06 = { by lemma 25 R->L }
% 13.10/2.06 join(zero, complement(join(zero, meet(X, X))))
% 13.10/2.06 = { by lemma 26 R->L }
% 13.10/2.06 join(zero, complement(join(zero, complement(complement(X)))))
% 13.10/2.06 = { by lemma 28 }
% 13.10/2.06 join(zero, meet(complement(X), top))
% 13.10/2.06 = { by lemma 32 }
% 13.10/2.07 complement(X)
% 13.10/2.07
% 13.10/2.07 Lemma 34: complement(complement(X)) = X.
% 13.10/2.07 Proof:
% 13.10/2.07 complement(complement(X))
% 13.10/2.07 = { by lemma 33 R->L }
% 13.10/2.07 join(zero, complement(complement(X)))
% 13.10/2.07 = { by lemma 26 }
% 13.10/2.07 join(zero, meet(X, X))
% 13.10/2.07 = { by lemma 25 }
% 13.10/2.07 X
% 13.10/2.07
% 13.10/2.07 Lemma 35: join(X, zero) = X.
% 13.10/2.07 Proof:
% 13.10/2.07 join(X, zero)
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.07 join(zero, X)
% 13.10/2.07 = { by lemma 34 R->L }
% 13.10/2.07 join(zero, complement(complement(X)))
% 13.10/2.07 = { by lemma 26 }
% 13.10/2.07 join(zero, meet(X, X))
% 13.10/2.07 = { by lemma 25 }
% 13.10/2.07 X
% 13.10/2.07
% 13.10/2.07 Lemma 36: meet(X, X) = X.
% 13.10/2.07 Proof:
% 13.10/2.07 meet(X, X)
% 13.10/2.07 = { by lemma 26 R->L }
% 13.10/2.07 complement(complement(X))
% 13.10/2.07 = { by lemma 34 }
% 13.10/2.07 X
% 13.10/2.07
% 13.10/2.07 Lemma 37: meet(X, top) = X.
% 13.10/2.07 Proof:
% 13.10/2.07 meet(X, top)
% 13.10/2.07 = { by lemma 28 R->L }
% 13.10/2.07 complement(join(zero, complement(X)))
% 13.10/2.07 = { by lemma 33 R->L }
% 13.10/2.07 join(zero, complement(join(zero, complement(X))))
% 13.10/2.07 = { by lemma 28 }
% 13.10/2.07 join(zero, meet(X, top))
% 13.10/2.07 = { by lemma 32 }
% 13.10/2.07 X
% 13.10/2.07
% 13.10/2.07 Lemma 38: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 13.10/2.07 Proof:
% 13.10/2.07 complement(join(complement(X), meet(Y, Z)))
% 13.10/2.07 = { by lemma 27 }
% 13.10/2.07 complement(join(complement(X), meet(Z, Y)))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.07 complement(join(meet(Z, Y), complement(X)))
% 13.10/2.07 = { by axiom 12 (maddux4_definiton_of_meet) }
% 13.10/2.07 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 13.10/2.07 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 13.10/2.07 meet(join(complement(Z), complement(Y)), X)
% 13.10/2.07 = { by lemma 27 R->L }
% 13.10/2.07 meet(X, join(complement(Z), complement(Y)))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) }
% 13.10/2.07 meet(X, join(complement(Y), complement(Z)))
% 13.10/2.07
% 13.10/2.07 Lemma 39: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 13.10/2.07 Proof:
% 13.10/2.07 join(complement(X), complement(Y))
% 13.10/2.07 = { by lemma 37 R->L }
% 13.10/2.07 meet(join(complement(X), complement(Y)), top)
% 13.10/2.07 = { by lemma 27 R->L }
% 13.10/2.07 meet(top, join(complement(X), complement(Y)))
% 13.10/2.07 = { by lemma 38 R->L }
% 13.10/2.07 complement(join(complement(top), meet(X, Y)))
% 13.10/2.07 = { by lemma 16 }
% 13.10/2.07 complement(join(zero, meet(X, Y)))
% 13.10/2.07 = { by lemma 27 R->L }
% 13.10/2.07 complement(join(zero, meet(Y, X)))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) }
% 13.10/2.07 complement(join(meet(Y, X), zero))
