%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL033+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 12:33:23 PM UTC 2026
% Result : Theorem 143.35s 18.51s
% Output : Proof 144.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : REL033+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/0.35 % Computer : n014.cluster.edu
% 0.10/0.35 % Model : x86_64 x86_64
% 0.10/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.35 % Memory : 8046.5625MB
% 0.10/0.35 % OS : Linux 6.8.0-71-generic
% 0.10/0.35 % CPULimit : 300
% 0.10/0.35 % WCLimit : 300
% 0.10/0.35 % DateTime : Sun Sep 27 22:54:45 UTC 2026
% 0.14/0.36 % CPUTime :
% 0.14/0.36 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 143.35/18.51 Command-line arguments: --no-flatten-goal
% 143.35/18.51
% 143.35/18.51 % SZS status Theorem
% 143.35/18.51
% 144.12/18.64 % SZS output start Proof
% 144.12/18.64 Axiom 1 (def_zero): zero = meet(X, complement(X)).
% 144.12/18.64 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 144.12/18.64 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 144.12/18.64 Axiom 4 (def_top): top = join(X, complement(X)).
% 144.12/18.64 Axiom 5 (composition_identity): composition(X, one) = X.
% 144.12/18.64 Axiom 6 (goals): composition(x0, top) = x0.
% 144.12/18.64 Axiom 7 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 144.12/18.64 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 144.12/18.64 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 144.12/18.64 Axiom 10 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 144.12/18.64 Axiom 11 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 144.12/18.64 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 144.12/18.64 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 144.12/18.64 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 144.12/18.64
% 144.12/18.64 Lemma 15: complement(top) = zero.
% 144.12/18.64 Proof:
% 144.12/18.64 complement(top)
% 144.12/18.64 = { by axiom 4 (def_top) }
% 144.12/18.64 complement(join(complement(X), complement(complement(X))))
% 144.12/18.64 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 144.12/18.64 meet(X, complement(X))
% 144.12/18.64 = { by axiom 1 (def_zero) R->L }
% 144.12/18.64 zero
% 144.12/18.64
% 144.12/18.64 Lemma 16: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 144.12/18.64 Proof:
% 144.12/18.64 join(meet(X, Y), complement(join(complement(X), Y)))
% 144.12/18.64 = { by axiom 7 (maddux4_definiton_of_meet) }
% 144.12/18.64 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 144.12/18.64 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 144.12/18.64 X
% 144.12/18.64
% 144.12/18.64 Lemma 17: join(meet(X, Y), meet(X, complement(Y))) = X.
% 144.12/18.64 Proof:
% 144.12/18.64 join(meet(X, Y), meet(X, complement(Y)))
% 144.12/18.64 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.64 join(meet(X, complement(Y)), meet(X, Y))
% 144.12/18.64 = { by axiom 7 (maddux4_definiton_of_meet) }
% 144.12/18.64 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 144.12/18.64 = { by lemma 16 }
% 144.12/18.64 X
% 144.12/18.64
% 144.12/18.64 Lemma 18: meet(Y, X) = meet(X, Y).
% 144.12/18.64 Proof:
% 144.12/18.64 meet(Y, X)
% 144.12/18.64 = { by axiom 7 (maddux4_definiton_of_meet) }
% 144.12/18.64 complement(join(complement(Y), complement(X)))
% 144.12/18.64 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.64 complement(join(complement(X), complement(Y)))
% 144.12/18.64 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 144.12/18.64 meet(X, Y)
% 144.12/18.64
% 144.12/18.64 Lemma 19: complement(join(zero, complement(X))) = meet(X, top).
% 144.12/18.64 Proof:
% 144.12/18.64 complement(join(zero, complement(X)))
% 144.12/18.64 = { by lemma 15 R->L }
% 144.12/18.64 complement(join(complement(top), complement(X)))
% 144.12/18.64 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 144.12/18.64 meet(top, X)
% 144.12/18.64 = { by lemma 18 R->L }
% 144.12/18.64 meet(X, top)
% 144.12/18.64
% 144.12/18.64 Lemma 20: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 144.12/18.64 Proof:
% 144.12/18.64 converse(composition(converse(X), Y))
% 144.12/18.64 = { by axiom 10 (converse_multiplicativity) }
% 144.12/18.64 composition(converse(Y), converse(converse(X)))
% 144.12/18.64 = { by axiom 2 (converse_idempotence) }
% 144.12/18.64 composition(converse(Y), X)
% 144.12/18.64
% 144.12/18.64 Lemma 21: composition(converse(one), X) = X.
% 144.12/18.64 Proof:
% 144.12/18.64 composition(converse(one), X)
% 144.12/18.64 = { by lemma 20 R->L }
% 144.12/18.64 converse(composition(converse(X), one))
% 144.12/18.64 = { by axiom 5 (composition_identity) }
% 144.12/18.64 converse(converse(X))
% 144.12/18.64 = { by axiom 2 (converse_idempotence) }
% 144.12/18.64 X
% 144.12/18.64
% 144.12/18.64 Lemma 22: composition(one, X) = X.
% 144.12/18.64 Proof:
% 144.12/18.64 composition(one, X)
% 144.12/18.64 = { by lemma 21 R->L }
% 144.12/18.64 composition(converse(one), composition(one, X))
% 144.12/18.64 = { by axiom 11 (composition_associativity) }
% 144.12/18.64 composition(composition(converse(one), one), X)
% 144.12/18.64 = { by axiom 5 (composition_identity) }
% 144.12/18.65 composition(converse(one), X)
% 144.12/18.65 = { by lemma 21 }
% 144.12/18.65 X
% 144.12/18.65
% 144.12/18.65 Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 144.12/18.65 Proof:
% 144.12/18.65 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 144.12/18.65 = { by axiom 14 (converse_cancellativity) }
% 144.12/18.65 complement(X)
% 144.12/18.65
% 144.12/18.65 Lemma 24: join(complement(X), complement(X)) = complement(X).
