%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL030+4 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n005.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 12:33:21 PM UTC 2026
% Result : Theorem 161.25s 20.89s
% Output : Proof 163.25s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : REL030+4 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.06 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.16/0.42 % Computer : n005.cluster.edu
% 0.16/0.42 % Model : x86_64 x86_64
% 0.16/0.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.42 % Memory : 8046.5625MB
% 0.16/0.42 % OS : Linux 6.8.0-71-generic
% 0.16/0.42 % CPULimit : 300
% 0.16/0.42 % WCLimit : 300
% 0.16/0.42 % DateTime : Sun Sep 27 22:54:31 UTC 2026
% 0.16/0.43 % CPUTime :
% 0.16/0.43 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 161.25/20.89 Command-line arguments: --flatten --complete-subsets
% 161.25/20.89
% 161.25/20.89 % SZS status Theorem
% 161.25/20.89
% 162.07/21.02 % SZS output start Proof
% 162.07/21.02 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 162.07/21.03 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 162.07/21.03 Axiom 3 (def_top): top = join(X, complement(X)).
% 162.07/21.03 Axiom 4 (goals): join(x0, one) = one.
% 162.07/21.03 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 162.07/21.03 Axiom 6 (composition_identity): composition(X, one) = X.
% 162.07/21.03 Axiom 7 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 162.07/21.03 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 162.07/21.03 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 162.07/21.03 Axiom 10 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 162.07/21.03 Axiom 11 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 162.07/21.03 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 162.07/21.03 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 162.07/21.03 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 162.07/21.03
% 162.07/21.03 Lemma 15: complement(top) = zero.
% 162.07/21.03 Proof:
% 162.07/21.03 complement(top)
% 162.07/21.03 = { by axiom 3 (def_top) }
% 162.07/21.03 complement(join(complement(X), complement(complement(X))))
% 162.07/21.03 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 162.07/21.03 meet(X, complement(X))
% 162.07/21.03 = { by axiom 5 (def_zero) R->L }
% 162.07/21.03 zero
% 162.07/21.03
% 162.07/21.03 Lemma 16: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 162.07/21.03 Proof:
% 162.07/21.03 join(meet(X, Y), complement(join(complement(X), Y)))
% 162.07/21.03 = { by axiom 7 (maddux4_definiton_of_meet) }
% 162.07/21.03 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 162.07/21.03 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 162.07/21.03 X
% 162.07/21.03
% 162.07/21.03 Lemma 17: join(meet(X, Y), meet(X, complement(Y))) = X.
% 162.07/21.03 Proof:
% 162.07/21.03 join(meet(X, Y), meet(X, complement(Y)))
% 162.07/21.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.03 join(meet(X, complement(Y)), meet(X, Y))
% 162.07/21.03 = { by axiom 7 (maddux4_definiton_of_meet) }
% 162.07/21.03 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 162.07/21.03 = { by lemma 16 }
% 162.07/21.03 X
% 162.07/21.03
% 162.07/21.03 Lemma 18: meet(Y, X) = meet(X, Y).
% 162.07/21.03 Proof:
% 162.07/21.03 meet(Y, X)
% 162.07/21.03 = { by axiom 7 (maddux4_definiton_of_meet) }
% 162.07/21.03 complement(join(complement(Y), complement(X)))
% 162.07/21.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.03 complement(join(complement(X), complement(Y)))
% 162.07/21.03 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 162.07/21.03 meet(X, Y)
% 162.07/21.03
% 162.07/21.03 Lemma 19: complement(join(zero, complement(X))) = meet(X, top).
% 162.07/21.03 Proof:
% 162.07/21.03 complement(join(zero, complement(X)))
% 162.07/21.03 = { by lemma 15 R->L }
% 162.07/21.03 complement(join(complement(top), complement(X)))
% 162.07/21.03 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 162.07/21.03 meet(top, X)
% 162.07/21.03 = { by lemma 18 R->L }
% 162.07/21.03 meet(X, top)
% 162.07/21.03
% 162.07/21.03 Lemma 20: composition(X, join(x0, one)) = X.
% 162.07/21.03 Proof:
% 162.07/21.03 composition(X, join(x0, one))
% 162.07/21.03 = { by axiom 4 (goals) }
% 162.07/21.03 composition(X, one)
% 162.07/21.03 = { by axiom 6 (composition_identity) }
% 162.07/21.03 X
% 162.07/21.03
% 162.07/21.03 Lemma 21: composition(converse(join(x0, one)), X) = X.
% 162.07/21.03 Proof:
% 162.07/21.03 composition(converse(join(x0, one)), X)
% 162.07/21.03 = { by axiom 1 (converse_idempotence) R->L }
% 162.07/21.03 composition(converse(join(x0, one)), converse(converse(X)))
% 162.07/21.03 = { by axiom 10 (converse_multiplicativity) R->L }
% 162.07/21.03 converse(composition(converse(X), join(x0, one)))
% 162.07/21.03 = { by lemma 20 }
% 162.07/21.03 converse(converse(X))
% 162.07/21.03 = { by axiom 1 (converse_idempotence) }
% 162.07/21.03 X
% 162.07/21.03
% 162.07/21.03 Lemma 22: composition(join(x0, one), X) = X.
% 162.07/21.03 Proof:
% 162.07/21.03 composition(join(x0, one), X)
% 162.07/21.03 = { by lemma 21 R->L }
% 162.07/21.03 composition(converse(join(x0, one)), composition(join(x0, one), X))
% 162.07/21.03 = { by axiom 11 (composition_associativity) }
% 162.07/21.03 composition(composition(converse(join(x0, one)), join(x0, one)), X)
% 162.07/21.03 = { by lemma 20 }
% 162.07/21.03 composition(converse(join(x0, one)), X)
% 162.07/21.03 = { by lemma 21 }
% 162.07/21.03 X
% 162.07/21.03
% 162.07/21.03 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 162.07/21.03 Proof:
% 162.07/21.03 join(complement(X), complement(X))
% 162.07/21.03 = { by lemma 21 R->L }
% 162.07/21.03 join(complement(X), composition(converse(join(x0, one)), complement(X)))
% 162.07/21.03 = { by lemma 22 R->L }
% 162.07/21.03 join(complement(X), composition(converse(join(x0, one)), complement(composition(join(x0, one), X))))
% 162.07/21.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.03 join(composition(converse(join(x0, one)), complement(composition(join(x0, one), X))), complement(X))
% 162.07/21.03 = { by axiom 14 (converse_cancellativity) }
% 162.07/21.03 complement(X)
% 162.07/21.03
% 162.07/21.03 Lemma 24: join(zero, complement(complement(X))) = X.