% 13.10/2.07 = { by lemma 35 }
% 13.10/2.07 complement(meet(Y, X))
% 13.10/2.07 = { by lemma 27 R->L }
% 13.10/2.07 complement(meet(X, Y))
% 13.10/2.07
% 13.10/2.07 Lemma 40: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 13.10/2.07 Proof:
% 13.10/2.07 complement(meet(X, complement(Y)))
% 13.10/2.07 = { by lemma 27 }
% 13.10/2.07 complement(meet(complement(Y), X))
% 13.10/2.07 = { by lemma 33 R->L }
% 13.10/2.07 complement(meet(join(zero, complement(Y)), X))
% 13.10/2.07 = { by lemma 39 R->L }
% 13.10/2.07 join(complement(join(zero, complement(Y))), complement(X))
% 13.10/2.07 = { by lemma 28 }
% 13.10/2.07 join(meet(Y, top), complement(X))
% 13.10/2.07 = { by lemma 37 }
% 13.10/2.07 join(Y, complement(X))
% 13.10/2.07
% 13.10/2.07 Lemma 41: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 13.10/2.07 Proof:
% 13.10/2.07 complement(join(X, complement(Y)))
% 13.10/2.07 = { by lemma 40 R->L }
% 13.10/2.07 complement(complement(meet(Y, complement(X))))
% 13.10/2.07 = { by lemma 26 }
% 13.10/2.07 meet(meet(Y, complement(X)), meet(Y, complement(X)))
% 13.10/2.07 = { by lemma 36 }
% 13.10/2.07 meet(Y, complement(X))
% 13.10/2.07
% 13.10/2.07 Lemma 42: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 13.10/2.07 Proof:
% 13.10/2.07 complement(join(complement(X), Y))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.07 complement(join(Y, complement(X)))
% 13.10/2.07 = { by lemma 41 }
% 13.10/2.07 meet(X, complement(Y))
% 13.10/2.07
% 13.10/2.07 Lemma 43: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 13.10/2.07 Proof:
% 13.10/2.07 complement(meet(complement(X), Y))
% 13.10/2.07 = { by lemma 27 }
% 13.10/2.07 complement(meet(Y, complement(X)))
% 13.10/2.07 = { by lemma 40 }
% 13.10/2.07 join(X, complement(Y))
% 13.10/2.07
% 13.10/2.07 Lemma 44: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 13.10/2.07 Proof:
% 13.10/2.07 meet(meet(X, Y), Z)
% 13.10/2.07 = { by lemma 27 }
% 13.10/2.07 meet(Z, meet(X, Y))
% 13.10/2.07 = { by lemma 36 R->L }
% 13.10/2.07 meet(meet(Z, meet(X, Y)), meet(Z, meet(X, Y)))
% 13.10/2.07 = { by lemma 26 R->L }
% 13.10/2.07 complement(complement(meet(Z, meet(X, Y))))
% 13.10/2.07 = { by lemma 27 }
% 13.10/2.07 complement(complement(meet(Z, meet(Y, X))))
% 13.10/2.07 = { by lemma 39 R->L }
% 13.10/2.07 complement(join(complement(Z), complement(meet(Y, X))))
% 13.10/2.07 = { by lemma 39 R->L }
% 13.10/2.07 complement(join(complement(Z), join(complement(Y), complement(X))))
% 13.10/2.07 = { by axiom 11 (maddux2_join_associativity) }
% 13.10/2.07 complement(join(join(complement(Z), complement(Y)), complement(X)))
% 13.10/2.07 = { by lemma 40 R->L }
% 13.10/2.07 complement(complement(meet(X, complement(join(complement(Z), complement(Y))))))
% 13.10/2.07 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 13.10/2.07 complement(complement(meet(X, meet(Z, Y))))
% 13.10/2.07 = { by lemma 27 R->L }
% 13.10/2.07 complement(complement(meet(X, meet(Y, Z))))
% 13.10/2.07 = { by lemma 34 }
% 13.10/2.07 meet(X, meet(Y, Z))
% 13.10/2.07 = { by lemma 27 R->L }
% 13.10/2.07 meet(X, meet(Z, Y))
% 13.10/2.07
% 13.10/2.07 Lemma 45: join(X, composition(x0, complement(X))) = join(X, composition(x0, top)).