% 144.12/18.65 Proof:
% 144.12/18.65 join(complement(X), complement(X))
% 144.12/18.65 = { by lemma 21 R->L }
% 144.12/18.65 join(complement(X), composition(converse(one), complement(X)))
% 144.12/18.65 = { by lemma 22 R->L }
% 144.12/18.65 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 144.12/18.65 = { by lemma 23 }
% 144.12/18.65 complement(X)
% 144.12/18.65
% 144.12/18.65 Lemma 25: join(zero, complement(complement(X))) = X.
% 144.12/18.65 Proof:
% 144.12/18.65 join(zero, complement(complement(X)))
% 144.12/18.65 = { by axiom 1 (def_zero) }
% 144.12/18.65 join(meet(X, complement(X)), complement(complement(X)))
% 144.12/18.65 = { by lemma 24 R->L }
% 144.12/18.65 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 144.12/18.65 = { by lemma 16 }
% 144.12/18.65 X
% 144.12/18.65
% 144.12/18.65 Lemma 26: meet(top, complement(X)) = complement(X).
% 144.12/18.65 Proof:
% 144.12/18.65 meet(top, complement(X))
% 144.12/18.65 = { by lemma 18 }
% 144.12/18.65 meet(complement(X), top)
% 144.12/18.65 = { by lemma 19 R->L }
% 144.12/18.65 complement(join(zero, complement(complement(X))))
% 144.12/18.65 = { by lemma 25 }
% 144.12/18.65 complement(X)
% 144.12/18.65
% 144.12/18.65 Lemma 27: complement(zero) = top.
% 144.12/18.65 Proof:
% 144.12/18.65 complement(zero)
% 144.12/18.65 = { by lemma 17 R->L }
% 144.12/18.65 join(meet(complement(zero), top), meet(complement(zero), complement(top)))
% 144.12/18.65 = { by lemma 15 }
% 144.12/18.65 join(meet(complement(zero), top), meet(complement(zero), zero))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.65 join(meet(complement(zero), zero), meet(complement(zero), top))
% 144.12/18.65 = { by lemma 18 R->L }
% 144.12/18.65 join(meet(zero, complement(zero)), meet(complement(zero), top))
% 144.12/18.65 = { by axiom 1 (def_zero) R->L }
% 144.12/18.65 join(zero, meet(complement(zero), top))
% 144.12/18.65 = { by lemma 18 R->L }
% 144.12/18.65 join(zero, meet(top, complement(zero)))
% 144.12/18.65 = { by lemma 26 }
% 144.12/18.65 join(zero, complement(zero))
% 144.12/18.65 = { by axiom 4 (def_top) R->L }
% 144.12/18.65 top
% 144.12/18.65
% 144.12/18.65 Lemma 28: join(X, join(Y, complement(X))) = join(Y, top).
% 144.12/18.65 Proof:
% 144.12/18.65 join(X, join(Y, complement(X)))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 join(X, join(complement(X), Y))
% 144.12/18.65 = { by axiom 9 (maddux2_join_associativity) }
% 144.12/18.65 join(join(X, complement(X)), Y)
% 144.12/18.65 = { by axiom 4 (def_top) R->L }
% 144.12/18.65 join(top, Y)
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.65 join(Y, top)
% 144.12/18.65
% 144.12/18.65 Lemma 29: join(top, complement(X)) = top.
% 144.12/18.65 Proof:
% 144.12/18.65 join(top, complement(X))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 join(complement(X), top)
% 144.12/18.65 = { by lemma 28 R->L }
% 144.12/18.65 join(X, join(complement(X), complement(X)))
% 144.12/18.65 = { by lemma 24 }
% 144.12/18.65 join(X, complement(X))
% 144.12/18.65 = { by axiom 4 (def_top) R->L }
% 144.12/18.65 top
% 144.12/18.65
% 144.12/18.65 Lemma 30: join(Y, top) = join(X, top).
% 144.12/18.65 Proof:
% 144.12/18.65 join(Y, top)
% 144.12/18.65 = { by lemma 29 R->L }
% 144.12/18.65 join(Y, join(top, complement(Y)))
% 144.12/18.65 = { by lemma 28 }
% 144.12/18.65 join(top, top)
% 144.12/18.65 = { by lemma 28 R->L }
% 144.12/18.65 join(X, join(top, complement(X)))
% 144.12/18.65 = { by lemma 29 }
% 144.12/18.65 join(X, top)
% 144.12/18.65
% 144.12/18.65 Lemma 31: join(X, top) = top.
% 144.12/18.65 Proof:
% 144.12/18.65 join(X, top)
% 144.12/18.65 = { by lemma 30 }
% 144.12/18.65 join(zero, top)
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 join(top, zero)
% 144.12/18.65 = { by lemma 15 R->L }
% 144.12/18.65 join(top, complement(top))
% 144.12/18.65 = { by axiom 4 (def_top) R->L }
% 144.12/18.65 top
% 144.12/18.65
% 144.12/18.65 Lemma 32: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 144.12/18.65 Proof:
% 144.12/18.65 converse(join(X, converse(Y)))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 converse(join(converse(Y), X))
% 144.12/18.65 = { by axiom 8 (converse_additivity) }
% 144.12/18.65 join(converse(converse(Y)), converse(X))
% 144.12/18.65 = { by axiom 2 (converse_idempotence) }
% 144.12/18.65 join(Y, converse(X))
% 144.12/18.65
% 144.12/18.65 Lemma 33: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 144.12/18.65 Proof:
% 144.12/18.65 converse(join(converse(X), Y))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 converse(join(Y, converse(X)))
% 144.12/18.65 = { by lemma 32 }
% 144.12/18.65 join(X, converse(Y))
% 144.12/18.65
% 144.12/18.65 Lemma 34: join(X, join(complement(X), Y)) = top.