% 162.07/21.03 Proof:
% 162.07/21.03 join(zero, complement(complement(X)))
% 162.07/21.03 = { by axiom 5 (def_zero) }
% 162.07/21.03 join(meet(X, complement(X)), complement(complement(X)))
% 162.07/21.03 = { by lemma 23 R->L }
% 162.07/21.03 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 162.07/21.03 = { by lemma 16 }
% 162.07/21.03 X
% 162.07/21.03
% 162.07/21.03 Lemma 25: meet(top, complement(X)) = complement(X).
% 162.07/21.03 Proof:
% 162.07/21.03 meet(top, complement(X))
% 162.07/21.03 = { by lemma 18 }
% 162.07/21.03 meet(complement(X), top)
% 162.07/21.03 = { by lemma 19 R->L }
% 162.07/21.03 complement(join(zero, complement(complement(X))))
% 162.07/21.03 = { by lemma 24 }
% 162.07/21.03 complement(X)
% 162.07/21.03
% 162.07/21.03 Lemma 26: complement(zero) = top.
% 162.07/21.03 Proof:
% 162.07/21.03 complement(zero)
% 162.07/21.03 = { by lemma 17 R->L }
% 162.07/21.03 join(meet(complement(zero), top), meet(complement(zero), complement(top)))
% 162.07/21.03 = { by lemma 15 }
% 162.07/21.03 join(meet(complement(zero), top), meet(complement(zero), zero))
% 162.07/21.03 = { by lemma 18 R->L }
% 162.07/21.03 join(meet(complement(zero), top), meet(zero, complement(zero)))
% 162.07/21.03 = { by axiom 5 (def_zero) R->L }
% 162.07/21.03 join(meet(complement(zero), top), zero)
% 162.07/21.03 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.03 join(zero, meet(complement(zero), top))
% 162.07/21.03 = { by lemma 18 R->L }
% 162.07/21.03 join(zero, meet(top, complement(zero)))
% 162.07/21.03 = { by lemma 25 }
% 162.07/21.03 join(zero, complement(zero))
% 162.07/21.03 = { by axiom 3 (def_top) R->L }
% 162.07/21.03 top
% 162.07/21.03
% 162.07/21.03 Lemma 27: join(X, join(Y, complement(X))) = join(Y, top).
% 162.07/21.03 Proof:
% 162.07/21.03 join(X, join(Y, complement(X)))
% 162.07/21.03 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.03 join(X, join(complement(X), Y))
% 162.07/21.03 = { by axiom 9 (maddux2_join_associativity) }
% 162.07/21.03 join(join(X, complement(X)), Y)
% 162.07/21.03 = { by axiom 3 (def_top) R->L }
% 162.07/21.03 join(top, Y)
% 162.07/21.03 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.03 join(Y, top)
% 162.07/21.03
% 162.07/21.03 Lemma 28: join(top, complement(X)) = top.
% 162.07/21.03 Proof:
% 162.07/21.03 join(top, complement(X))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 join(complement(X), top)
% 162.07/21.04 = { by lemma 27 R->L }
% 162.07/21.04 join(X, join(complement(X), complement(X)))
% 162.07/21.04 = { by lemma 23 }
% 162.07/21.04 join(X, complement(X))
% 162.07/21.04 = { by axiom 3 (def_top) R->L }
% 162.07/21.04 top
% 162.07/21.04
% 162.07/21.04 Lemma 29: join(Y, top) = join(X, top).
% 162.07/21.04 Proof:
% 162.07/21.04 join(Y, top)
% 162.07/21.04 = { by lemma 28 R->L }
% 162.07/21.04 join(Y, join(top, complement(Y)))
% 162.07/21.04 = { by lemma 27 }
% 162.07/21.04 join(top, top)
% 162.07/21.04 = { by lemma 27 R->L }
% 162.07/21.04 join(X, join(top, complement(X)))
% 162.07/21.04 = { by lemma 28 }
% 162.07/21.04 join(X, top)
% 162.07/21.04
% 162.07/21.04 Lemma 30: join(complement(X), X) = top.
% 162.07/21.04 Proof:
% 162.07/21.04 join(complement(X), X)
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 join(X, complement(X))
% 162.07/21.04 = { by axiom 3 (def_top) R->L }
% 162.07/21.04 top
% 162.07/21.04
% 162.07/21.04 Lemma 31: join(X, top) = top.
% 162.07/21.04 Proof:
% 162.07/21.04 join(X, top)
% 162.07/21.04 = { by lemma 29 }
% 162.07/21.04 join(complement(top), top)
% 162.07/21.04 = { by lemma 30 }
% 162.07/21.04 top
% 162.07/21.04
% 162.07/21.04 Lemma 32: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 162.07/21.04 Proof:
% 162.07/21.04 converse(join(X, converse(Y)))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 converse(join(converse(Y), X))
% 162.07/21.04 = { by axiom 8 (converse_additivity) }
% 162.07/21.04 join(converse(converse(Y)), converse(X))
% 162.07/21.04 = { by axiom 1 (converse_idempotence) }
% 162.07/21.04 join(Y, converse(X))
% 162.07/21.04
% 162.07/21.04 Lemma 33: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 162.07/21.04 Proof:
% 162.07/21.04 converse(join(converse(X), Y))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 converse(join(Y, converse(X)))
% 162.07/21.04 = { by lemma 32 }
% 162.07/21.04 join(X, converse(Y))
% 162.07/21.04
% 162.07/21.04 Lemma 34: join(X, join(complement(X), Y)) = top.
% 162.07/21.04 Proof:
% 162.07/21.04 join(X, join(complement(X), Y))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 join(X, join(Y, complement(X)))
% 162.07/21.04 = { by lemma 27 }
% 162.07/21.04 join(Y, top)
% 162.07/21.04 = { by lemma 29 R->L }
% 162.07/21.04 join(Z, top)
% 162.07/21.04 = { by lemma 31 }
% 162.07/21.04 top
% 162.07/21.04
% 162.07/21.04 Lemma 35: converse(top) = top.
% 162.07/21.04 Proof:
% 162.07/21.04 converse(top)
% 162.07/21.04 = { by lemma 31 R->L }
% 162.07/21.04 converse(join(X, top))
% 162.07/21.04 = { by axiom 8 (converse_additivity) }
% 162.07/21.04 join(converse(X), converse(top))
% 162.07/21.04 = { by axiom 3 (def_top) }
% 162.07/21.04 join(converse(X), converse(join(converse(complement(converse(X))), complement(converse(complement(converse(X)))))))
% 162.07/21.04 = { by lemma 33 }
% 162.07/21.04 join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X)))))))
% 162.07/21.04 = { by lemma 34 }
% 162.07/21.04 top
% 162.07/21.04
% 162.07/21.04 Lemma 36: meet(X, zero) = zero.