% 13.10/2.07 Proof:
% 13.10/2.07 join(X, composition(x0, complement(X)))
% 13.10/2.07 = { by lemma 19 R->L }
% 13.10/2.07 join(composition(one, X), composition(x0, complement(X)))
% 13.10/2.07 = { by axiom 5 (goals_1) R->L }
% 13.10/2.07 join(composition(join(x0, one), X), composition(x0, complement(X)))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.07 join(composition(join(one, x0), X), composition(x0, complement(X)))
% 13.10/2.07 = { by axiom 13 (composition_distributivity) }
% 13.10/2.07 join(join(composition(one, X), composition(x0, X)), composition(x0, complement(X)))
% 13.10/2.07 = { by lemma 19 }
% 13.10/2.07 join(join(X, composition(x0, X)), composition(x0, complement(X)))
% 13.10/2.07 = { by axiom 11 (maddux2_join_associativity) R->L }
% 13.10/2.07 join(X, join(composition(x0, X), composition(x0, complement(X))))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) }
% 13.10/2.07 join(X, join(composition(x0, complement(X)), composition(x0, X)))
% 13.10/2.07 = { by axiom 1 (converse_idempotence) R->L }
% 13.10/2.07 join(X, join(composition(x0, complement(X)), composition(x0, converse(converse(X)))))
% 13.10/2.07 = { by axiom 1 (converse_idempotence) R->L }
% 13.10/2.07 join(X, converse(converse(join(composition(x0, complement(X)), composition(x0, converse(converse(X)))))))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.07 join(X, converse(converse(join(composition(x0, converse(converse(X))), composition(x0, complement(X))))))
% 13.10/2.07 = { by axiom 10 (converse_additivity) }
% 13.10/2.07 join(X, converse(join(converse(composition(x0, converse(converse(X)))), converse(composition(x0, complement(X))))))
% 13.10/2.07 = { by axiom 8 (converse_multiplicativity) }
% 13.10/2.07 join(X, converse(join(composition(converse(converse(converse(X))), converse(x0)), converse(composition(x0, complement(X))))))
% 13.10/2.07 = { by axiom 1 (converse_idempotence) }
% 13.10/2.07 join(X, converse(join(composition(converse(X), converse(x0)), converse(composition(x0, complement(X))))))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) }
% 13.10/2.07 join(X, converse(join(converse(composition(x0, complement(X))), composition(converse(X), converse(x0)))))
% 13.10/2.07 = { by axiom 8 (converse_multiplicativity) }
% 13.10/2.07 join(X, converse(join(composition(converse(complement(X)), converse(x0)), composition(converse(X), converse(x0)))))
% 13.10/2.07 = { by axiom 13 (composition_distributivity) R->L }
% 13.10/2.07 join(X, converse(composition(join(converse(complement(X)), converse(X)), converse(x0))))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.07 join(X, converse(composition(join(converse(X), converse(complement(X))), converse(x0))))
% 13.10/2.07 = { by axiom 1 (converse_idempotence) R->L }
% 13.10/2.07 join(X, converse(composition(join(converse(converse(converse(X))), converse(complement(X))), converse(x0))))
% 13.10/2.07 = { by axiom 10 (converse_additivity) R->L }
% 13.10/2.07 join(X, converse(composition(converse(join(converse(converse(X)), complement(X))), converse(x0))))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) }
% 13.10/2.07 join(X, converse(composition(converse(join(complement(X), converse(converse(X)))), converse(x0))))
% 13.10/2.07 = { by axiom 8 (converse_multiplicativity) R->L }
% 13.10/2.07 join(X, converse(converse(composition(x0, join(complement(X), converse(converse(X)))))))
% 13.10/2.07 = { by axiom 1 (converse_idempotence) }
% 13.10/2.07 join(X, composition(x0, join(complement(X), converse(converse(X)))))
% 13.10/2.07 = { by axiom 1 (converse_idempotence) }
% 13.10/2.07 join(X, composition(x0, join(complement(X), X)))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) }
% 13.10/2.07 join(X, composition(x0, join(X, complement(X))))
% 13.10/2.07 = { by axiom 7 (def_top) R->L }
% 13.10/2.07 join(X, composition(x0, top))
% 13.10/2.07
% 13.10/2.07 Lemma 46: meet(X, join(X, complement(Y))) = X.
