%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL029-4 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n004.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 12:33:19 PM UTC 2026
% Result : Unsatisfiable 53.82s 7.28s
% Output : Proof 55.42s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : REL029-4 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/0.37 % Computer : n004.cluster.edu
% 0.10/0.37 % Model : x86_64 x86_64
% 0.10/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.37 % Memory : 8046.5625MB
% 0.10/0.37 % OS : Linux 6.8.0-71-generic
% 0.10/0.37 % CPULimit : 300
% 0.10/0.37 % WCLimit : 300
% 0.10/0.37 % DateTime : Sun Sep 27 22:54:07 UTC 2026
% 0.10/0.38 % CPUTime :
% 0.10/0.38 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 53.82/7.28 Command-line arguments: --no-flatten-goal
% 53.82/7.28
% 53.82/7.28 % SZS status Unsatisfiable
% 53.82/7.28
% 54.60/7.39 % SZS output start Proof
% 54.60/7.39 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 54.60/7.39 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 54.60/7.39 Axiom 3 (goals_17): join(sk1, one) = one.
% 54.60/7.39 Axiom 4 (goals_18): join(sk2, one) = one.
% 54.60/7.39 Axiom 5 (composition_identity_6): composition(X, one) = X.
% 54.60/7.39 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 54.60/7.39 Axiom 7 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 54.60/7.39 Axiom 8 (def_top_12): top = join(X, complement(X)).
% 54.60/7.39 Axiom 9 (def_zero_13): zero = meet(X, complement(X)).
% 54.60/7.39 Axiom 10 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 54.60/7.39 Axiom 11 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 54.60/7.39 Axiom 12 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 54.60/7.39 Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 54.60/7.39 Axiom 14 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 54.60/7.39 Axiom 15 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 54.60/7.39
% 54.60/7.39 Lemma 16: complement(top) = zero.
% 54.60/7.39 Proof:
% 54.60/7.39 complement(top)
% 54.60/7.39 = { by axiom 8 (def_top_12) }
% 54.60/7.39 complement(join(complement(X), complement(complement(X))))
% 54.60/7.39 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 54.60/7.39 meet(X, complement(X))
% 54.60/7.39 = { by axiom 9 (def_zero_13) R->L }
% 54.60/7.39 zero
% 54.60/7.39
% 54.60/7.39 Lemma 17: composition(converse(one), X) = X.
% 54.60/7.39 Proof:
% 54.60/7.39 composition(converse(one), X)
% 54.60/7.39 = { by axiom 1 (converse_idempotence_8) R->L }
% 54.60/7.39 composition(converse(one), converse(converse(X)))
% 54.60/7.39 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 54.60/7.39 converse(composition(converse(X), one))
% 54.60/7.39 = { by axiom 5 (composition_identity_6) }
% 54.60/7.39 converse(converse(X))
% 54.60/7.39 = { by axiom 1 (converse_idempotence_8) }
% 54.60/7.40 X
% 54.60/7.40
% 54.60/7.40 Lemma 18: converse(one) = one.
% 54.60/7.40 Proof:
% 54.60/7.40 converse(one)
% 54.60/7.40 = { by axiom 5 (composition_identity_6) R->L }
% 54.60/7.40 composition(converse(one), one)
% 54.60/7.40 = { by lemma 17 }
% 54.60/7.40 one
% 54.60/7.40
% 54.60/7.40 Lemma 19: composition(one, X) = X.
% 54.60/7.40 Proof:
% 54.60/7.40 composition(one, X)
% 54.60/7.40 = { by lemma 18 R->L }
% 54.60/7.40 composition(converse(one), X)
% 54.60/7.40 = { by lemma 17 }
% 54.60/7.40 X
% 54.60/7.40
% 54.60/7.40 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 54.60/7.40 Proof:
% 54.60/7.40 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 54.60/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 54.60/7.40 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 54.60/7.40 = { by axiom 14 (converse_cancellativity_11) }
% 54.60/7.40 complement(X)
% 54.60/7.40
% 54.60/7.40 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 54.60/7.40 Proof:
% 54.60/7.40 join(complement(X), complement(X))
% 54.60/7.40 = { by lemma 17 R->L }
% 54.60/7.40 join(complement(X), composition(converse(one), complement(X)))
% 54.60/7.40 = { by lemma 19 R->L }
% 54.60/7.40 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 54.60/7.40 = { by lemma 20 }
% 54.60/7.40 complement(X)
% 54.60/7.40
% 54.60/7.40 Lemma 22: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 54.60/7.40 Proof:
% 54.60/7.40 join(meet(X, Y), complement(join(complement(X), Y)))
% 54.60/7.40 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 54.60/7.40 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 54.60/7.40 = { by axiom 15 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 54.60/7.40 X
% 54.60/7.40
% 54.60/7.40 Lemma 23: join(zero, complement(complement(X))) = X.
% 54.60/7.40 Proof:
% 54.60/7.40 join(zero, complement(complement(X)))
% 54.60/7.40 = { by axiom 9 (def_zero_13) }
% 54.60/7.40 join(meet(X, complement(X)), complement(complement(X)))
% 54.60/7.40 = { by lemma 21 R->L }
% 54.60/7.40 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 54.60/7.40 = { by lemma 22 }
% 54.60/7.40 X
% 54.60/7.40
% 54.60/7.40 Lemma 24: meet(Y, X) = meet(X, Y).
% 54.60/7.40 Proof:
% 54.60/7.40 meet(Y, X)
% 54.60/7.40 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 54.60/7.40 complement(join(complement(Y), complement(X)))
% 54.60/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 54.60/7.40 complement(join(complement(X), complement(Y)))
% 54.60/7.40 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 54.60/7.40 meet(X, Y)
% 54.60/7.40
% 54.60/7.40 Lemma 25: complement(join(zero, complement(X))) = meet(X, top).
% 54.60/7.40 Proof:
% 54.60/7.40 complement(join(zero, complement(X)))
% 54.60/7.40 = { by lemma 16 R->L }
% 54.60/7.40 complement(join(complement(top), complement(X)))
% 54.60/7.40 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 54.60/7.40 meet(top, X)
% 54.60/7.40 = { by lemma 24 R->L }
% 54.60/7.40 meet(X, top)
% 54.60/7.40
% 54.60/7.40 Lemma 26: join(X, join(Y, complement(X))) = join(Y, top).