% 162.07/21.04 Proof:
% 162.07/21.04 meet(X, zero)
% 162.07/21.04 = { by lemma 18 }
% 162.07/21.04 meet(zero, X)
% 162.07/21.04 = { by axiom 7 (maddux4_definiton_of_meet) }
% 162.07/21.04 complement(join(complement(zero), complement(X)))
% 162.07/21.04 = { by lemma 26 }
% 162.07/21.04 complement(join(top, complement(X)))
% 162.07/21.04 = { by lemma 28 }
% 162.07/21.04 complement(top)
% 162.07/21.04 = { by lemma 15 }
% 162.07/21.04 zero
% 162.07/21.04
% 162.07/21.04 Lemma 37: join(zero, meet(X, top)) = X.
% 162.07/21.04 Proof:
% 162.07/21.04 join(zero, meet(X, top))
% 162.07/21.04 = { by lemma 26 R->L }
% 162.07/21.04 join(zero, meet(X, complement(zero)))
% 162.07/21.04 = { by lemma 36 R->L }
% 162.07/21.04 join(meet(X, zero), meet(X, complement(zero)))
% 162.07/21.04 = { by lemma 17 }
% 162.07/21.04 X
% 162.07/21.04
% 162.07/21.04 Lemma 38: join(zero, complement(X)) = complement(X).
% 162.07/21.04 Proof:
% 162.07/21.04 join(zero, complement(X))
% 162.07/21.04 = { by lemma 25 R->L }
% 162.07/21.04 join(zero, meet(top, complement(X)))
% 162.07/21.04 = { by lemma 18 }
% 162.07/21.04 join(zero, meet(complement(X), top))
% 162.07/21.04 = { by lemma 37 }
% 162.07/21.04 complement(X)
% 162.07/21.04
% 162.07/21.04 Lemma 39: meet(X, top) = X.
% 162.07/21.04 Proof:
% 162.07/21.04 meet(X, top)
% 162.07/21.04 = { by lemma 19 R->L }
% 162.07/21.04 complement(join(zero, complement(X)))
% 162.07/21.04 = { by lemma 38 R->L }
% 162.07/21.04 join(zero, complement(join(zero, complement(X))))
% 162.07/21.04 = { by lemma 19 }
% 162.07/21.04 join(zero, meet(X, top))
% 162.07/21.04 = { by lemma 37 }
% 162.07/21.04 X
% 162.07/21.04
% 162.07/21.04 Lemma 40: join(X, zero) = X.
% 162.07/21.04 Proof:
% 162.07/21.04 join(X, zero)
% 162.07/21.04 = { by lemma 15 R->L }
% 162.07/21.04 join(X, complement(top))
% 162.07/21.04 = { by lemma 31 R->L }
% 162.07/21.04 join(X, complement(join(complement(X), top)))
% 162.07/21.04 = { by lemma 39 R->L }
% 162.07/21.04 join(meet(X, top), complement(join(complement(X), top)))
% 162.07/21.04 = { by lemma 16 }
% 162.07/21.04 X
% 162.07/21.04
% 162.07/21.04 Lemma 41: join(meet(X, complement(Y)), meet(X, Y)) = X.
% 162.07/21.04 Proof:
% 162.07/21.04 join(meet(X, complement(Y)), meet(X, Y))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 join(meet(X, Y), meet(X, complement(Y)))
% 162.07/21.04 = { by lemma 17 }
% 162.07/21.04 X
% 162.07/21.04
% 162.07/21.04 Lemma 42: join(meet(complement(X), Y), meet(Y, X)) = Y.
% 162.07/21.04 Proof:
% 162.07/21.04 join(meet(complement(X), Y), meet(Y, X))
% 162.07/21.04 = { by lemma 18 }
% 162.07/21.04 join(meet(Y, complement(X)), meet(Y, X))
% 162.07/21.04 = { by lemma 41 }
% 162.07/21.04 Y
% 162.07/21.04
% 162.07/21.04 Lemma 43: complement(complement(X)) = X.
% 162.07/21.04 Proof:
% 162.07/21.04 complement(complement(X))
% 162.07/21.04 = { by lemma 42 R->L }
% 162.07/21.04 join(meet(complement(X), complement(complement(X))), meet(complement(complement(X)), X))
% 162.07/21.04 = { by axiom 5 (def_zero) R->L }
% 162.07/21.04 join(zero, meet(complement(complement(X)), X))
% 162.07/21.04 = { by lemma 18 R->L }
% 162.07/21.04 join(zero, meet(X, complement(complement(X))))
% 162.07/21.04 = { by axiom 5 (def_zero) }
% 162.07/21.04 join(meet(X, complement(X)), meet(X, complement(complement(X))))
% 162.07/21.04 = { by lemma 17 }
% 162.07/21.04 X
% 162.07/21.04
% 162.07/21.04 Lemma 44: join(zero, X) = X.
% 162.07/21.04 Proof:
% 162.07/21.04 join(zero, X)
% 162.07/21.04 = { by lemma 43 R->L }
% 162.07/21.04 join(zero, complement(complement(X)))
% 162.07/21.04 = { by lemma 24 }
% 162.07/21.04 X
% 162.07/21.04
% 162.07/21.04 Lemma 45: meet(top, X) = X.
% 162.07/21.04 Proof:
% 162.07/21.04 meet(top, X)
% 162.07/21.04 = { by lemma 18 }
% 162.07/21.04 meet(X, top)
% 162.07/21.04 = { by lemma 39 }
% 162.07/21.04 X
% 162.07/21.04
% 162.07/21.04 Lemma 46: complement(join(meet(X, Y), complement(Z))) = meet(Z, join(complement(X), complement(Y))).
% 162.07/21.04 Proof:
% 162.07/21.04 complement(join(meet(X, Y), complement(Z)))
% 162.07/21.04 = { by axiom 7 (maddux4_definiton_of_meet) }
% 162.07/21.04 complement(join(complement(join(complement(X), complement(Y))), complement(Z)))
% 162.07/21.04 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 162.07/21.04 meet(join(complement(X), complement(Y)), Z)
% 162.07/21.04 = { by lemma 18 R->L }
% 162.07/21.04 meet(Z, join(complement(X), complement(Y)))
% 162.07/21.04
% 162.07/21.04 Lemma 47: complement(meet(X, Y)) = join(complement(X), complement(Y)).