% 13.10/2.07 Proof:
% 13.10/2.07 meet(X, join(X, complement(Y)))
% 13.10/2.07 = { by lemma 40 R->L }
% 13.10/2.07 meet(X, complement(meet(Y, complement(X))))
% 13.10/2.07 = { by lemma 39 R->L }
% 13.10/2.07 meet(X, join(complement(Y), complement(complement(X))))
% 13.10/2.07 = { by lemma 38 R->L }
% 13.10/2.07 complement(join(complement(X), meet(Y, complement(X))))
% 13.10/2.07 = { by lemma 33 R->L }
% 13.10/2.07 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 13.10/2.07 = { by lemma 16 R->L }
% 13.10/2.07 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 13.10/2.07 = { by lemma 23 R->L }
% 13.10/2.07 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 13.10/2.07 = { by lemma 17 R->L }
% 13.10/2.07 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 13.10/2.07 = { by lemma 39 }
% 13.10/2.07 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 13.10/2.07 = { by lemma 27 R->L }
% 13.10/2.07 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 13.10/2.07 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 13.10/2.07 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 13.10/2.07 = { by lemma 27 R->L }
% 13.10/2.07 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 13.10/2.07 = { by lemma 24 }
% 13.10/2.07 X
% 13.10/2.07
% 13.10/2.07 Lemma 47: join(meet(X, Y), meet(complement(Y), X)) = X.
% 13.10/2.07 Proof:
% 13.10/2.07 join(meet(X, Y), meet(complement(Y), X))
% 13.10/2.07 = { by lemma 27 }
% 13.10/2.07 join(meet(X, Y), meet(X, complement(Y)))
% 13.10/2.07 = { by lemma 31 }
% 13.10/2.07 X
% 13.10/2.07
% 13.10/2.07 Lemma 48: join(X, join(Y, complement(join(X, Y)))) = top.
% 13.10/2.07 Proof:
% 13.10/2.07 join(X, join(Y, complement(join(X, Y))))
% 13.10/2.07 = { by axiom 11 (maddux2_join_associativity) }
% 13.10/2.07 join(join(X, Y), complement(join(X, Y)))
% 13.10/2.07 = { by axiom 7 (def_top) R->L }
% 13.10/2.07 top
% 13.10/2.07
% 13.10/2.07 Lemma 49: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 13.10/2.07 Proof:
% 13.10/2.07 meet(X, meet(Y, complement(meet(X, Y))))
% 13.10/2.07 = { by lemma 39 R->L }
% 13.10/2.07 meet(X, meet(Y, join(complement(X), complement(Y))))
% 13.10/2.07 = { by axiom 12 (maddux4_definiton_of_meet) }
% 13.10/2.07 complement(join(complement(X), complement(meet(Y, join(complement(X), complement(Y))))))
% 13.10/2.07 = { by lemma 39 R->L }
% 13.10/2.07 complement(join(complement(X), join(complement(Y), complement(join(complement(X), complement(Y))))))
% 13.10/2.07 = { by lemma 48 }
% 13.10/2.07 complement(top)
% 13.10/2.07 = { by lemma 16 }
% 13.10/2.07 zero
% 13.10/2.07
% 13.10/2.07 Lemma 50: meet(X, join(Y, complement(X))) = meet(X, Y).