% 54.60/7.40 Proof:
% 54.60/7.40 join(X, join(Y, complement(X)))
% 54.60/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 54.60/7.40 join(X, join(complement(X), Y))
% 54.60/7.40 = { by axiom 10 (maddux2_join_associativity_2) }
% 54.60/7.40 join(join(X, complement(X)), Y)
% 54.60/7.40 = { by axiom 8 (def_top_12) R->L }
% 54.60/7.40 join(top, Y)
% 54.60/7.40 = { by axiom 2 (maddux1_join_commutativity_1) }
% 54.60/7.40 join(Y, top)
% 54.60/7.40
% 54.60/7.40 Lemma 27: join(top, complement(X)) = top.
% 54.60/7.40 Proof:
% 54.60/7.40 join(top, complement(X))
% 54.60/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 54.60/7.40 join(complement(X), top)
% 54.60/7.40 = { by lemma 26 R->L }
% 54.60/7.40 join(X, join(complement(X), complement(X)))
% 54.60/7.40 = { by lemma 21 }
% 54.60/7.40 join(X, complement(X))
% 54.60/7.40 = { by axiom 8 (def_top_12) R->L }
% 54.60/7.40 top
% 54.60/7.40
% 54.60/7.40 Lemma 28: join(X, join(complement(X), Y)) = join(Z, top).
% 54.60/7.40 Proof:
% 54.60/7.40 join(X, join(complement(X), Y))
% 54.60/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 54.60/7.40 join(X, join(Y, complement(X)))
% 54.60/7.40 = { by lemma 26 }
% 54.60/7.40 join(Y, top)
% 54.60/7.40 = { by lemma 27 R->L }
% 54.60/7.40 join(Y, join(top, complement(Y)))
% 54.60/7.40 = { by lemma 26 }
% 54.60/7.40 join(top, top)
% 54.60/7.40 = { by lemma 26 R->L }
% 54.60/7.40 join(Z, join(top, complement(Z)))
% 54.60/7.40 = { by lemma 27 }
% 54.60/7.40 join(Z, top)
% 54.60/7.40
% 54.60/7.40 Lemma 29: join(X, top) = top.
% 54.60/7.40 Proof:
% 54.60/7.40 join(X, top)
% 54.60/7.40 = { by lemma 28 R->L }
% 54.60/7.40 join(Y, join(complement(Y), composition(converse(Z), complement(composition(Z, Y)))))
% 54.60/7.40 = { by lemma 20 }
% 54.60/7.40 join(Y, complement(Y))
% 54.60/7.40 = { by axiom 8 (def_top_12) R->L }
% 54.60/7.40 top
% 54.60/7.40
% 54.60/7.40 Lemma 30: join(X, converse(top)) = top.
% 54.60/7.40 Proof:
% 54.60/7.40 join(X, converse(top))
% 54.60/7.40 = { by axiom 8 (def_top_12) }
% 54.60/7.40 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 54.60/7.40 = { by axiom 6 (converse_additivity_9) }
% 54.60/7.40 join(X, join(converse(converse(complement(X))), converse(complement(converse(complement(X))))))
% 54.60/7.40 = { by axiom 1 (converse_idempotence_8) }
% 54.60/7.40 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 54.60/7.40 = { by lemma 28 }
% 54.60/7.40 join(Y, top)
% 54.60/7.40 = { by lemma 29 }
% 54.60/7.40 top
% 54.60/7.40
% 54.60/7.40 Lemma 31: join(zero, meet(X, top)) = X.
% 54.60/7.40 Proof:
% 54.60/7.40 join(zero, meet(X, top))
% 54.60/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 54.60/7.40 join(meet(X, top), zero)
% 54.60/7.40 = { by lemma 16 R->L }
% 54.60/7.40 join(meet(X, top), complement(top))
% 54.60/7.40 = { by lemma 29 R->L }
% 54.60/7.40 join(meet(X, join(converse(Y), top)), complement(top))
% 54.60/7.40 = { by axiom 1 (converse_idempotence_8) R->L }
% 54.60/7.40 join(meet(X, join(converse(Y), converse(converse(top)))), complement(top))
% 54.60/7.40 = { by axiom 6 (converse_additivity_9) R->L }
% 54.60/7.40 join(meet(X, converse(join(Y, converse(top)))), complement(top))
% 54.60/7.40 = { by lemma 30 }
% 54.60/7.40 join(meet(X, converse(top)), complement(top))
% 54.60/7.40 = { by lemma 30 R->L }
% 54.60/7.40 join(meet(X, converse(top)), complement(join(complement(X), converse(top))))
% 54.60/7.40 = { by lemma 22 }
% 54.60/7.40 X
% 54.60/7.40
% 54.60/7.40 Lemma 32: join(zero, complement(X)) = complement(X).
% 54.60/7.40 Proof:
% 54.60/7.40 join(zero, complement(X))
% 54.60/7.40 = { by lemma 23 R->L }
% 55.42/7.40 join(zero, complement(join(zero, complement(complement(X)))))
% 55.42/7.40 = { by lemma 25 }
% 55.42/7.40 join(zero, meet(complement(X), top))
% 55.42/7.40 = { by lemma 31 }
% 55.42/7.40 complement(X)
% 55.42/7.40
% 55.42/7.40 Lemma 33: complement(complement(X)) = X.
% 55.42/7.40 Proof:
% 55.42/7.40 complement(complement(X))
% 55.42/7.40 = { by lemma 32 R->L }
% 55.42/7.40 join(zero, complement(complement(X)))
% 55.42/7.40 = { by lemma 23 }
% 55.42/7.40 X
% 55.42/7.40
% 55.42/7.40 Lemma 34: join(zero, meet(X, X)) = X.