% 144.12/18.65 Proof:
% 144.12/18.65 join(X, join(complement(X), Y))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 join(X, join(Y, complement(X)))
% 144.12/18.65 = { by lemma 28 }
% 144.12/18.65 join(Y, top)
% 144.12/18.65 = { by lemma 30 R->L }
% 144.12/18.65 join(Z, top)
% 144.12/18.65 = { by lemma 31 }
% 144.12/18.65 top
% 144.12/18.65
% 144.12/18.65 Lemma 35: join(X, converse(top)) = top.
% 144.12/18.65 Proof:
% 144.12/18.65 join(X, converse(top))
% 144.12/18.65 = { by axiom 4 (def_top) }
% 144.12/18.65 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 144.12/18.65 = { by lemma 33 }
% 144.12/18.65 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 144.12/18.65 = { by lemma 34 }
% 144.12/18.65 top
% 144.12/18.65
% 144.12/18.65 Lemma 36: converse(top) = top.
% 144.12/18.65 Proof:
% 144.12/18.65 converse(top)
% 144.12/18.65 = { by lemma 31 R->L }
% 144.12/18.65 converse(join(X, top))
% 144.12/18.65 = { by axiom 8 (converse_additivity) }
% 144.12/18.65 join(converse(X), converse(top))
% 144.12/18.65 = { by lemma 35 }
% 144.12/18.65 top
% 144.12/18.65
% 144.12/18.65 Lemma 37: join(zero, meet(X, top)) = X.
% 144.12/18.65 Proof:
% 144.12/18.65 join(zero, meet(X, top))
% 144.12/18.65 = { by lemma 27 R->L }
% 144.12/18.65 join(zero, meet(X, complement(zero)))
% 144.12/18.65 = { by lemma 15 R->L }
% 144.12/18.65 join(complement(top), meet(X, complement(zero)))
% 144.12/18.65 = { by lemma 29 R->L }
% 144.12/18.65 join(complement(join(top, complement(X))), meet(X, complement(zero)))
% 144.12/18.65 = { by lemma 27 R->L }
% 144.12/18.65 join(complement(join(complement(zero), complement(X))), meet(X, complement(zero)))
% 144.12/18.65 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 144.12/18.65 join(meet(zero, X), meet(X, complement(zero)))
% 144.12/18.65 = { by lemma 18 R->L }
% 144.12/18.65 join(meet(X, zero), meet(X, complement(zero)))
% 144.12/18.65 = { by lemma 17 }
% 144.12/18.65 X
% 144.12/18.65
% 144.12/18.65 Lemma 38: join(zero, complement(X)) = complement(X).
% 144.12/18.65 Proof:
% 144.12/18.65 join(zero, complement(X))
% 144.12/18.65 = { by lemma 26 R->L }
% 144.12/18.65 join(zero, meet(top, complement(X)))
% 144.12/18.65 = { by lemma 18 }
% 144.12/18.65 join(zero, meet(complement(X), top))
% 144.12/18.65 = { by lemma 37 }
% 144.12/18.65 complement(X)
% 144.12/18.65
% 144.12/18.65 Lemma 39: meet(X, top) = X.
% 144.12/18.65 Proof:
% 144.12/18.65 meet(X, top)
% 144.12/18.65 = { by lemma 19 R->L }
% 144.12/18.65 complement(join(zero, complement(X)))
% 144.12/18.65 = { by lemma 38 R->L }
% 144.12/18.65 join(zero, complement(join(zero, complement(X))))
% 144.12/18.65 = { by lemma 19 }
% 144.12/18.65 join(zero, meet(X, top))
% 144.12/18.65 = { by lemma 37 }
% 144.12/18.65 X
% 144.12/18.65
% 144.12/18.65 Lemma 40: join(X, X) = X.
% 144.12/18.65 Proof:
% 144.12/18.65 join(X, X)
% 144.12/18.65 = { by lemma 39 R->L }
% 144.12/18.65 join(X, meet(X, top))
% 144.12/18.65 = { by lemma 39 R->L }
% 144.12/18.65 join(meet(X, top), meet(X, top))
% 144.12/18.65 = { by lemma 18 }
% 144.12/18.65 join(meet(top, X), meet(X, top))
% 144.12/18.65 = { by lemma 18 }
% 144.12/18.65 join(meet(top, X), meet(top, X))
% 144.12/18.65 = { by axiom 7 (maddux4_definiton_of_meet) }
% 144.12/18.65 join(meet(top, X), complement(join(complement(top), complement(X))))
% 144.12/18.65 = { by axiom 7 (maddux4_definiton_of_meet) }
% 144.12/18.65 join(complement(join(complement(top), complement(X))), complement(join(complement(top), complement(X))))
% 144.12/18.65 = { by lemma 24 }
% 144.12/18.65 complement(join(complement(top), complement(X)))
% 144.12/18.65 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 144.12/18.65 meet(top, X)
% 144.12/18.65 = { by lemma 18 R->L }
% 144.12/18.65 meet(X, top)
% 144.12/18.65 = { by lemma 39 }
% 144.12/18.65 X
% 144.12/18.65
% 144.12/18.65 Lemma 41: complement(complement(X)) = X.