% 13.10/2.07 Proof:
% 13.10/2.07 meet(X, join(Y, complement(X)))
% 13.10/2.07 = { by lemma 43 R->L }
% 13.10/2.07 meet(X, complement(meet(complement(Y), X)))
% 13.10/2.07 = { by lemma 47 R->L }
% 13.10/2.07 join(meet(meet(X, complement(meet(complement(Y), X))), Y), meet(complement(Y), meet(X, complement(meet(complement(Y), X)))))
% 13.10/2.07 = { by lemma 49 }
% 13.10/2.07 join(meet(meet(X, complement(meet(complement(Y), X))), Y), zero)
% 13.10/2.07 = { by lemma 35 }
% 13.10/2.07 meet(meet(X, complement(meet(complement(Y), X))), Y)
% 13.10/2.07 = { by lemma 44 }
% 13.10/2.07 meet(X, meet(Y, complement(meet(complement(Y), X))))
% 13.10/2.07 = { by lemma 43 }
% 13.10/2.07 meet(X, meet(Y, join(Y, complement(X))))
% 13.10/2.07 = { by lemma 46 }
% 13.10/2.07 meet(X, Y)
% 13.10/2.07
% 13.10/2.07 Lemma 51: meet(X, join(complement(X), Y)) = meet(X, Y).
% 13.10/2.07 Proof:
% 13.10/2.07 meet(X, join(complement(X), Y))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.07 meet(X, join(Y, complement(X)))
% 13.10/2.07 = { by lemma 50 }
% 13.10/2.07 meet(X, Y)
% 13.10/2.07
% 13.10/2.07 Lemma 52: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 13.10/2.07 Proof:
% 13.10/2.07 join(meet(X, Y), meet(Y, complement(X)))
% 13.10/2.07 = { by lemma 27 }
% 13.10/2.07 join(meet(Y, X), meet(Y, complement(X)))
% 13.10/2.07 = { by lemma 31 }
% 13.10/2.07 Y
% 13.10/2.07
% 13.10/2.07 Lemma 53: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 13.10/2.07 Proof:
% 13.10/2.07 meet(X, complement(meet(X, Y)))
% 13.10/2.07 = { by lemma 27 }
% 13.10/2.07 meet(X, complement(meet(Y, X)))
% 13.10/2.07 = { by lemma 52 R->L }
% 13.10/2.07 join(meet(Y, meet(X, complement(meet(Y, X)))), meet(meet(X, complement(meet(Y, X))), complement(Y)))
% 13.10/2.07 = { by lemma 49 }
% 13.10/2.07 join(zero, meet(meet(X, complement(meet(Y, X))), complement(Y)))
% 13.10/2.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.07 join(meet(meet(X, complement(meet(Y, X))), complement(Y)), zero)
% 13.10/2.07 = { by lemma 35 }
% 13.10/2.07 meet(meet(X, complement(meet(Y, X))), complement(Y))
% 13.10/2.07 = { by lemma 44 }
% 13.10/2.07 meet(X, meet(complement(Y), complement(meet(Y, X))))
% 13.10/2.07 = { by lemma 27 }
% 13.10/2.07 meet(X, meet(complement(meet(Y, X)), complement(Y)))
% 13.10/2.07 = { by lemma 41 R->L }
% 13.10/2.07 meet(X, complement(join(Y, complement(complement(meet(Y, X))))))
% 13.10/2.07 = { by lemma 26 }
% 13.10/2.07 meet(X, complement(join(Y, meet(meet(Y, X), meet(Y, X)))))
% 13.10/2.07 = { by lemma 36 }
% 13.10/2.07 meet(X, complement(join(Y, meet(Y, X))))
% 13.10/2.07 = { by axiom 12 (maddux4_definiton_of_meet) }
% 13.10/2.07 meet(X, complement(join(Y, complement(join(complement(Y), complement(X))))))
% 13.10/2.07 = { by lemma 43 R->L }
% 13.10/2.07 meet(X, complement(complement(meet(complement(Y), join(complement(Y), complement(X))))))
% 13.10/2.07 = { by lemma 46 }
% 13.10/2.07 meet(X, complement(complement(complement(Y))))
% 13.10/2.07 = { by lemma 34 }
% 13.10/2.08 meet(X, complement(Y))
% 13.10/2.08
% 13.10/2.08 Lemma 54: meet(x1, join(complement(one), X)) = meet(X, x1).