% 55.42/7.40 Proof:
% 55.42/7.40 join(zero, meet(X, X))
% 55.42/7.40 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.42/7.40 join(zero, complement(join(complement(X), complement(X))))
% 55.42/7.40 = { by axiom 9 (def_zero_13) }
% 55.42/7.40 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 55.42/7.40 = { by lemma 22 }
% 55.42/7.40 X
% 55.42/7.40
% 55.42/7.40 Lemma 35: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 55.42/7.40 Proof:
% 55.42/7.40 join(zero, join(X, complement(complement(Y))))
% 55.42/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.40 join(zero, join(complement(complement(Y)), X))
% 55.42/7.40 = { by lemma 21 R->L }
% 55.42/7.40 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 55.42/7.40 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 55.42/7.40 join(zero, join(meet(Y, Y), X))
% 55.42/7.40 = { by axiom 10 (maddux2_join_associativity_2) }
% 55.42/7.40 join(join(zero, meet(Y, Y)), X)
% 55.42/7.40 = { by lemma 34 }
% 55.42/7.40 join(Y, X)
% 55.42/7.40 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.40 join(X, Y)
% 55.42/7.40
% 55.42/7.40 Lemma 36: join(X, X) = X.
% 55.42/7.40 Proof:
% 55.42/7.40 join(X, X)
% 55.42/7.40 = { by lemma 33 R->L }
% 55.42/7.40 join(X, complement(complement(X)))
% 55.42/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.40 join(complement(complement(X)), X)
% 55.42/7.40 = { by lemma 35 R->L }
% 55.42/7.40 join(zero, join(complement(complement(X)), complement(complement(X))))
% 55.42/7.40 = { by lemma 21 }
% 55.42/7.40 join(zero, complement(complement(X)))
% 55.42/7.40 = { by lemma 23 }
% 55.42/7.40 X
% 55.42/7.40
% 55.42/7.40 Lemma 37: join(X, zero) = X.
% 55.42/7.40 Proof:
% 55.42/7.40 join(X, zero)
% 55.42/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.40 join(zero, X)
% 55.42/7.40 = { by lemma 35 R->L }
% 55.42/7.40 join(zero, join(zero, complement(complement(X))))
% 55.42/7.40 = { by lemma 32 }
% 55.42/7.40 join(zero, complement(complement(X)))
% 55.42/7.40 = { by lemma 23 }
% 55.42/7.40 X
% 55.42/7.40
% 55.42/7.40 Lemma 38: join(one, sk1) = one.
% 55.42/7.40 Proof:
% 55.42/7.40 join(one, sk1)
% 55.42/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.40 join(sk1, one)
% 55.42/7.40 = { by axiom 3 (goals_17) }
% 55.42/7.40 one
% 55.42/7.40
% 55.42/7.40 Lemma 39: join(one, sk2) = one.
% 55.42/7.40 Proof:
% 55.42/7.40 join(one, sk2)
% 55.42/7.40 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.40 join(sk2, one)
% 55.42/7.40 = { by axiom 4 (goals_18) }
% 55.42/7.40 one
% 55.42/7.40
% 55.42/7.40 Lemma 40: meet(X, top) = X.
% 55.42/7.40 Proof:
% 55.42/7.40 meet(X, top)
% 55.42/7.40 = { by lemma 25 R->L }
% 55.42/7.40 complement(join(zero, complement(X)))
% 55.42/7.40 = { by lemma 32 R->L }
% 55.42/7.40 join(zero, complement(join(zero, complement(X))))
% 55.42/7.40 = { by lemma 25 }
% 55.42/7.40 join(zero, meet(X, top))
% 55.42/7.40 = { by lemma 31 }
% 55.42/7.40 X
% 55.42/7.40
% 55.42/7.40 Lemma 41: meet(top, X) = X.
% 55.42/7.40 Proof:
% 55.42/7.40 meet(top, X)
% 55.42/7.40 = { by lemma 24 }
% 55.42/7.40 meet(X, top)
% 55.42/7.40 = { by lemma 40 }
% 55.42/7.40 X
% 55.42/7.40
% 55.42/7.40 Lemma 42: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 55.42/7.40 Proof:
% 55.42/7.40 composition(join(one, Y), X)
% 55.42/7.40 = { by lemma 18 R->L }
% 55.42/7.40 composition(join(converse(one), Y), X)
% 55.42/7.40 = { by axiom 13 (composition_distributivity_7) }
% 55.42/7.40 join(composition(converse(one), X), composition(Y, X))
% 55.42/7.40 = { by lemma 17 }
% 55.42/7.40 join(X, composition(Y, X))
% 55.42/7.40
% 55.42/7.40 Lemma 43: join(X, composition(X, sk1)) = X.
% 55.42/7.40 Proof:
% 55.42/7.40 join(X, composition(X, sk1))
% 55.42/7.40 = { by axiom 1 (converse_idempotence_8) R->L }
% 55.42/7.40 join(X, composition(converse(converse(X)), sk1))
% 55.42/7.40 = { by axiom 1 (converse_idempotence_8) R->L }
% 55.42/7.40 join(converse(converse(X)), composition(converse(converse(X)), sk1))
% 55.42/7.40 = { by axiom 1 (converse_idempotence_8) R->L }
% 55.42/7.40 join(converse(converse(X)), composition(converse(converse(X)), converse(converse(sk1))))
% 55.42/7.40 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 55.42/7.40 join(converse(converse(X)), converse(composition(converse(sk1), converse(X))))
% 55.42/7.40 = { by axiom 6 (converse_additivity_9) R->L }
% 55.42/7.40 converse(join(converse(X), composition(converse(sk1), converse(X))))
% 55.42/7.40 = { by lemma 42 R->L }
% 55.42/7.40 converse(composition(join(one, converse(sk1)), converse(X)))
% 55.42/7.40 = { by lemma 18 R->L }
% 55.42/7.40 converse(composition(join(converse(one), converse(sk1)), converse(X)))
% 55.42/7.40 = { by axiom 6 (converse_additivity_9) R->L }
% 55.42/7.40 converse(composition(converse(join(one, sk1)), converse(X)))
% 55.42/7.40 = { by lemma 38 }
% 55.42/7.40 converse(composition(converse(one), converse(X)))
% 55.42/7.40 = { by lemma 18 }
% 55.42/7.40 converse(composition(one, converse(X)))
% 55.42/7.40 = { by lemma 19 }
% 55.42/7.40 converse(converse(X))
% 55.42/7.40 = { by axiom 1 (converse_idempotence_8) }
% 55.42/7.40 X
% 55.42/7.40
% 55.42/7.40 Lemma 44: join(X, composition(X, sk2)) = X.