% 162.07/21.04 Proof:
% 162.07/21.04 complement(meet(X, Y))
% 162.07/21.04 = { by lemma 18 }
% 162.07/21.04 complement(meet(Y, X))
% 162.07/21.04 = { by lemma 40 R->L }
% 162.07/21.04 complement(join(meet(Y, X), zero))
% 162.07/21.04 = { by lemma 15 R->L }
% 162.07/21.04 complement(join(meet(Y, X), complement(top)))
% 162.07/21.04 = { by lemma 46 }
% 162.07/21.04 meet(top, join(complement(Y), complement(X)))
% 162.07/21.04 = { by lemma 45 }
% 162.07/21.04 join(complement(Y), complement(X))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.04 join(complement(X), complement(Y))
% 162.07/21.04
% 162.07/21.04 Lemma 48: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 162.07/21.04 Proof:
% 162.07/21.04 join(X, join(Y, Z))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 join(join(Y, Z), X)
% 162.07/21.04 = { by axiom 9 (maddux2_join_associativity) R->L }
% 162.07/21.04 join(Y, join(Z, X))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.04 join(Y, join(X, Z))
% 162.07/21.04
% 162.07/21.04 Lemma 49: join(composition(x0, X), X) = X.
% 162.07/21.04 Proof:
% 162.07/21.04 join(composition(x0, X), X)
% 162.07/21.04 = { by lemma 22 R->L }
% 162.07/21.04 join(composition(x0, X), composition(join(x0, one), X))
% 162.07/21.04 = { by axiom 4 (goals) }
% 162.07/21.04 join(composition(x0, X), composition(one, X))
% 162.07/21.04 = { by axiom 12 (composition_distributivity) R->L }
% 162.07/21.04 composition(join(x0, one), X)
% 162.07/21.04 = { by lemma 22 }
% 162.07/21.04 X
% 162.07/21.04
% 162.07/21.04 Lemma 50: complement(meet(Y, complement(X))) = join(X, complement(Y)).
% 162.07/21.04 Proof:
% 162.07/21.04 complement(meet(Y, complement(X)))
% 162.07/21.04 = { by lemma 47 }
% 162.07/21.04 join(complement(Y), complement(complement(X)))
% 162.07/21.04 = { by lemma 43 }
% 162.07/21.04 join(complement(Y), X)
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.04 join(X, complement(Y))
% 162.07/21.04
% 162.07/21.04 Lemma 51: meet(X, join(X, Y)) = X.
% 162.07/21.04 Proof:
% 162.07/21.04 meet(X, join(X, Y))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 meet(X, join(Y, X))
% 162.07/21.04 = { by lemma 40 R->L }
% 162.07/21.04 join(meet(X, join(Y, X)), zero)
% 162.07/21.04 = { by lemma 15 R->L }
% 162.07/21.04 join(meet(X, join(Y, X)), complement(top))
% 162.07/21.04 = { by lemma 43 R->L }
% 162.07/21.04 join(meet(X, join(Y, complement(complement(X)))), complement(top))
% 162.07/21.04 = { by lemma 31 R->L }
% 162.07/21.04 join(meet(X, join(Y, complement(complement(X)))), complement(join(Y, top)))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 join(meet(X, join(Y, complement(complement(X)))), complement(join(top, Y)))
% 162.07/21.04 = { by axiom 3 (def_top) }
% 162.07/21.04 join(meet(X, join(Y, complement(complement(X)))), complement(join(join(complement(X), complement(complement(X))), Y)))
% 162.07/21.04 = { by axiom 9 (maddux2_join_associativity) R->L }
% 162.07/21.04 join(meet(X, join(Y, complement(complement(X)))), complement(join(complement(X), join(complement(complement(X)), Y))))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.04 join(meet(X, join(Y, complement(complement(X)))), complement(join(complement(X), join(Y, complement(complement(X))))))
% 162.07/21.04 = { by lemma 16 }
% 162.07/21.04 X
% 162.07/21.04
% 162.07/21.04 Lemma 52: meet(composition(x0, X), X) = composition(x0, X).
% 162.07/21.04 Proof:
% 162.07/21.04 meet(composition(x0, X), X)
% 162.07/21.04 = { by axiom 7 (maddux4_definiton_of_meet) }
% 162.07/21.04 complement(join(complement(composition(x0, X)), complement(X)))
% 162.07/21.04 = { by lemma 38 R->L }
% 162.07/21.04 join(zero, complement(join(complement(composition(x0, X)), complement(X))))
% 162.07/21.04 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 162.07/21.04 join(zero, meet(composition(x0, X), X))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 join(meet(composition(x0, X), X), zero)
% 162.07/21.04 = { by lemma 15 R->L }
% 162.07/21.04 join(meet(composition(x0, X), X), complement(top))
% 162.07/21.04 = { by lemma 31 R->L }
% 162.07/21.04 join(meet(composition(x0, X), X), complement(join(X, top)))
% 162.07/21.04 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.04 join(meet(composition(x0, X), X), complement(join(top, X)))
% 162.07/21.04 = { by lemma 30 R->L }
% 162.07/21.04 join(meet(composition(x0, X), X), complement(join(join(complement(composition(x0, X)), composition(x0, X)), X)))
% 162.07/21.04 = { by axiom 9 (maddux2_join_associativity) R->L }
% 162.07/21.04 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), join(composition(x0, X), X))))
% 162.07/21.04 = { by lemma 49 }
% 162.07/21.04 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), X)))
% 162.07/21.04 = { by lemma 16 }
% 162.07/21.04 composition(x0, X)
% 162.07/21.04
% 162.07/21.04 Lemma 53: join(X, converse(complement(converse(X)))) = top.
% 162.07/21.04 Proof:
% 162.07/21.04 join(X, converse(complement(converse(X))))
% 162.07/21.04 = { by lemma 33 R->L }
% 162.07/21.04 converse(join(converse(X), complement(converse(X))))
% 162.07/21.05 = { by axiom 3 (def_top) R->L }
% 162.07/21.05 converse(top)
% 162.07/21.05 = { by lemma 35 }
% 162.07/21.05 top
% 162.07/21.05
% 162.07/21.05 Lemma 54: complement(join(meet(X, complement(Y)), complement(Z))) = meet(Z, complement(meet(X, complement(Y)))).