% 144.12/18.65 Proof:
% 144.12/18.65 complement(complement(X))
% 144.12/18.65 = { by lemma 38 R->L }
% 144.12/18.65 join(zero, complement(complement(X)))
% 144.12/18.65 = { by lemma 25 }
% 144.12/18.65 X
% 144.12/18.65
% 144.12/18.65 Lemma 42: join(zero, X) = X.
% 144.12/18.65 Proof:
% 144.12/18.65 join(zero, X)
% 144.12/18.65 = { by lemma 41 R->L }
% 144.12/18.65 join(zero, complement(complement(X)))
% 144.12/18.65 = { by lemma 25 }
% 144.12/18.65 X
% 144.12/18.65
% 144.12/18.65 Lemma 43: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 144.12/18.65 Proof:
% 144.12/18.65 converse(composition(X, converse(Y)))
% 144.12/18.65 = { by axiom 10 (converse_multiplicativity) }
% 144.12/18.65 composition(converse(converse(Y)), converse(X))
% 144.12/18.65 = { by axiom 2 (converse_idempotence) }
% 144.12/18.65 composition(Y, converse(X))
% 144.12/18.65
% 144.12/18.65 Lemma 44: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 144.12/18.65 Proof:
% 144.12/18.65 join(X, join(Y, Z))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 join(join(Y, Z), X)
% 144.12/18.65 = { by axiom 9 (maddux2_join_associativity) R->L }
% 144.12/18.65 join(Y, join(Z, X))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.65 join(Y, join(X, Z))
% 144.12/18.65
% 144.12/18.65 Lemma 45: join(complement(converse(X)), converse(join(X, Y))) = top.
% 144.12/18.65 Proof:
% 144.12/18.65 join(complement(converse(X)), converse(join(X, Y)))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 join(complement(converse(X)), converse(join(Y, X)))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.65 join(converse(join(Y, X)), complement(converse(X)))
% 144.12/18.65 = { by axiom 8 (converse_additivity) }
% 144.12/18.65 join(join(converse(Y), converse(X)), complement(converse(X)))
% 144.12/18.65 = { by axiom 9 (maddux2_join_associativity) R->L }
% 144.12/18.65 join(converse(Y), join(converse(X), complement(converse(X))))
% 144.12/18.65 = { by lemma 44 }
% 144.12/18.65 join(converse(X), join(converse(Y), complement(converse(X))))
% 144.12/18.65 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.65 join(converse(X), join(complement(converse(X)), converse(Y)))
% 144.12/18.65 = { by lemma 34 }
% 144.12/18.65 top
% 144.12/18.65
% 144.12/18.65 Lemma 46: composition(converse(x0), complement(x0)) = zero.
% 144.12/18.65 Proof:
% 144.12/18.65 composition(converse(x0), complement(x0))
% 144.12/18.65 = { by lemma 42 R->L }
% 144.12/18.65 join(zero, composition(converse(x0), complement(x0)))
% 144.12/18.65 = { by lemma 15 R->L }
% 144.12/18.65 join(complement(top), composition(converse(x0), complement(x0)))
% 144.12/18.65 = { by axiom 6 (goals) R->L }
% 144.12/18.65 join(complement(top), composition(converse(x0), complement(composition(x0, top))))
% 144.12/18.65 = { by lemma 23 }
% 144.12/18.65 complement(top)
% 144.12/18.65 = { by lemma 15 }
% 144.12/18.65 zero
% 144.12/18.65
% 144.12/18.65 Lemma 47: composition(complement(x0), top) = complement(x0).
% 144.12/18.65 Proof:
% 144.12/18.65 composition(complement(x0), top)
% 144.12/18.65 = { by lemma 36 R->L }
% 144.12/18.65 composition(complement(x0), converse(top))
% 144.12/18.65 = { by lemma 43 R->L }
% 144.12/18.65 converse(composition(top, converse(complement(x0))))
% 144.12/18.65 = { by lemma 35 R->L }
% 144.12/18.65 converse(composition(join(one, converse(top)), converse(complement(x0))))
% 144.12/18.65 = { by axiom 12 (composition_distributivity) }
% 144.12/18.65 converse(join(composition(one, converse(complement(x0))), composition(converse(top), converse(complement(x0)))))
% 144.12/18.65 = { by lemma 22 }
% 144.12/18.65 converse(join(converse(complement(x0)), composition(converse(top), converse(complement(x0)))))
% 144.12/18.65 = { by lemma 36 }
% 144.12/18.65 converse(join(converse(complement(x0)), composition(top, converse(complement(x0)))))
% 144.12/18.65 = { by lemma 33 }
% 144.12/18.65 join(complement(x0), converse(composition(top, converse(complement(x0)))))
% 144.12/18.65 = { by lemma 43 }
% 144.12/18.65 join(complement(x0), composition(complement(x0), converse(top)))
% 144.12/18.65 = { by lemma 36 }
% 144.12/18.65 join(complement(x0), composition(complement(x0), top))
% 144.12/18.65 = { by lemma 27 R->L }
% 144.12/18.65 join(complement(x0), composition(complement(x0), complement(zero)))
% 144.12/18.65 = { by lemma 15 R->L }
% 144.12/18.66 join(complement(x0), composition(complement(x0), complement(complement(top))))
% 144.12/18.66 = { by lemma 45 R->L }
% 144.12/18.66 join(complement(x0), composition(complement(x0), complement(complement(join(complement(converse(zero)), converse(join(zero, meet(converse(zero), converse(zero)))))))))
% 144.12/18.66 = { by axiom 7 (maddux4_definiton_of_meet) }
% 144.12/18.66 join(complement(x0), composition(complement(x0), complement(complement(join(complement(converse(zero)), converse(join(zero, complement(join(complement(converse(zero)), complement(converse(zero)))))))))))
% 144.12/18.66 = { by axiom 1 (def_zero) }
% 144.12/18.66 join(complement(x0), composition(complement(x0), complement(complement(join(complement(converse(zero)), converse(join(meet(converse(zero), complement(converse(zero))), complement(join(complement(converse(zero)), complement(converse(zero)))))))))))
% 144.12/18.66 = { by lemma 16 }