% 55.42/7.40 Proof:
% 55.42/7.40 join(X, composition(X, sk2))
% 55.42/7.40 = { by axiom 1 (converse_idempotence_8) R->L }
% 55.42/7.40 join(X, composition(converse(converse(X)), sk2))
% 55.42/7.40 = { by axiom 1 (converse_idempotence_8) R->L }
% 55.42/7.40 join(converse(converse(X)), composition(converse(converse(X)), sk2))
% 55.42/7.40 = { by axiom 1 (converse_idempotence_8) R->L }
% 55.42/7.40 join(converse(converse(X)), composition(converse(converse(X)), converse(converse(sk2))))
% 55.42/7.40 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 55.42/7.41 join(converse(converse(X)), converse(composition(converse(sk2), converse(X))))
% 55.42/7.41 = { by axiom 6 (converse_additivity_9) R->L }
% 55.42/7.41 converse(join(converse(X), composition(converse(sk2), converse(X))))
% 55.42/7.41 = { by lemma 42 R->L }
% 55.42/7.41 converse(composition(join(one, converse(sk2)), converse(X)))
% 55.42/7.41 = { by lemma 18 R->L }
% 55.42/7.41 converse(composition(join(converse(one), converse(sk2)), converse(X)))
% 55.42/7.41 = { by axiom 6 (converse_additivity_9) R->L }
% 55.42/7.41 converse(composition(converse(join(one, sk2)), converse(X)))
% 55.42/7.41 = { by lemma 39 }
% 55.42/7.41 converse(composition(converse(one), converse(X)))
% 55.42/7.41 = { by lemma 18 }
% 55.42/7.41 converse(composition(one, converse(X)))
% 55.42/7.41 = { by lemma 19 }
% 55.42/7.41 converse(converse(X))
% 55.42/7.41 = { by axiom 1 (converse_idempotence_8) }
% 55.42/7.41 X
% 55.42/7.41
% 55.42/7.41 Lemma 45: join(X, composition(sk2, X)) = X.
% 55.42/7.41 Proof:
% 55.42/7.41 join(X, composition(sk2, X))
% 55.42/7.41 = { by lemma 42 R->L }
% 55.42/7.41 composition(join(one, sk2), X)
% 55.42/7.41 = { by lemma 39 }
% 55.42/7.41 composition(one, X)
% 55.42/7.41 = { by lemma 19 }
% 55.42/7.41 X
% 55.42/7.41
% 55.42/7.41 Lemma 46: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 55.42/7.41 Proof:
% 55.42/7.41 join(Y, join(X, Z))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.41 join(join(X, Z), Y)
% 55.42/7.41 = { by axiom 10 (maddux2_join_associativity_2) R->L }
% 55.42/7.41 join(X, join(Z, Y))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.41 join(X, join(Y, Z))
% 55.42/7.41
% 55.42/7.41 Lemma 47: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 55.42/7.41 Proof:
% 55.42/7.41 meet(X, join(complement(Y), complement(Z)))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.41 meet(X, join(complement(Z), complement(Y)))
% 55.42/7.41 = { by lemma 24 }
% 55.42/7.41 meet(join(complement(Z), complement(Y)), X)
% 55.42/7.41 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.42/7.41 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 55.42/7.41 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 55.42/7.41 complement(join(meet(Z, Y), complement(X)))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.41 complement(join(complement(X), meet(Z, Y)))
% 55.42/7.41 = { by lemma 24 R->L }
% 55.42/7.41 complement(join(complement(X), meet(Y, Z)))
% 55.42/7.41
% 55.42/7.41 Lemma 48: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 55.42/7.41 Proof:
% 55.42/7.41 join(complement(X), complement(Y))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.41 join(complement(Y), complement(X))
% 55.42/7.41 = { by lemma 34 R->L }
% 55.42/7.41 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 55.42/7.41 = { by lemma 47 }
% 55.42/7.41 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 55.42/7.41 = { by lemma 32 }
% 55.42/7.41 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 55.42/7.41 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 55.42/7.41 complement(join(meet(Y, X), meet(Y, X)))
% 55.42/7.41 = { by lemma 36 }
% 55.42/7.41 complement(meet(Y, X))
% 55.42/7.41 = { by lemma 24 R->L }
% 55.42/7.41 complement(meet(X, Y))
% 55.42/7.41
% 55.42/7.41 Lemma 49: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 55.42/7.41 Proof:
% 55.42/7.41 complement(join(X, complement(Y)))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.41 complement(join(complement(Y), X))
% 55.42/7.41 = { by lemma 41 R->L }
% 55.42/7.41 complement(join(complement(Y), meet(top, X)))
% 55.42/7.41 = { by lemma 47 R->L }
% 55.42/7.41 meet(Y, join(complement(top), complement(X)))
% 55.42/7.41 = { by lemma 16 }
% 55.42/7.41 meet(Y, join(zero, complement(X)))
% 55.42/7.41 = { by lemma 32 }
% 55.42/7.41 meet(Y, complement(X))
% 55.42/7.41
% 55.42/7.41 Lemma 50: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 55.42/7.41 Proof:
% 55.42/7.41 complement(join(complement(X), Y))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.41 complement(join(Y, complement(X)))
% 55.42/7.41 = { by lemma 49 }
% 55.42/7.41 meet(X, complement(Y))
% 55.42/7.41
% 55.42/7.41 Lemma 51: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 55.42/7.41 Proof:
% 55.42/7.41 complement(meet(X, complement(Y)))
% 55.42/7.41 = { by lemma 37 R->L }
% 55.42/7.41 complement(join(meet(X, complement(Y)), zero))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.41 complement(join(zero, meet(X, complement(Y))))
% 55.42/7.41 = { by lemma 49 R->L }
% 55.42/7.41 complement(join(zero, complement(join(Y, complement(X)))))
% 55.42/7.41 = { by lemma 25 }
% 55.42/7.41 meet(join(Y, complement(X)), top)
% 55.42/7.41 = { by lemma 40 }
% 55.42/7.41 join(Y, complement(X))
% 55.42/7.41
% 55.42/7.41 Lemma 52: meet(X, join(X, complement(Y))) = X.