% 13.10/2.08 Proof:
% 13.10/2.08 meet(x1, join(complement(one), X))
% 13.10/2.08 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.10/2.08 meet(x1, join(X, complement(one)))
% 13.10/2.08 = { by lemma 40 R->L }
% 13.10/2.08 meet(x1, complement(meet(one, complement(X))))
% 13.45/2.08 = { by lemma 53 R->L }
% 13.45/2.08 meet(x1, complement(meet(x1, meet(one, complement(X)))))
% 13.45/2.08 = { by axiom 4 (goals) R->L }
% 13.45/2.08 meet(x1, complement(meet(x1, meet(join(x1, one), complement(X)))))
% 13.45/2.08 = { by lemma 27 }
% 13.45/2.08 meet(x1, complement(meet(x1, meet(complement(X), join(x1, one)))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.45/2.08 meet(x1, complement(meet(x1, meet(complement(X), join(one, x1)))))
% 13.45/2.08 = { by lemma 27 }
% 13.45/2.08 meet(x1, complement(meet(x1, meet(join(one, x1), complement(X)))))
% 13.45/2.08 = { by lemma 44 R->L }
% 13.45/2.08 meet(x1, complement(meet(meet(x1, complement(X)), join(one, x1))))
% 13.45/2.08 = { by lemma 42 R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, x1))))
% 13.45/2.08 = { by lemma 24 R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(x1, X), complement(join(complement(x1), X)))))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(complement(join(complement(x1), X)), meet(x1, X))))))
% 13.45/2.08 = { by lemma 34 R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(complement(join(complement(x1), X)), complement(complement(meet(x1, X))))))))
% 13.45/2.08 = { by lemma 52 R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(meet(x1, X), join(complement(join(complement(x1), X)), complement(complement(meet(x1, X))))), meet(join(complement(join(complement(x1), X)), complement(complement(meet(x1, X)))), complement(meet(x1, X))))))))
% 13.45/2.08 = { by lemma 27 R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(meet(x1, X), join(complement(join(complement(x1), X)), complement(complement(meet(x1, X))))), meet(complement(meet(x1, X)), join(complement(join(complement(x1), X)), complement(complement(meet(x1, X))))))))))
% 13.45/2.08 = { by lemma 50 }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(meet(x1, X), join(complement(join(complement(x1), X)), complement(complement(meet(x1, X))))), meet(complement(meet(x1, X)), complement(join(complement(x1), X))))))))
% 13.45/2.08 = { by lemma 34 }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(meet(x1, X), join(complement(join(complement(x1), X)), meet(x1, X))), meet(complement(meet(x1, X)), complement(join(complement(x1), X))))))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(meet(x1, X), join(meet(x1, X), complement(join(complement(x1), X)))), meet(complement(meet(x1, X)), complement(join(complement(x1), X))))))))
% 13.45/2.08 = { by lemma 36 R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(meet(x1, X), join(meet(x1, X), meet(complement(join(complement(x1), X)), complement(join(complement(x1), X))))), meet(complement(meet(x1, X)), complement(join(complement(x1), X))))))))
% 13.45/2.08 = { by lemma 26 R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(meet(x1, X), join(meet(x1, X), complement(complement(complement(join(complement(x1), X)))))), meet(complement(meet(x1, X)), complement(join(complement(x1), X))))))))
% 13.45/2.08 = { by lemma 46 }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(x1, X), meet(complement(meet(x1, X)), complement(join(complement(x1), X))))))))
% 13.45/2.08 = { by lemma 27 R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(x1, X), meet(complement(join(complement(x1), X)), complement(meet(x1, X))))))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(one, join(meet(complement(join(complement(x1), X)), complement(meet(x1, X))), meet(x1, X))))))