% 162.07/21.05 Proof:
% 162.07/21.05 complement(join(meet(X, complement(Y)), complement(Z)))
% 162.07/21.05 = { by lemma 46 }
% 162.07/21.05 meet(Z, join(complement(X), complement(complement(Y))))
% 162.07/21.05 = { by lemma 43 }
% 162.07/21.05 meet(Z, join(complement(X), Y))
% 162.07/21.05 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.05 meet(Z, join(Y, complement(X)))
% 162.07/21.05 = { by lemma 50 R->L }
% 162.07/21.05 meet(Z, complement(meet(X, complement(Y))))
% 162.07/21.05
% 162.07/21.05 Lemma 55: complement(join(X, meet(Y, complement(Z)))) = meet(complement(X), complement(meet(Y, complement(Z)))).
% 162.07/21.05 Proof:
% 162.07/21.05 complement(join(X, meet(Y, complement(Z))))
% 162.07/21.05 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.05 complement(join(meet(Y, complement(Z)), X))
% 162.07/21.05 = { by lemma 39 R->L }
% 162.07/21.05 complement(join(meet(Y, complement(Z)), meet(X, top)))
% 162.07/21.05 = { by lemma 19 R->L }
% 162.07/21.05 complement(join(meet(Y, complement(Z)), complement(join(zero, complement(X)))))
% 162.07/21.05 = { by lemma 54 }
% 162.07/21.05 meet(join(zero, complement(X)), complement(meet(Y, complement(Z))))
% 162.07/21.05 = { by lemma 38 }
% 162.07/21.05 meet(complement(X), complement(meet(Y, complement(Z))))
% 162.07/21.05
% 162.07/21.05 Lemma 56: meet(X, converse(complement(converse(complement(X))))) = X.
% 162.07/21.05 Proof:
% 162.07/21.05 meet(X, converse(complement(converse(complement(X)))))
% 162.07/21.05 = { by lemma 40 R->L }
% 162.07/21.05 join(meet(X, converse(complement(converse(complement(X))))), zero)
% 162.07/21.05 = { by lemma 15 R->L }
% 162.07/21.05 join(meet(X, converse(complement(converse(complement(X))))), complement(top))
% 162.07/21.05 = { by lemma 53 R->L }
% 162.07/21.05 join(meet(X, converse(complement(converse(complement(X))))), complement(join(complement(X), converse(complement(converse(complement(X)))))))
% 162.07/21.05 = { by lemma 16 }
% 162.07/21.05 X
% 162.07/21.05
% 162.07/21.05 Lemma 57: meet(complement(X), complement(join(X, Y))) = complement(join(X, Y)).
% 162.07/21.05 Proof:
% 162.07/21.05 meet(complement(X), complement(join(X, Y)))
% 162.07/21.05 = { by lemma 18 }
% 162.07/21.05 meet(complement(join(X, Y)), complement(X))
% 162.07/21.05 = { by lemma 51 R->L }
% 162.07/21.05 meet(complement(join(X, Y)), complement(meet(X, join(X, Y))))
% 162.07/21.05 = { by lemma 47 }
% 162.07/21.05 meet(complement(join(X, Y)), join(complement(X), complement(join(X, Y))))
% 162.07/21.05 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.05 meet(complement(join(X, Y)), join(complement(join(X, Y)), complement(X)))
% 162.07/21.05 = { by lemma 51 }
% 162.07/21.05 complement(join(X, Y))
% 162.07/21.05
% 162.07/21.05 Lemma 58: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 162.07/21.05 Proof:
% 162.07/21.05 join(meet(X, Y), meet(Y, complement(X)))
% 162.07/21.05 = { by lemma 18 }
% 162.07/21.05 join(meet(Y, X), meet(Y, complement(X)))
% 162.07/21.05 = { by lemma 17 }
% 162.07/21.05 Y
% 162.07/21.05
% 162.07/21.05 Lemma 59: join(composition(X, Y), composition(X, complement(Y))) = composition(X, top).
% 162.07/21.05 Proof:
% 162.07/21.05 join(composition(X, Y), composition(X, complement(Y)))
% 162.07/21.05 = { by lemma 56 R->L }
% 162.07/21.05 join(composition(X, Y), composition(X, meet(complement(Y), converse(complement(converse(complement(complement(Y))))))))
% 162.07/21.05 = { by lemma 43 }
% 162.07/21.05 join(composition(X, Y), composition(X, meet(complement(Y), converse(complement(converse(Y))))))
% 162.07/21.05 = { by lemma 40 R->L }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), zero)))
% 162.07/21.05 = { by axiom 5 (def_zero) }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(converse(zero), complement(converse(zero))))))
% 162.07/21.05 = { by axiom 1 (converse_idempotence) R->L }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(converse(zero), converse(converse(complement(converse(zero))))))))
% 162.07/21.05 = { by lemma 45 R->L }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(converse(zero), converse(meet(top, converse(complement(converse(zero)))))))))
% 162.07/21.05 = { by lemma 15 R->L }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(converse(zero), converse(meet(top, converse(complement(converse(complement(top))))))))))
% 162.07/21.05 = { by lemma 56 }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(converse(zero), converse(top)))))
% 162.07/21.05 = { by lemma 35 }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(converse(zero), top))))
% 162.07/21.05 = { by lemma 39 }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(zero))))
% 162.07/21.05 = { by lemma 36 R->L }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), zero)))))
% 162.07/21.05 = { by lemma 15 R->L }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(top))))))
% 162.07/21.05 = { by lemma 34 R->L }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(join(converse(meet(Y, converse(complement(converse(Y))))), join(complement(converse(meet(Y, converse(complement(converse(Y)))))), converse(meet(Y, complement(converse(complement(converse(Y))))))))))))))
% 162.07/21.05 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.05 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(join(converse(meet(Y, converse(complement(converse(Y))))), join(converse(meet(Y, complement(converse(complement(converse(Y)))))), complement(converse(meet(Y, converse(complement(converse(Y))))))))))))))
% 162.07/21.06 = { by lemma 48 R->L }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(join(converse(meet(Y, complement(converse(complement(converse(Y)))))), join(converse(meet(Y, converse(complement(converse(Y))))), complement(converse(meet(Y, converse(complement(converse(Y))))))))))))))
% 162.07/21.06 = { by axiom 9 (maddux2_join_associativity) }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(join(join(converse(meet(Y, complement(converse(complement(converse(Y)))))), converse(meet(Y, converse(complement(converse(Y)))))), complement(converse(meet(Y, converse(complement(converse(Y)))))))))))))
% 162.07/21.06 = { by axiom 8 (converse_additivity) R->L }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(join(converse(join(meet(Y, complement(converse(complement(converse(Y))))), meet(Y, converse(complement(converse(Y)))))), complement(converse(meet(Y, converse(complement(converse(Y)))))))))))))