% 55.42/7.41 Proof:
% 55.42/7.41 meet(X, join(X, complement(Y)))
% 55.42/7.41 = { by lemma 51 R->L }
% 55.42/7.41 meet(X, complement(meet(Y, complement(X))))
% 55.42/7.41 = { by lemma 50 R->L }
% 55.42/7.41 complement(join(complement(X), meet(Y, complement(X))))
% 55.42/7.41 = { by lemma 32 R->L }
% 55.42/7.41 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 55.42/7.41 = { by lemma 16 R->L }
% 55.42/7.41 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 55.42/7.41 = { by lemma 29 R->L }
% 55.42/7.41 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 55.42/7.41 = { by lemma 26 R->L }
% 55.42/7.41 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 55.42/7.41 = { by lemma 48 }
% 55.42/7.41 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 55.42/7.41 = { by lemma 24 R->L }
% 55.42/7.41 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 55.42/7.41 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 55.42/7.41 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 55.42/7.41 = { by lemma 24 R->L }
% 55.42/7.41 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 55.42/7.41 = { by lemma 22 }
% 55.42/7.41 X
% 55.42/7.41
% 55.42/7.41 Lemma 53: meet(X, join(Y, X)) = X.
% 55.42/7.41 Proof:
% 55.42/7.41 meet(X, join(Y, X))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.41 meet(X, join(X, Y))
% 55.42/7.41 = { by lemma 40 R->L }
% 55.42/7.41 meet(X, join(X, meet(Y, top)))
% 55.42/7.41 = { by lemma 25 R->L }
% 55.42/7.41 meet(X, join(X, complement(join(zero, complement(Y)))))
% 55.42/7.41 = { by lemma 52 }
% 55.42/7.41 X
% 55.42/7.41
% 55.42/7.41 Lemma 54: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 55.42/7.41 Proof:
% 55.42/7.41 meet(complement(X), join(Y, complement(Z)))
% 55.42/7.41 = { by lemma 24 }
% 55.42/7.41 meet(join(Y, complement(Z)), complement(X))
% 55.42/7.41 = { by lemma 49 R->L }
% 55.42/7.41 complement(join(X, complement(join(Y, complement(Z)))))
% 55.42/7.41 = { by lemma 49 }
% 55.42/7.41 complement(join(X, meet(Z, complement(Y))))
% 55.42/7.41
% 55.42/7.41 Lemma 55: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 55.42/7.41 Proof:
% 55.42/7.41 meet(complement(X), complement(Y))
% 55.42/7.41 = { by lemma 24 }
% 55.42/7.41 meet(complement(Y), complement(X))
% 55.42/7.41 = { by lemma 32 R->L }
% 55.42/7.41 meet(join(zero, complement(Y)), complement(X))
% 55.42/7.41 = { by lemma 49 R->L }
% 55.42/7.41 complement(join(X, complement(join(zero, complement(Y)))))
% 55.42/7.41 = { by lemma 25 }
% 55.42/7.41 complement(join(X, meet(Y, top)))
% 55.42/7.41 = { by lemma 40 }
% 55.42/7.41 complement(join(X, Y))
% 55.42/7.41
% 55.42/7.41 Lemma 56: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 55.42/7.41 Proof:
% 55.42/7.41 join(meet(X, Y), meet(Y, complement(X)))
% 55.42/7.41 = { by lemma 24 }
% 55.42/7.41 join(meet(Y, X), meet(Y, complement(X)))
% 55.42/7.41 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.41 join(meet(Y, complement(X)), meet(Y, X))
% 55.42/7.41 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.42/7.41 join(meet(Y, complement(X)), complement(join(complement(Y), complement(X))))
% 55.42/7.41 = { by lemma 22 }
% 55.42/7.41 Y
% 55.42/7.41
% 55.42/7.41 Lemma 57: meet(complement(Z), meet(X, Y)) = meet(X, meet(Y, complement(Z))).
% 55.42/7.41 Proof:
% 55.42/7.41 meet(complement(Z), meet(X, Y))
% 55.42/7.41 = { by lemma 24 }
% 55.42/7.41 meet(meet(X, Y), complement(Z))
% 55.42/7.41 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.42/7.41 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 55.42/7.41 = { by lemma 55 }
% 55.42/7.41 complement(join(join(complement(X), complement(Y)), Z))
% 55.42/7.41 = { by axiom 10 (maddux2_join_associativity_2) R->L }
% 55.42/7.41 complement(join(complement(X), join(complement(Y), Z)))
% 55.42/7.41 = { by lemma 50 }
% 55.42/7.41 meet(X, complement(join(complement(Y), Z)))
% 55.42/7.41 = { by lemma 50 }
% 55.42/7.41 meet(X, meet(Y, complement(Z)))
% 55.42/7.41
% 55.42/7.42 Lemma 58: meet(X, join(complement(X), Y)) = meet(X, Y).