% 13.45/2.08 = { by axiom 11 (maddux2_join_associativity) }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(join(one, meet(complement(join(complement(x1), X)), complement(meet(x1, X)))), meet(x1, X)))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) }
% 13.45/2.08 meet(x1, complement(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, meet(complement(join(complement(x1), X)), complement(meet(x1, X))))))))
% 13.45/2.08 = { by lemma 35 R->L }
% 13.45/2.08 meet(x1, complement(join(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, meet(complement(join(complement(x1), X)), complement(meet(x1, X)))))), zero)))
% 13.45/2.08 = { by lemma 16 R->L }
% 13.45/2.08 meet(x1, complement(join(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, meet(complement(join(complement(x1), X)), complement(meet(x1, X)))))), complement(top))))
% 13.45/2.08 = { by lemma 42 R->L }
% 13.45/2.08 meet(x1, complement(join(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, complement(join(complement(complement(join(complement(x1), X))), meet(x1, X)))))), complement(top))))
% 13.45/2.08 = { by lemma 23 R->L }
% 13.45/2.08 meet(x1, complement(join(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, complement(join(complement(complement(join(complement(x1), X))), meet(x1, X)))))), complement(join(one, top)))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) }
% 13.45/2.08 meet(x1, complement(join(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, complement(join(complement(complement(join(complement(x1), X))), meet(x1, X)))))), complement(join(top, one)))))
% 13.45/2.08 = { by lemma 48 R->L }
% 13.45/2.08 meet(x1, complement(join(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, complement(join(complement(complement(join(complement(x1), X))), meet(x1, X)))))), complement(join(join(complement(complement(join(complement(x1), X))), join(meet(x1, X), complement(join(complement(complement(join(complement(x1), X))), meet(x1, X))))), one)))))
% 13.45/2.08 = { by axiom 11 (maddux2_join_associativity) R->L }
% 13.45/2.08 meet(x1, complement(join(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, complement(join(complement(complement(join(complement(x1), X))), meet(x1, X)))))), complement(join(complement(complement(join(complement(x1), X))), join(join(meet(x1, X), complement(join(complement(complement(join(complement(x1), X))), meet(x1, X)))), one))))))
% 13.45/2.08 = { by axiom 11 (maddux2_join_associativity) R->L }
% 13.45/2.08 meet(x1, complement(join(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, complement(join(complement(complement(join(complement(x1), X))), meet(x1, X)))))), complement(join(complement(complement(join(complement(x1), X))), join(meet(x1, X), join(complement(join(complement(complement(join(complement(x1), X))), meet(x1, X))), one)))))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) }
% 13.45/2.08 meet(x1, complement(join(meet(complement(join(complement(x1), X)), join(meet(x1, X), join(one, complement(join(complement(complement(join(complement(x1), X))), meet(x1, X)))))), complement(join(complement(complement(join(complement(x1), X))), join(meet(x1, X), join(one, complement(join(complement(complement(join(complement(x1), X))), meet(x1, X))))))))))
% 13.45/2.08 = { by lemma 24 }
% 13.45/2.08 meet(x1, complement(complement(join(complement(x1), X))))
% 13.45/2.08 = { by lemma 42 }
% 13.45/2.08 meet(x1, complement(meet(x1, complement(X))))
% 13.45/2.08 = { by lemma 53 }
% 13.45/2.08 meet(x1, complement(complement(X)))
% 13.45/2.08 = { by lemma 34 }
% 13.45/2.08 meet(x1, X)
% 13.45/2.08 = { by lemma 27 R->L }
% 13.45/2.08 meet(X, x1)
% 13.45/2.08
% 13.45/2.08 Goal 1 (goals_2): composition(x0, x1) = meet(x0, x1).