% 162.07/21.06 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(join(complement(converse(meet(Y, converse(complement(converse(Y)))))), converse(join(meet(Y, complement(converse(complement(converse(Y))))), meet(Y, converse(complement(converse(Y)))))))))))))
% 162.07/21.06 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(join(complement(converse(meet(Y, converse(complement(converse(Y)))))), converse(join(meet(Y, converse(complement(converse(Y)))), meet(Y, complement(converse(complement(converse(Y))))))))))))))
% 162.07/21.06 = { by lemma 17 }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(join(complement(converse(meet(Y, converse(complement(converse(Y)))))), converse(Y))))))))
% 162.07/21.06 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), complement(join(converse(Y), complement(converse(meet(Y, converse(complement(converse(Y)))))))))))))
% 162.07/21.06 = { by lemma 57 }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(complement(join(converse(Y), complement(converse(meet(Y, converse(complement(converse(Y))))))))))))
% 162.07/21.06 = { by lemma 43 R->L }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(complement(join(complement(complement(converse(Y))), complement(converse(meet(Y, converse(complement(converse(Y))))))))))))
% 162.07/21.06 = { by lemma 23 R->L }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(complement(join(complement(join(complement(converse(Y)), complement(converse(Y)))), complement(converse(meet(Y, converse(complement(converse(Y))))))))))))
% 162.07/21.06 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(complement(join(meet(converse(Y), converse(Y)), complement(converse(meet(Y, converse(complement(converse(Y))))))))))))
% 162.07/21.06 = { by lemma 46 }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(converse(meet(Y, converse(complement(converse(Y))))), join(complement(converse(Y)), complement(converse(Y))))))))
% 162.07/21.06 = { by lemma 23 }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(converse(meet(Y, converse(complement(converse(Y))))), complement(converse(Y)))))))
% 162.07/21.06 = { by lemma 18 R->L }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), converse(meet(Y, converse(complement(converse(Y))))))))))
% 162.07/21.06 = { by lemma 18 }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(complement(converse(Y)), converse(meet(converse(complement(converse(Y))), Y)))))))
% 162.07/21.06 = { by axiom 1 (converse_idempotence) R->L }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(converse(converse(complement(converse(Y)))), converse(meet(converse(complement(converse(Y))), Y)))))))
% 162.07/21.06 = { by lemma 18 }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(converse(meet(converse(complement(converse(Y))), Y)), converse(converse(complement(converse(Y)))))))))
% 162.07/21.06 = { by lemma 16 R->L }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(converse(meet(converse(complement(converse(Y))), Y)), converse(join(meet(converse(complement(converse(Y))), Y), complement(join(complement(converse(complement(converse(Y)))), Y)))))))))
% 162.07/21.06 = { by axiom 8 (converse_additivity) }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(meet(converse(meet(converse(complement(converse(Y))), Y)), join(converse(meet(converse(complement(converse(Y))), Y)), converse(complement(join(complement(converse(complement(converse(Y)))), Y)))))))))
% 162.07/21.06 = { by lemma 51 }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(converse(meet(converse(complement(converse(Y))), Y))))))
% 162.07/21.06 = { by lemma 18 R->L }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), converse(converse(meet(Y, converse(complement(converse(Y)))))))))
% 162.07/21.06 = { by axiom 1 (converse_idempotence) }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(Y, converse(complement(converse(Y)))))))
% 162.07/21.06 = { by lemma 18 }
% 162.07/21.06 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(Y)))), meet(converse(complement(converse(Y))), Y))))
% 162.07/21.06 = { by lemma 42 }
% 162.07/21.06 join(composition(X, Y), composition(X, converse(complement(converse(Y)))))
% 162.07/21.06 = { by axiom 1 (converse_idempotence) R->L }
% 162.07/21.06 converse(converse(join(composition(X, Y), composition(X, converse(complement(converse(Y)))))))
% 162.07/21.06 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.06 converse(converse(join(composition(X, converse(complement(converse(Y)))), composition(X, Y))))
% 162.07/21.06 = { by axiom 8 (converse_additivity) }
% 162.07/21.06 converse(join(converse(composition(X, converse(complement(converse(Y))))), converse(composition(X, Y))))
% 162.07/21.06 = { by axiom 10 (converse_multiplicativity) }
% 162.07/21.06 converse(join(composition(converse(converse(complement(converse(Y)))), converse(X)), converse(composition(X, Y))))
% 162.07/21.06 = { by axiom 1 (converse_idempotence) }
% 162.07/21.06 converse(join(composition(complement(converse(Y)), converse(X)), converse(composition(X, Y))))
% 162.07/21.06 = { by axiom 2 (maddux1_join_commutativity) }
% 162.07/21.06 converse(join(converse(composition(X, Y)), composition(complement(converse(Y)), converse(X))))
% 162.07/21.06 = { by axiom 10 (converse_multiplicativity) }
% 162.07/21.06 converse(join(composition(converse(Y), converse(X)), composition(complement(converse(Y)), converse(X))))
% 162.07/21.06 = { by axiom 12 (composition_distributivity) R->L }
% 162.07/21.06 converse(composition(join(converse(Y), complement(converse(Y))), converse(X)))
% 162.07/21.06 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.06 converse(composition(join(complement(converse(Y)), converse(Y)), converse(X)))
% 162.07/21.06 = { by lemma 32 R->L }
% 162.07/21.06 converse(composition(converse(join(Y, converse(complement(converse(Y))))), converse(X)))
% 162.07/21.06 = { by axiom 10 (converse_multiplicativity) R->L }
% 162.07/21.06 converse(converse(composition(X, join(Y, converse(complement(converse(Y)))))))
% 162.07/21.06 = { by axiom 1 (converse_idempotence) }
% 162.07/21.06 composition(X, join(Y, converse(complement(converse(Y)))))
% 162.07/21.06 = { by lemma 53 }
% 162.07/21.06 composition(X, top)
% 162.07/21.06
% 162.07/21.06 Lemma 60: meet(meet(X, complement(Y)), complement(meet(X, complement(composition(x0, Y))))) = zero.