% 144.12/18.66 join(complement(x0), composition(complement(x0), complement(complement(join(complement(converse(zero)), converse(converse(zero)))))))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.66 join(complement(x0), composition(complement(x0), complement(complement(join(converse(converse(zero)), complement(converse(zero)))))))
% 144.12/18.66 = { by axiom 2 (converse_idempotence) }
% 144.12/18.66 join(complement(x0), composition(complement(x0), complement(complement(join(zero, complement(converse(zero)))))))
% 144.12/18.66 = { by lemma 19 }
% 144.12/18.66 join(complement(x0), composition(complement(x0), complement(meet(converse(zero), top))))
% 144.12/18.66 = { by lemma 39 }
% 144.12/18.66 join(complement(x0), composition(complement(x0), complement(converse(zero))))
% 144.12/18.66 = { by axiom 2 (converse_idempotence) R->L }
% 144.12/18.66 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(zero))))
% 144.12/18.66 = { by lemma 46 R->L }
% 144.12/18.66 join(complement(x0), composition(converse(converse(complement(x0))), complement(converse(composition(converse(x0), complement(x0))))))
% 144.12/18.66 = { by lemma 20 }
% 144.12/18.66 join(complement(x0), composition(converse(converse(complement(x0))), complement(composition(converse(complement(x0)), x0))))
% 144.12/18.66 = { by lemma 23 }
% 144.12/18.66 complement(x0)
% 144.12/18.66
% 144.12/18.66 Lemma 48: meet(join(X, Y), join(X, complement(Y))) = X.
% 144.12/18.66 Proof:
% 144.12/18.66 meet(join(X, Y), join(X, complement(Y)))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.66 meet(join(X, Y), join(complement(Y), X))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.66 meet(join(Y, X), join(complement(Y), X))
% 144.12/18.66 = { by lemma 39 R->L }
% 144.12/18.66 meet(join(Y, X), join(complement(Y), meet(X, top)))
% 144.12/18.66 = { by lemma 19 R->L }
% 144.12/18.66 meet(join(Y, X), join(complement(Y), complement(join(zero, complement(X)))))
% 144.12/18.66 = { by lemma 39 R->L }
% 144.12/18.66 meet(join(Y, meet(X, top)), join(complement(Y), complement(join(zero, complement(X)))))
% 144.12/18.66 = { by lemma 19 R->L }
% 144.12/18.66 meet(join(Y, complement(join(zero, complement(X)))), join(complement(Y), complement(join(zero, complement(X)))))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.66 meet(join(Y, complement(join(zero, complement(X)))), join(complement(join(zero, complement(X))), complement(Y)))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.66 meet(join(complement(join(zero, complement(X))), Y), join(complement(join(zero, complement(X))), complement(Y)))
% 144.12/18.66 = { by lemma 18 }
% 144.12/18.66 meet(join(complement(join(zero, complement(X))), complement(Y)), join(complement(join(zero, complement(X))), Y))
% 144.12/18.66 = { by axiom 7 (maddux4_definiton_of_meet) }
% 144.12/18.66 complement(join(complement(join(complement(join(zero, complement(X))), complement(Y))), complement(join(complement(join(zero, complement(X))), Y))))
% 144.12/18.66 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 144.12/18.66 complement(join(meet(join(zero, complement(X)), Y), complement(join(complement(join(zero, complement(X))), Y))))
% 144.12/18.66 = { by lemma 16 }
% 144.12/18.66 complement(join(zero, complement(X)))
% 144.12/18.66 = { by lemma 19 }
% 144.12/18.66 meet(X, top)
% 144.12/18.66 = { by lemma 39 }
% 144.12/18.66 X
% 144.12/18.66
% 144.12/18.66 Lemma 49: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 144.12/18.66 Proof:
% 144.12/18.66 join(composition(X, Y), composition(X, Z))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.66 join(composition(X, Z), composition(X, Y))
% 144.12/18.66 = { by axiom 2 (converse_idempotence) R->L }
% 144.12/18.66 join(composition(X, Z), composition(X, converse(converse(Y))))
% 144.12/18.66 = { by axiom 2 (converse_idempotence) R->L }
% 144.12/18.66 converse(converse(join(composition(X, Z), composition(X, converse(converse(Y))))))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.66 converse(converse(join(composition(X, converse(converse(Y))), composition(X, Z))))
% 144.12/18.66 = { by axiom 8 (converse_additivity) }
% 144.12/18.66 converse(join(converse(composition(X, converse(converse(Y)))), converse(composition(X, Z))))
% 144.12/18.66 = { by lemma 43 }
% 144.12/18.66 converse(join(composition(converse(Y), converse(X)), converse(composition(X, Z))))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.66 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 144.12/18.66 = { by axiom 10 (converse_multiplicativity) }
% 144.12/18.66 converse(join(composition(converse(Z), converse(X)), composition(converse(Y), converse(X))))
% 144.12/18.66 = { by axiom 12 (composition_distributivity) R->L }
% 144.12/18.66 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.66 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 144.12/18.66 = { by lemma 32 R->L }
% 144.12/18.66 converse(composition(converse(join(Z, converse(converse(Y)))), converse(X)))
% 144.12/18.66 = { by axiom 10 (converse_multiplicativity) R->L }
% 144.12/18.66 converse(converse(composition(X, join(Z, converse(converse(Y))))))
% 144.12/18.66 = { by axiom 2 (converse_idempotence) }
% 144.12/18.66 composition(X, join(Z, converse(converse(Y))))
% 144.12/18.66 = { by axiom 2 (converse_idempotence) }
% 144.12/18.66 composition(X, join(Z, Y))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.66 composition(X, join(Y, Z))
% 144.12/18.66
% 144.12/18.66 Lemma 50: join(complement(x0), composition(join(complement(x0), X), Y)) = join(complement(x0), composition(X, Y)).