% 55.42/7.42 Proof:
% 55.42/7.42 meet(X, join(complement(X), Y))
% 55.42/7.42 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.42 meet(X, join(Y, complement(X)))
% 55.42/7.42 = { by lemma 51 R->L }
% 55.42/7.42 meet(X, complement(meet(X, complement(Y))))
% 55.42/7.42 = { by lemma 50 R->L }
% 55.42/7.42 complement(join(complement(X), meet(X, complement(Y))))
% 55.42/7.42 = { by lemma 47 R->L }
% 55.42/7.42 meet(X, join(complement(X), complement(complement(Y))))
% 55.42/7.42 = { by lemma 51 R->L }
% 55.42/7.42 meet(X, complement(meet(complement(Y), complement(complement(X)))))
% 55.42/7.42 = { by lemma 50 R->L }
% 55.42/7.42 complement(join(complement(X), meet(complement(Y), complement(complement(X)))))
% 55.42/7.42 = { by lemma 54 R->L }
% 55.42/7.42 meet(complement(complement(X)), join(complement(X), complement(complement(Y))))
% 55.42/7.42 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.42 meet(complement(complement(X)), join(complement(complement(Y)), complement(X)))
% 55.42/7.42 = { by lemma 24 }
% 55.42/7.42 meet(join(complement(complement(Y)), complement(X)), complement(complement(X)))
% 55.42/7.42 = { by lemma 49 R->L }
% 55.42/7.42 complement(join(complement(X), complement(join(complement(complement(Y)), complement(X)))))
% 55.42/7.42 = { by lemma 55 R->L }
% 55.42/7.42 complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))
% 55.42/7.42 = { by lemma 56 R->L }
% 55.42/7.42 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X))))), complement(complement(complement(Y)))))
% 55.42/7.42 = { by lemma 24 R->L }
% 55.42/7.42 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(complement(Y))), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))))
% 55.42/7.42 = { by lemma 54 R->L }
% 55.42/7.42 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(complement(Y))), meet(complement(complement(X)), join(complement(X), complement(complement(complement(complement(Y))))))))
% 55.42/7.42 = { by lemma 24 }
% 55.42/7.42 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(complement(Y))), meet(join(complement(X), complement(complement(complement(complement(Y))))), complement(complement(X)))))
% 55.42/7.42 = { by lemma 57 R->L }
% 55.42/7.42 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(X)), meet(complement(complement(complement(Y))), join(complement(X), complement(complement(complement(complement(Y))))))))
% 55.42/7.42 = { by lemma 24 }
% 55.42/7.42 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(complement(complement(X)), meet(join(complement(X), complement(complement(complement(complement(Y))))), complement(complement(complement(Y))))))
% 55.42/7.42 = { by lemma 57 }
% 55.42/7.42 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(join(complement(X), complement(complement(complement(complement(Y))))), meet(complement(complement(complement(Y))), complement(complement(X)))))
% 55.42/7.42 = { by lemma 49 R->L }
% 55.42/7.42 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), meet(join(complement(X), complement(complement(complement(complement(Y))))), complement(join(complement(X), complement(complement(complement(complement(Y))))))))
% 55.42/7.42 = { by axiom 9 (def_zero_13) R->L }
% 55.42/7.42 join(meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X)))))), zero)
% 55.42/7.42 = { by lemma 37 }
% 55.42/7.42 meet(complement(complement(Y)), complement(join(complement(X), meet(complement(complement(complement(Y))), complement(complement(X))))))
% 55.42/7.42 = { by lemma 55 }
% 55.42/7.42 meet(complement(complement(Y)), complement(join(complement(X), complement(join(complement(complement(Y)), complement(X))))))
% 55.42/7.42 = { by lemma 49 }
% 55.42/7.42 meet(complement(complement(Y)), meet(join(complement(complement(Y)), complement(X)), complement(complement(X))))
% 55.42/7.42 = { by lemma 24 R->L }
% 55.42/7.42 meet(complement(complement(Y)), meet(complement(complement(X)), join(complement(complement(Y)), complement(X))))
% 55.42/7.42 = { by lemma 48 }
% 55.42/7.42 meet(complement(complement(Y)), meet(complement(complement(X)), complement(meet(complement(Y), X))))
% 55.42/7.42 = { by lemma 55 }
% 55.42/7.42 meet(complement(complement(Y)), complement(join(complement(X), meet(complement(Y), X))))
% 55.42/7.42 = { by lemma 55 }
% 55.42/7.42 complement(join(complement(Y), join(complement(X), meet(complement(Y), X))))
% 55.42/7.42 = { by lemma 46 }
% 55.42/7.42 complement(join(complement(X), join(complement(Y), meet(complement(Y), X))))
% 55.42/7.42 = { by lemma 50 }
% 55.42/7.42 meet(X, complement(join(complement(Y), meet(complement(Y), X))))
% 55.42/7.42 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.42/7.42 meet(X, complement(join(complement(Y), complement(join(complement(complement(Y)), complement(X))))))
% 55.42/7.42 = { by lemma 51 R->L }
% 55.42/7.42 meet(X, complement(complement(meet(join(complement(complement(Y)), complement(X)), complement(complement(Y))))))
% 55.42/7.42 = { by lemma 24 R->L }
% 55.42/7.42 meet(X, complement(complement(meet(complement(complement(Y)), join(complement(complement(Y)), complement(X))))))
% 55.42/7.42 = { by lemma 52 }
% 55.42/7.42 meet(X, complement(complement(complement(complement(Y)))))
% 55.42/7.42 = { by lemma 33 }
% 55.42/7.42 meet(X, complement(complement(Y)))
% 55.42/7.42 = { by lemma 33 }
% 55.42/7.42 meet(X, Y)
% 55.42/7.42
% 55.42/7.42 Lemma 59: join(composition(X, Y), join(Z, composition(W, Y))) = join(Z, composition(join(X, W), Y)).
% 55.42/7.42 Proof:
% 55.42/7.42 join(composition(X, Y), join(Z, composition(W, Y)))
% 55.42/7.42 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.42 join(composition(X, Y), join(composition(W, Y), Z))
% 55.42/7.42 = { by lemma 46 }
% 55.42/7.42 join(composition(W, Y), join(composition(X, Y), Z))
% 55.42/7.42 = { by axiom 10 (maddux2_join_associativity_2) }
% 55.42/7.42 join(join(composition(W, Y), composition(X, Y)), Z)
% 55.42/7.42 = { by axiom 13 (composition_distributivity_7) R->L }
% 55.42/7.42 join(composition(join(W, X), Y), Z)
% 55.42/7.42 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.42 join(Z, composition(join(W, X), Y))
% 55.42/7.43 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.43 join(Z, composition(join(X, W), Y))
% 55.42/7.43
% 55.42/7.43 Lemma 60: meet(X, composition(top, sk1)) = composition(X, sk1).
% 55.42/7.43 Proof:
% 55.42/7.43 meet(X, composition(top, sk1))
% 55.42/7.43 = { by lemma 58 R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(top, sk1)))
% 55.42/7.43 = { by axiom 8 (def_top_12) }
% 55.42/7.43 meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), sk1)))
% 55.42/7.43 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(join(complement(complement(X)), complement(X)), sk1)))
% 55.42/7.43 = { by lemma 59 R->L }
% 55.42/7.43 meet(X, join(composition(complement(complement(X)), sk1), join(complement(X), composition(complement(X), sk1))))
% 55.42/7.43 = { by lemma 43 }
% 55.42/7.43 meet(X, join(composition(complement(complement(X)), sk1), complement(X)))
% 55.42/7.43 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.43 meet(X, join(complement(X), composition(complement(complement(X)), sk1)))
% 55.42/7.43 = { by lemma 58 }
% 55.42/7.43 meet(X, composition(complement(complement(X)), sk1))
% 55.42/7.43 = { by lemma 33 }
% 55.42/7.43 meet(X, composition(X, sk1))
% 55.42/7.43 = { by lemma 24 }
% 55.42/7.43 meet(composition(X, sk1), X)
% 55.42/7.43 = { by lemma 43 R->L }
% 55.42/7.43 meet(composition(X, sk1), join(X, composition(X, sk1)))
% 55.42/7.43 = { by lemma 53 }
% 55.42/7.43 composition(X, sk1)
% 55.42/7.43
% 55.42/7.43 Lemma 61: meet(X, composition(top, sk2)) = composition(X, sk2).