% 162.07/21.06 Proof:
% 162.07/21.06 meet(meet(X, complement(Y)), complement(meet(X, complement(composition(x0, Y)))))
% 162.07/21.06 = { by lemma 54 R->L }
% 162.07/21.06 complement(join(meet(X, complement(composition(x0, Y))), complement(meet(X, complement(Y)))))
% 162.07/21.06 = { by lemma 50 }
% 162.07/21.06 complement(join(meet(X, complement(composition(x0, Y))), join(Y, complement(X))))
% 162.07/21.07 = { by lemma 49 R->L }
% 162.07/21.07 complement(join(meet(X, complement(composition(x0, Y))), join(join(composition(x0, Y), Y), complement(X))))
% 162.07/21.07 = { by axiom 9 (maddux2_join_associativity) R->L }
% 162.07/21.07 complement(join(meet(X, complement(composition(x0, Y))), join(composition(x0, Y), join(Y, complement(X)))))
% 162.07/21.07 = { by lemma 50 R->L }
% 162.07/21.07 complement(join(meet(X, complement(composition(x0, Y))), join(composition(x0, Y), complement(meet(X, complement(Y))))))
% 162.07/21.07 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 162.07/21.07 complement(join(meet(X, complement(composition(x0, Y))), join(complement(meet(X, complement(Y))), composition(x0, Y))))
% 162.07/21.07 = { by lemma 50 }
% 162.07/21.07 complement(join(meet(X, complement(composition(x0, Y))), join(join(Y, complement(X)), composition(x0, Y))))
% 163.25/21.07 = { by axiom 9 (maddux2_join_associativity) R->L }
% 163.25/21.07 complement(join(meet(X, complement(composition(x0, Y))), join(Y, join(complement(X), composition(x0, Y)))))
% 163.25/21.07 = { by axiom 2 (maddux1_join_commutativity) }
% 163.25/21.07 complement(join(meet(X, complement(composition(x0, Y))), join(Y, join(composition(x0, Y), complement(X)))))
% 163.25/21.07 = { by lemma 50 R->L }
% 163.25/21.07 complement(join(meet(X, complement(composition(x0, Y))), join(Y, complement(meet(X, complement(composition(x0, Y)))))))
% 163.25/21.07 = { by lemma 27 }
% 163.25/21.07 complement(join(Y, top))
% 163.25/21.07 = { by lemma 29 R->L }
% 163.25/21.07 complement(join(Z, top))
% 163.25/21.07 = { by lemma 31 }
% 163.25/21.07 complement(top)
% 163.25/21.07 = { by lemma 15 }
% 163.25/21.07 zero
% 163.25/21.07
% 163.25/21.07 Lemma 61: complement(join(meet(X, complement(composition(x0, Y))), meet(X, complement(Y)))) = complement(meet(X, complement(composition(x0, Y)))).
% 163.25/21.07 Proof:
% 163.25/21.07 complement(join(meet(X, complement(composition(x0, Y))), meet(X, complement(Y))))
% 163.25/21.07 = { by lemma 55 }
% 163.25/21.07 meet(complement(meet(X, complement(composition(x0, Y)))), complement(meet(X, complement(Y))))
% 163.25/21.07 = { by lemma 44 R->L }
% 163.25/21.07 join(zero, meet(complement(meet(X, complement(composition(x0, Y)))), complement(meet(X, complement(Y)))))
% 163.25/21.07 = { by lemma 60 R->L }
% 163.25/21.07 join(meet(meet(X, complement(Y)), complement(meet(X, complement(composition(x0, Y))))), meet(complement(meet(X, complement(composition(x0, Y)))), complement(meet(X, complement(Y)))))
% 163.25/21.07 = { by lemma 58 }
% 163.25/21.07 complement(meet(X, complement(composition(x0, Y))))
% 163.25/21.07
% 163.25/21.07 Lemma 62: join(meet(X, complement(composition(x0, Y))), meet(X, complement(Y))) = meet(X, complement(composition(x0, Y))).
% 163.25/21.07 Proof:
% 163.25/21.07 join(meet(X, complement(composition(x0, Y))), meet(X, complement(Y)))
% 163.25/21.07 = { by lemma 39 R->L }
% 163.25/21.07 meet(join(meet(X, complement(composition(x0, Y))), meet(X, complement(Y))), top)
% 163.25/21.07 = { by lemma 19 R->L }
% 163.25/21.07 complement(join(zero, complement(join(meet(X, complement(composition(x0, Y))), meet(X, complement(Y))))))
% 163.25/21.07 = { by lemma 61 }
% 163.25/21.07 complement(join(zero, complement(meet(X, complement(composition(x0, Y))))))
% 163.25/21.07 = { by lemma 19 }
% 163.25/21.07 meet(meet(X, complement(composition(x0, Y))), top)
% 163.25/21.07 = { by lemma 39 }
% 163.25/21.07 meet(X, complement(composition(x0, Y)))
% 163.25/21.07
% 163.25/21.07 Goal 1 (goals_1): tuple(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2))))) = tuple(meet(composition(x0, x1), complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))).
% 163.25/21.07 Proof:
% 163.25/21.07 tuple(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.07 = { by lemma 51 R->L }
% 163.25/21.07 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(join(complement(x2), composition(x0, complement(x1))), meet(composition(x0, x1), composition(x0, x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.07 = { by lemma 62 }
% 163.25/21.07 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(meet(composition(x0, x1), complement(composition(x0, x2))), join(join(complement(x2), composition(x0, complement(x1))), meet(composition(x0, x1), composition(x0, x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.07 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 163.25/21.07 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(meet(composition(x0, x1), complement(composition(x0, x2))), join(meet(composition(x0, x1), composition(x0, x2)), join(complement(x2), composition(x0, complement(x1)))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.07 = { by axiom 9 (maddux2_join_associativity) }
% 163.25/21.07 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), composition(x0, x2))), join(complement(x2), composition(x0, complement(x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.07 = { by lemma 41 }
% 163.25/21.07 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(composition(x0, x1), join(complement(x2), composition(x0, complement(x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.07 = { by lemma 48 }
% 163.25/21.07 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(complement(x2), join(composition(x0, x1), composition(x0, complement(x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.07 = { by lemma 59 }
% 163.25/21.07 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(complement(x2), composition(x0, top))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.07 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 163.25/21.07 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(composition(x0, top), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by lemma 59 R->L }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(join(composition(x0, x2), composition(x0, complement(x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 9 (maddux2_join_associativity) R->L }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(composition(x0, x2), join(composition(x0, complement(x2)), complement(x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 2 (maddux1_join_commutativity) }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(composition(x0, x2), join(complement(x2), composition(x0, complement(x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by lemma 22 R->L }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(composition(x0, x2), join(composition(join(x0, one), complement(x2)), composition(x0, complement(x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 12 (composition_distributivity) R->L }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(composition(x0, x2), composition(join(join(x0, one), x0), complement(x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(composition(x0, x2), composition(join(x0, join(x0, one)), complement(x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 4 (goals) }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(composition(x0, x2), composition(join(x0, one), complement(x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by lemma 22 }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(composition(x0, x2), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by lemma 43 R->L }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(complement(complement(composition(x0, x2))), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 163.25/21.08 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), join(complement(x2), complement(complement(composition(x0, x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 7 (maddux4_definiton_of_meet) }
% 163.25/21.08 tuple(complement(join(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))), complement(join(complement(x2), complement(complement(composition(x0, x2))))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 163.25/21.08 tuple(complement(join(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))), meet(x2, complement(composition(x0, x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by lemma 18 R->L }