% 144.12/18.66 Proof:
% 144.12/18.66 join(complement(x0), composition(join(complement(x0), X), Y))
% 144.12/18.66 = { by lemma 47 R->L }
% 144.12/18.66 join(composition(complement(x0), top), composition(join(complement(x0), X), Y))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.66 join(composition(join(complement(x0), X), Y), composition(complement(x0), top))
% 144.12/18.66 = { by axiom 12 (composition_distributivity) }
% 144.12/18.66 join(join(composition(complement(x0), Y), composition(X, Y)), composition(complement(x0), top))
% 144.12/18.66 = { by axiom 9 (maddux2_join_associativity) R->L }
% 144.12/18.66 join(composition(complement(x0), Y), join(composition(X, Y), composition(complement(x0), top)))
% 144.12/18.66 = { by lemma 44 }
% 144.12/18.66 join(composition(X, Y), join(composition(complement(x0), Y), composition(complement(x0), top)))
% 144.12/18.66 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.66 join(composition(X, Y), join(composition(complement(x0), top), composition(complement(x0), Y)))
% 144.12/18.66 = { by lemma 49 }
% 144.12/18.67 join(composition(X, Y), composition(complement(x0), join(top, Y)))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.67 join(composition(X, Y), composition(complement(x0), join(Y, top)))
% 144.12/18.67 = { by lemma 30 R->L }
% 144.12/18.67 join(composition(X, Y), composition(complement(x0), join(Z, top)))
% 144.12/18.67 = { by lemma 31 }
% 144.12/18.67 join(composition(X, Y), composition(complement(x0), top))
% 144.12/18.67 = { by lemma 47 }
% 144.12/18.67 join(composition(X, Y), complement(x0))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.67 join(complement(x0), composition(X, Y))
% 144.12/18.67
% 144.12/18.67 Lemma 51: meet(X, join(Y, complement(X))) = meet(X, Y).
% 144.12/18.67 Proof:
% 144.12/18.67 meet(X, join(Y, complement(X)))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.67 meet(X, join(complement(X), Y))
% 144.12/18.67 = { by axiom 7 (maddux4_definiton_of_meet) }
% 144.12/18.67 complement(join(complement(X), complement(join(complement(X), Y))))
% 144.12/18.67 = { by lemma 16 R->L }
% 144.12/18.67 complement(join(complement(X), complement(join(complement(X), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X))))))))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.67 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), meet(Y, complement(X)))))))
% 144.12/18.67 = { by lemma 24 R->L }
% 144.12/18.67 complement(join(complement(X), complement(join(complement(X), join(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X)))), meet(Y, complement(X)))))))
% 144.12/18.67 = { by axiom 9 (maddux2_join_associativity) R->L }
% 144.12/18.67 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), join(complement(join(complement(Y), complement(X))), meet(Y, complement(X))))))))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.67 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X)))))))))
% 144.12/18.67 = { by lemma 16 }
% 144.12/18.67 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), Y)))))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.67 complement(join(complement(X), complement(join(complement(X), join(Y, complement(join(complement(Y), complement(X))))))))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.67 complement(join(complement(X), complement(join(complement(X), join(Y, complement(join(complement(X), complement(Y))))))))
% 144.12/18.67 = { by axiom 9 (maddux2_join_associativity) }
% 144.12/18.67 complement(join(complement(X), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 144.12/18.67 = { by lemma 48 R->L }
% 144.12/18.67 complement(join(meet(join(complement(X), Y), join(complement(X), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 144.12/18.67 = { by lemma 18 }
% 144.12/18.67 complement(join(meet(join(complement(X), complement(Y)), join(complement(X), Y)), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.67 complement(join(meet(join(complement(X), complement(Y)), join(complement(X), Y)), complement(join(complement(join(complement(X), complement(Y))), join(complement(X), Y)))))
% 144.12/18.67 = { by lemma 16 }
% 144.12/18.67 complement(join(complement(X), complement(Y)))
% 144.12/18.67 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 144.12/18.67 meet(X, Y)
% 144.12/18.67
% 144.12/18.67 Lemma 52: join(x0, composition(meet(X, x0), top)) = x0.