% 55.42/7.43 Proof:
% 55.42/7.43 meet(X, composition(top, sk2))
% 55.42/7.43 = { by lemma 58 R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(top, sk2)))
% 55.42/7.43 = { by axiom 8 (def_top_12) }
% 55.42/7.43 meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), sk2)))
% 55.42/7.43 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(join(complement(complement(X)), complement(X)), sk2)))
% 55.42/7.43 = { by lemma 59 R->L }
% 55.42/7.43 meet(X, join(composition(complement(complement(X)), sk2), join(complement(X), composition(complement(X), sk2))))
% 55.42/7.43 = { by lemma 44 }
% 55.42/7.43 meet(X, join(composition(complement(complement(X)), sk2), complement(X)))
% 55.42/7.43 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.43 meet(X, join(complement(X), composition(complement(complement(X)), sk2)))
% 55.42/7.43 = { by lemma 58 }
% 55.42/7.43 meet(X, composition(complement(complement(X)), sk2))
% 55.42/7.43 = { by lemma 33 }
% 55.42/7.43 meet(X, composition(X, sk2))
% 55.42/7.43 = { by lemma 24 }
% 55.42/7.43 meet(composition(X, sk2), X)
% 55.42/7.43 = { by lemma 44 R->L }
% 55.42/7.43 meet(composition(X, sk2), join(X, composition(X, sk2)))
% 55.42/7.43 = { by lemma 53 }
% 55.42/7.43 composition(X, sk2)
% 55.42/7.43
% 55.42/7.43 Lemma 62: meet(X, composition(sk2, top)) = composition(sk2, X).
% 55.42/7.43 Proof:
% 55.42/7.43 meet(X, composition(sk2, top))
% 55.42/7.43 = { by lemma 58 R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(sk2, top)))
% 55.42/7.43 = { by axiom 8 (def_top_12) }
% 55.42/7.43 meet(X, join(complement(X), composition(sk2, join(complement(X), complement(complement(X))))))
% 55.42/7.43 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(sk2, join(complement(complement(X)), complement(X)))))
% 55.42/7.43 = { by axiom 1 (converse_idempotence_8) R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(sk2, join(complement(complement(X)), converse(converse(complement(X)))))))
% 55.42/7.43 = { by axiom 1 (converse_idempotence_8) R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(sk2, join(converse(converse(complement(complement(X)))), converse(converse(complement(X)))))))
% 55.42/7.43 = { by axiom 1 (converse_idempotence_8) R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(converse(converse(sk2)), join(converse(converse(complement(complement(X)))), converse(converse(complement(X)))))))
% 55.42/7.43 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(converse(converse(sk2)), join(converse(converse(complement(X))), converse(converse(complement(complement(X))))))))
% 55.42/7.43 = { by axiom 6 (converse_additivity_9) R->L }
% 55.42/7.43 meet(X, join(complement(X), composition(converse(converse(sk2)), converse(join(converse(complement(X)), converse(complement(complement(X))))))))
% 55.42/7.43 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 55.42/7.43 meet(X, join(complement(X), converse(composition(join(converse(complement(X)), converse(complement(complement(X)))), converse(sk2)))))
% 55.42/7.43 = { by axiom 13 (composition_distributivity_7) }
% 55.42/7.43 meet(X, join(complement(X), converse(join(composition(converse(complement(X)), converse(sk2)), composition(converse(complement(complement(X))), converse(sk2))))))
% 55.42/7.43 = { by axiom 6 (converse_additivity_9) }
% 55.42/7.43 meet(X, join(complement(X), join(converse(composition(converse(complement(X)), converse(sk2))), converse(composition(converse(complement(complement(X))), converse(sk2))))))
% 55.42/7.43 = { by axiom 7 (converse_multiplicativity_10) }
% 55.42/7.43 meet(X, join(complement(X), join(composition(converse(converse(sk2)), converse(converse(complement(X)))), converse(composition(converse(complement(complement(X))), converse(sk2))))))
% 55.42/7.43 = { by axiom 7 (converse_multiplicativity_10) }
% 55.42/7.43 meet(X, join(complement(X), join(composition(converse(converse(sk2)), converse(converse(complement(X)))), composition(converse(converse(sk2)), converse(converse(complement(complement(X))))))))
% 55.42/7.43 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.43 meet(X, join(complement(X), join(composition(converse(converse(sk2)), converse(converse(complement(complement(X))))), composition(converse(converse(sk2)), converse(converse(complement(X)))))))
% 55.42/7.43 = { by axiom 1 (converse_idempotence_8) }
% 55.42/7.43 meet(X, join(complement(X), join(composition(sk2, converse(converse(complement(complement(X))))), composition(converse(converse(sk2)), converse(converse(complement(X)))))))
% 55.42/7.43 = { by axiom 1 (converse_idempotence_8) }
% 55.42/7.43 meet(X, join(complement(X), join(composition(sk2, converse(converse(complement(complement(X))))), composition(sk2, converse(converse(complement(X)))))))
% 55.42/7.43 = { by axiom 1 (converse_idempotence_8) }
% 55.42/7.43 meet(X, join(complement(X), join(composition(sk2, complement(complement(X))), composition(sk2, converse(converse(complement(X)))))))
% 55.42/7.43 = { by axiom 1 (converse_idempotence_8) }
% 55.42/7.43 meet(X, join(complement(X), join(composition(sk2, complement(complement(X))), composition(sk2, complement(X)))))