% 13.45/2.08 Proof:
% 13.45/2.08 composition(x0, x1)
% 13.45/2.08 = { by lemma 34 R->L }
% 13.45/2.08 composition(x0, complement(complement(x1)))
% 13.45/2.08 = { by lemma 47 R->L }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), meet(complement(x1), composition(x0, complement(complement(x1)))))
% 13.45/2.08 = { by axiom 12 (maddux4_definiton_of_meet) }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), complement(join(complement(complement(x1)), complement(composition(x0, complement(complement(x1)))))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), complement(join(complement(composition(x0, complement(complement(x1)))), complement(complement(x1)))))
% 13.45/2.08 = { by lemma 19 R->L }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), complement(join(complement(composition(x0, complement(complement(x1)))), composition(one, complement(complement(x1))))))
% 13.45/2.08 = { by axiom 5 (goals_1) R->L }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), complement(join(complement(composition(x0, complement(complement(x1)))), composition(join(x0, one), complement(complement(x1))))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), complement(join(composition(join(x0, one), complement(complement(x1))), complement(composition(x0, complement(complement(x1)))))))
% 13.45/2.08 = { by axiom 13 (composition_distributivity) }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), complement(join(join(composition(x0, complement(complement(x1))), composition(one, complement(complement(x1)))), complement(composition(x0, complement(complement(x1)))))))
% 13.45/2.08 = { by axiom 11 (maddux2_join_associativity) R->L }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), complement(join(composition(x0, complement(complement(x1))), join(composition(one, complement(complement(x1))), complement(composition(x0, complement(complement(x1))))))))
% 13.45/2.08 = { by axiom 3 (maddux1_join_commutativity) }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), complement(join(composition(x0, complement(complement(x1))), join(complement(composition(x0, complement(complement(x1)))), composition(one, complement(complement(x1)))))))
% 13.45/2.08 = { by lemma 29 }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), complement(top))
% 13.45/2.08 = { by lemma 16 }
% 13.45/2.08 join(meet(composition(x0, complement(complement(x1))), x1), zero)
% 13.45/2.08 = { by lemma 35 }
% 13.45/2.08 meet(composition(x0, complement(complement(x1))), x1)
% 13.45/2.08 = { by lemma 34 }
% 13.45/2.08 meet(composition(x0, x1), x1)
% 13.45/2.08 = { by lemma 27 R->L }
% 13.45/2.08 meet(x1, composition(x0, x1))
% 13.45/2.08 = { by lemma 34 R->L }
% 13.45/2.08 meet(x1, composition(x0, complement(complement(x1))))
% 13.45/2.08 = { by lemma 51 R->L }
% 13.45/2.08 meet(x1, join(complement(x1), composition(x0, complement(complement(x1)))))
% 13.45/2.08 = { by lemma 45 }
% 13.45/2.08 meet(x1, join(complement(x1), composition(x0, top)))
% 13.45/2.08 = { by lemma 51 }
% 13.45/2.08 meet(x1, composition(x0, top))
% 13.45/2.08 = { by lemma 27 }
% 13.45/2.08 meet(composition(x0, top), x1)
% 13.45/2.08 = { by lemma 54 R->L }
% 13.45/2.08 meet(x1, join(complement(one), composition(x0, top)))
% 13.45/2.08 = { by lemma 45 R->L }
% 13.45/2.08 meet(x1, join(complement(one), composition(x0, complement(complement(one)))))
% 13.45/2.08 = { by lemma 54 }
% 13.45/2.08 meet(composition(x0, complement(complement(one))), x1)
% 13.45/2.08 = { by lemma 34 }
% 13.45/2.08 meet(composition(x0, one), x1)
% 13.45/2.08 = { by axiom 2 (composition_identity) }
% 13.45/2.08 meet(x0, x1)
% 13.45/2.08 % SZS output end Proof
% 13.45/2.08
% 13.45/2.08 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------