% 144.12/18.67 Proof:
% 144.12/18.67 join(x0, composition(meet(X, x0), top))
% 144.12/18.67 = { by lemma 27 R->L }
% 144.12/18.67 join(x0, composition(meet(X, x0), complement(zero)))
% 144.12/18.67 = { by axiom 2 (converse_idempotence) R->L }
% 144.12/18.67 join(x0, composition(converse(converse(meet(X, x0))), complement(zero)))
% 144.12/18.67 = { by lemma 41 R->L }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(zero)))
% 144.12/18.67 = { by lemma 46 R->L }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(converse(x0), complement(x0)))))
% 144.12/18.67 = { by lemma 48 R->L }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(meet(join(converse(x0), complement(converse(meet(X, converse(converse(x0)))))), join(converse(x0), complement(complement(converse(meet(X, converse(converse(x0)))))))), complement(x0)))))
% 144.12/18.67 = { by lemma 18 }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(meet(join(converse(x0), complement(converse(meet(converse(converse(x0)), X)))), join(converse(x0), complement(complement(converse(meet(X, converse(converse(x0)))))))), complement(x0)))))
% 144.12/18.67 = { by axiom 2 (converse_idempotence) R->L }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(meet(join(converse(converse(converse(x0))), complement(converse(meet(converse(converse(x0)), X)))), join(converse(x0), complement(complement(converse(meet(X, converse(converse(x0)))))))), complement(x0)))))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(meet(join(complement(converse(meet(converse(converse(x0)), X))), converse(converse(converse(x0)))), join(converse(x0), complement(complement(converse(meet(X, converse(converse(x0)))))))), complement(x0)))))
% 144.12/18.67 = { by lemma 16 R->L }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(meet(join(complement(converse(meet(converse(converse(x0)), X))), converse(join(meet(converse(converse(x0)), X), complement(join(complement(converse(converse(x0))), X))))), join(converse(x0), complement(complement(converse(meet(X, converse(converse(x0)))))))), complement(x0)))))
% 144.12/18.67 = { by lemma 45 }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(meet(top, join(converse(x0), complement(complement(converse(meet(X, converse(converse(x0)))))))), complement(x0)))))
% 144.12/18.67 = { by lemma 18 }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(meet(join(converse(x0), complement(complement(converse(meet(X, converse(converse(x0))))))), top), complement(x0)))))
% 144.12/18.67 = { by lemma 39 }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(join(converse(x0), complement(complement(converse(meet(X, converse(converse(x0))))))), complement(x0)))))
% 144.12/18.67 = { by lemma 41 }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(join(converse(x0), converse(meet(X, converse(converse(x0))))), complement(x0)))))
% 144.12/18.67 = { by axiom 12 (composition_distributivity) }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(join(composition(converse(x0), complement(x0)), composition(converse(meet(X, converse(converse(x0)))), complement(x0))))))
% 144.12/18.67 = { by lemma 46 }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(join(zero, composition(converse(meet(X, converse(converse(x0)))), complement(x0))))))
% 144.12/18.67 = { by lemma 42 }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(converse(meet(X, converse(converse(x0)))), complement(x0)))))
% 144.12/18.67 = { by axiom 2 (converse_idempotence) }
% 144.12/18.67 join(complement(complement(x0)), composition(converse(converse(meet(X, x0))), complement(composition(converse(meet(X, x0)), complement(x0)))))
% 144.12/18.67 = { by lemma 23 }
% 144.12/18.67 complement(complement(x0))
% 144.12/18.67 = { by lemma 41 }
% 144.12/18.67 x0
% 144.12/18.67
% 144.12/18.67 Lemma 53: composition(meet(X, x0), Y) = meet(x0, composition(X, Y)).
% 144.12/18.67 Proof:
% 144.12/18.67 composition(meet(X, x0), Y)
% 144.12/18.67 = { by lemma 48 R->L }
% 144.12/18.67 meet(join(composition(meet(X, x0), Y), x0), join(composition(meet(X, x0), Y), complement(x0)))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.67 meet(join(x0, composition(meet(X, x0), Y)), join(composition(meet(X, x0), Y), complement(x0)))
% 144.12/18.67 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.67 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(meet(X, x0), Y)))
% 144.12/18.67 = { by lemma 41 R->L }
% 144.12/18.67 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(meet(X, complement(complement(x0))), Y)))
% 144.12/18.67 = { by lemma 50 R->L }
% 144.12/18.67 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(complement(x0), meet(X, complement(complement(x0)))), Y)))
% 144.12/18.67 = { by lemma 18 }
% 144.12/18.67 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(complement(x0), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.67 = { by lemma 16 R->L }
% 144.12/18.67 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(join(meet(complement(x0), join(meet(X, Z), join(complement(complement(complement(x0))), complement(join(complement(X), Z))))), complement(join(complement(complement(x0)), join(meet(X, Z), join(complement(complement(complement(x0))), complement(join(complement(X), Z))))))), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.67 = { by lemma 44 R->L }
% 144.12/18.67 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(join(meet(complement(x0), join(meet(X, Z), join(complement(complement(complement(x0))), complement(join(complement(X), Z))))), complement(join(meet(X, Z), join(complement(complement(x0)), join(complement(complement(complement(x0))), complement(join(complement(X), Z))))))), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 34 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(join(meet(complement(x0), join(meet(X, Z), join(complement(complement(complement(x0))), complement(join(complement(X), Z))))), complement(join(meet(X, Z), top))), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 31 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(join(meet(complement(x0), join(meet(X, Z), join(complement(complement(complement(x0))), complement(join(complement(X), Z))))), complement(top)), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 41 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(join(meet(complement(x0), join(meet(X, Z), join(complement(x0), complement(join(complement(X), Z))))), complement(top)), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 