% 163.25/21.08 tuple(complement(join(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))), meet(complement(composition(x0, x2)), x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by lemma 57 R->L }
% 163.25/21.08 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))), meet(complement(composition(x0, x2)), x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by lemma 61 }
% 163.25/21.08 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), meet(complement(composition(x0, x2)), x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by lemma 50 }
% 163.25/21.08 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(join(composition(x0, x2), complement(composition(x0, x1))), meet(complement(composition(x0, x2)), x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 9 (maddux2_join_associativity) R->L }
% 163.25/21.08 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(composition(x0, x2), join(complement(composition(x0, x1)), meet(complement(composition(x0, x2)), x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 163.25/21.08 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(composition(x0, x2), join(meet(complement(composition(x0, x2)), x2), complement(composition(x0, x1)))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 9 (maddux2_join_associativity) }
% 163.25/21.08 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(join(composition(x0, x2), meet(complement(composition(x0, x2)), x2)), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by lemma 18 }
% 163.25/21.08 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(join(composition(x0, x2), meet(x2, complement(composition(x0, x2)))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.08 = { by axiom 7 (maddux4_definiton_of_meet) }
% 163.25/21.08 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(join(composition(x0, x2), complement(join(complement(x2), complement(complement(composition(x0, x2)))))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 43 }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(join(composition(x0, x2), complement(join(complement(x2), composition(x0, x2)))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 43 R->L }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(join(complement(complement(composition(x0, x2))), complement(join(complement(x2), composition(x0, x2)))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 47 R->L }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(meet(complement(composition(x0, x2)), join(complement(x2), composition(x0, x2)))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 52 R->L }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(meet(complement(composition(x0, x2)), join(complement(x2), meet(composition(x0, x2), x2)))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 52 R->L }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(meet(complement(meet(composition(x0, x2), x2)), join(complement(x2), meet(composition(x0, x2), x2)))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 47 }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(meet(join(complement(composition(x0, x2)), complement(x2)), join(complement(x2), meet(composition(x0, x2), x2)))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 23 R->L }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(meet(join(complement(composition(x0, x2)), join(complement(x2), complement(x2))), join(complement(x2), meet(composition(x0, x2), x2)))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by axiom 9 (maddux2_join_associativity) }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(meet(join(join(complement(composition(x0, x2)), complement(x2)), complement(x2)), join(complement(x2), meet(composition(x0, x2), x2)))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by axiom 7 (maddux4_definiton_of_meet) }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(meet(join(join(complement(composition(x0, x2)), complement(x2)), complement(x2)), join(complement(x2), complement(join(complement(composition(x0, x2)), complement(x2)))))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(meet(join(complement(x2), join(complement(composition(x0, x2)), complement(x2))), join(complement(x2), complement(join(complement(composition(x0, x2)), complement(x2)))))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 46 R->L }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(complement(join(meet(x2, join(complement(composition(x0, x2)), complement(x2))), complement(join(complement(x2), join(complement(composition(x0, x2)), complement(x2))))))), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 16 }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(complement(complement(x2)), complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 43 }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(join(x2, complement(composition(x0, x1))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 50 R->L }
% 163.25/21.09 tuple(meet(complement(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))))), complement(complement(meet(composition(x0, x1), complement(x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 43 }
% 163.25/21.09 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), complement(complement(meet(composition(x0, x1), complement(x2))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 43 }
% 163.25/21.09 tuple(meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2))), meet(composition(x0, x1), complement(x2))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 18 }
% 163.25/21.09 tuple(meet(meet(composition(x0, x1), complement(x2)), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), complement(x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 62 }
% 163.25/21.09 tuple(meet(meet(composition(x0, x1), complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by axiom 7 (maddux4_definiton_of_meet) }
% 163.25/21.09 tuple(complement(join(complement(meet(composition(x0, x1), complement(x2))), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.09 = { by lemma 38 R->L }
% 163.25/21.09 tuple(join(zero, complement(join(complement(meet(composition(x0, x1), complement(x2))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.10 = { by lemma 60 R->L }
% 163.25/21.10 tuple(join(meet(meet(composition(x0, x1), complement(x2)), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), complement(join(complement(meet(composition(x0, x1), complement(x2))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))))), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.10 = { by lemma 16 }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))))
% 163.25/21.10 = { by lemma 39 R->L }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), meet(join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2)))), top))
% 163.25/21.10 = { by lemma 19 R->L }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, complement(join(meet(composition(x0, x1), complement(x2_2)), meet(composition(x0, x1), complement(composition(x0, x2_2))))))))
% 163.25/21.10 = { by axiom 2 (maddux1_join_commutativity) }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, complement(join(meet(composition(x0, x1), complement(composition(x0, x2_2))), meet(composition(x0, x1), complement(x2_2)))))))
% 163.25/21.10 = { by lemma 55 }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, meet(complement(meet(composition(x0, x1), complement(composition(x0, x2_2)))), complement(meet(composition(x0, x1), complement(x2_2)))))))
% 163.25/21.10 = { by lemma 44 R->L }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, join(zero, meet(complement(meet(composition(x0, x1), complement(composition(x0, x2_2)))), complement(meet(composition(x0, x1), complement(x2_2))))))))
% 163.25/21.10 = { by lemma 60 R->L }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, join(meet(meet(composition(x0, x1), complement(x2_2)), complement(meet(composition(x0, x1), complement(composition(x0, x2_2))))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2_2)))), complement(meet(composition(x0, x1), complement(x2_2))))))))
% 163.25/21.10 = { by lemma 58 }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), complement(join(zero, complement(meet(composition(x0, x1), complement(composition(x0, x2_2)))))))
% 163.25/21.10 = { by lemma 19 }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), meet(meet(composition(x0, x1), complement(composition(x0, x2_2))), top))
% 163.25/21.10 = { by lemma 39 }
% 163.25/21.10 tuple(meet(composition(x0, x1), complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2_2))))
% 163.25/21.10 % SZS output end Proof
% 163.25/21.10
% 163.25/21.10 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------