31 R->L }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(join(meet(complement(x0), join(meet(X, Z), join(complement(x0), complement(join(complement(X), Z))))), complement(join(complement(meet(complement(x0), join(meet(X, Z), join(complement(x0), complement(join(complement(X), Z)))))), top))), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 39 R->L }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(join(meet(meet(complement(x0), join(meet(X, Z), join(complement(x0), complement(join(complement(X), Z))))), top), complement(join(complement(meet(complement(x0), join(meet(X, Z), join(complement(x0), complement(join(complement(X), Z)))))), top))), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 16 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(meet(complement(x0), join(meet(X, Z), join(complement(x0), complement(join(complement(X), Z))))), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 44 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(meet(complement(x0), join(complement(x0), join(meet(X, Z), complement(join(complement(X), Z))))), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 16 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(meet(complement(x0), join(complement(x0), X)), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(meet(complement(x0), join(X, complement(x0))), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 41 R->L }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(meet(complement(x0), join(X, complement(complement(complement(x0))))), meet(complement(complement(x0)), X)), Y)))
% 144.12/18.68 = { by lemma 51 R->L }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(meet(complement(x0), join(X, complement(complement(complement(x0))))), meet(complement(complement(x0)), join(X, complement(complement(complement(x0)))))), Y)))
% 144.12/18.68 = { by lemma 18 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(meet(complement(x0), join(X, complement(complement(complement(x0))))), meet(join(X, complement(complement(complement(x0)))), complement(complement(x0)))), Y)))
% 144.12/18.68 = { by lemma 18 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(meet(join(X, complement(complement(complement(x0)))), complement(x0)), meet(join(X, complement(complement(complement(x0)))), complement(complement(x0)))), Y)))
% 144.12/18.68 = { by lemma 17 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(X, complement(complement(complement(x0)))), Y)))
% 144.12/18.68 = { by lemma 41 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(X, complement(x0)), Y)))
% 144.12/18.68 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(join(complement(x0), X), Y)))
% 144.12/18.68 = { by lemma 50 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), Y)), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.68 meet(join(composition(meet(X, x0), Y), x0), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by lemma 52 R->L }
% 144.12/18.68 meet(join(composition(meet(X, x0), Y), join(x0, composition(meet(X, x0), top))), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.68 meet(join(composition(meet(X, x0), Y), join(composition(meet(X, x0), top), x0)), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by lemma 44 R->L }
% 144.12/18.68 meet(join(composition(meet(X, x0), top), join(composition(meet(X, x0), Y), x0)), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by axiom 9 (maddux2_join_associativity) }
% 144.12/18.68 meet(join(join(composition(meet(X, x0), top), composition(meet(X, x0), Y)), x0), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by lemma 49 }
% 144.12/18.68 meet(join(composition(meet(X, x0), join(top, Y)), x0), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), join(top, Y))), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), join(Y, top))), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by lemma 30 R->L }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), join(W, top))), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by lemma 31 }
% 144.12/18.68 meet(join(x0, composition(meet(X, x0), top)), join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by lemma 52 }
% 144.12/18.68 meet(x0, join(complement(x0), composition(X, Y)))
% 144.12/18.68 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 144.12/18.68 meet(x0, join(composition(X, Y), complement(x0)))
% 144.12/18.68 = { by lemma 51 }
% 144.12/18.68 meet(x0, composition(X, Y))
% 144.12/18.68
% 144.12/18.68 Goal 1 (goals_1): tuple(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2)))) = tuple(composition(meet(x0, x1), x2), meet(x0, composition(x1, x2_2))).
% 144.12/18.68 Proof:
% 144.12/18.68 tuple(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 144.12/18.68 = { by axiom 3 (maddux1_join_commutativity) }
% 144.12/18.68 tuple(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(meet(x0, composition(x1, x2_2)), composition(meet(x0, x1), x2_2)))
% 144.12/18.68 = { by lemma 18 R->L }
% 144.12/18.68 tuple(join(meet(x0, composition(x1, x2)), composition(meet(x1, x0), x2)), join(meet(x0, composition(x1, x2_2)), composition(meet(x0, x1), x2_2)))
% 144.12/18.68 = { by lemma 18 R->L }
% 144.12/18.68 tuple(join(meet(x0, composition(x1, x2)), composition(meet(x1, x0), x2)), join(meet(x0, composition(x1, x2_2)), composition(meet(x1, x0), x2_2)))
% 144.12/18.68 = { by lemma 53 }
% 144.12/18.68 tuple(join(meet(x0, composition(x1, x2)), meet(x0, composition(x1, x2))), join(meet(x0, composition(x1, x2_2)), composition(meet(x1, x0), x2_2)))
% 144.12/18.68 = { by lemma 40 }
% 144.12/18.68 tuple(meet(x0, composition(x1, x2)), join(meet(x0, composition(x1, x2_2)), composition(meet(x1, x0), x2_2)))
% 144.12/18.68 = { by lemma 53 }
% 144.12/18.68 tuple(meet(x0, composition(x1, x2)), join(meet(x0, composition(x1, x2_2)), meet(x0, composition(x1, x2_2))))
% 144.12/18.68 = { by lemma 40 }
% 144.12/18.68 tuple(meet(x0, composition(x1, x2)), meet(x0, composition(x1, x2_2)))
% 144.12/18.68 = { by lemma 53 R->L }
% 144.12/18.68 tuple(composition(meet(x1, x0), x2), meet(x0, composition(x1, x2_2)))
% 144.12/18.68 = { by lemma 18 }
% 144.12/18.68 tuple(composition(meet(x0, x1), x2), meet(x0, composition(x1, x2_2)))
% 144.12/18.68 % SZS output end Proof
% 144.12/18.68
% 144.12/18.68 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------