% 55.42/7.43 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.42/7.43 meet(X, join(complement(X), join(composition(sk2, complement(X)), composition(sk2, complement(complement(X))))))
% 55.42/7.43 = { by axiom 10 (maddux2_join_associativity_2) }
% 55.42/7.43 meet(X, join(join(complement(X), composition(sk2, complement(X))), composition(sk2, complement(complement(X)))))
% 55.42/7.43 = { by lemma 45 }
% 55.42/7.43 meet(X, join(complement(X), composition(sk2, complement(complement(X)))))
% 55.42/7.43 = { by lemma 58 }
% 55.42/7.43 meet(X, composition(sk2, complement(complement(X))))
% 55.42/7.43 = { by lemma 33 }
% 55.42/7.43 meet(X, composition(sk2, X))
% 55.42/7.43 = { by lemma 24 }
% 55.42/7.43 meet(composition(sk2, X), X)
% 55.42/7.43 = { by lemma 45 R->L }
% 55.42/7.43 meet(composition(sk2, X), join(X, composition(sk2, X)))
% 55.42/7.43 = { by lemma 53 }
% 55.42/7.43 composition(sk2, X)
% 55.42/7.43
% 55.42/7.43 Lemma 63: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 55.42/7.43 Proof:
% 55.42/7.43 meet(Y, meet(X, Z))
% 55.42/7.43 = { by lemma 40 R->L }
% 55.42/7.43 meet(meet(Y, top), meet(X, Z))
% 55.42/7.43 = { by lemma 25 R->L }
% 55.42/7.43 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 55.42/7.43 = { by lemma 57 }
% 55.42/7.43 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 55.42/7.43 = { by lemma 25 }
% 55.42/7.43 meet(X, meet(Z, meet(Y, top)))
% 55.42/7.43 = { by lemma 40 }
% 55.42/7.43 meet(X, meet(Z, Y))
% 55.42/7.43 = { by lemma 24 R->L }
% 55.42/7.43 meet(X, meet(Y, Z))
% 55.42/7.43
% 55.42/7.43 Lemma 64: meet(Y, composition(sk2, X)) = meet(X, composition(sk2, Y)).
% 55.42/7.43 Proof:
% 55.42/7.43 meet(Y, composition(sk2, X))
% 55.42/7.43 = { by lemma 62 R->L }
% 55.42/7.43 meet(Y, meet(X, composition(sk2, top)))
% 55.42/7.43 = { by lemma 63 R->L }
% 55.42/7.43 meet(X, meet(Y, composition(sk2, top)))
% 55.42/7.43 = { by lemma 62 }
% 55.42/7.43 meet(X, composition(sk2, Y))
% 55.42/7.43
% 55.42/7.43 Lemma 65: meet(X, composition(sk2, composition(sk1, X))) = composition(sk2, composition(sk1, X)).
% 55.42/7.43 Proof:
% 55.42/7.43 meet(X, composition(sk2, composition(sk1, X)))
% 55.42/7.43 = { by lemma 24 }
% 55.42/7.43 meet(composition(sk2, composition(sk1, X)), X)
% 55.42/7.43 = { by lemma 19 R->L }
% 55.42/7.43 meet(composition(sk2, composition(sk1, X)), composition(one, X))
% 55.42/7.43 = { by lemma 38 R->L }
% 55.42/7.43 meet(composition(sk2, composition(sk1, X)), composition(join(one, sk1), X))
% 55.42/7.43 = { by lemma 45 R->L }
% 55.42/7.43 meet(composition(sk2, composition(sk1, X)), composition(join(one, join(sk1, composition(sk2, sk1))), X))
% 55.42/7.43 = { by axiom 10 (maddux2_join_associativity_2) }
% 55.42/7.43 meet(composition(sk2, composition(sk1, X)), composition(join(join(one, sk1), composition(sk2, sk1)), X))
% 55.42/7.43 = { by lemma 38 }
% 55.42/7.43 meet(composition(sk2, composition(sk1, X)), composition(join(one, composition(sk2, sk1)), X))
% 55.42/7.43 = { by lemma 42 }
% 55.42/7.43 meet(composition(sk2, composition(sk1, X)), join(X, composition(composition(sk2, sk1), X)))
% 55.42/7.43 = { by axiom 11 (composition_associativity_5) R->L }
% 55.42/7.43 meet(composition(sk2, composition(sk1, X)), join(X, composition(sk2, composition(sk1, X))))
% 55.42/7.43 = { by lemma 53 }
% 55.42/7.43 composition(sk2, composition(sk1, X))
% 55.42/7.43
% 55.42/7.43 Lemma 66: meet(X, composition(meet(one, Y), X)) = composition(meet(Y, one), X).
% 55.42/7.44 Proof:
% 55.42/7.44 meet(X, composition(meet(one, Y), X))
% 55.42/7.44 = { by lemma 24 }
% 55.42/7.44 meet(X, composition(meet(Y, one), X))
% 55.42/7.44 = { by lemma 24 }
% 55.42/7.44 meet(composition(meet(Y, one), X), X)
% 55.42/7.44 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.42/7.44 complement(join(complement(composition(meet(Y, one), X)), complement(X)))
% 55.42/7.44 = { by lemma 41 R->L }
% 55.42/7.44 complement(meet(top, join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by lemma 29 R->L }
% 55.42/7.44 complement(meet(join(composition(meet(one, complement(Y)), X), top), join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by lemma 26 R->L }
% 55.42/7.44 complement(meet(join(composition(meet(Y, one), X), join(composition(meet(one, complement(Y)), X), complement(composition(meet(Y, one), X)))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.44 complement(meet(join(composition(meet(Y, one), X), join(complement(composition(meet(Y, one), X)), composition(meet(one, complement(Y)), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by lemma 59 }
% 55.42/7.44 complement(meet(join(complement(composition(meet(Y, one), X)), composition(join(meet(Y, one), meet(one, complement(Y))), X)), join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by lemma 56 }
% 55.42/7.44 complement(meet(join(complement(composition(meet(Y, one), X)), composition(one, X)), join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.44 complement(meet(join(composition(one, X), complement(composition(meet(Y, one), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by lemma 24 R->L }
% 55.42/7.44 complement(meet(join(composition(one, X), complement(composition(meet(one, Y), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by lemma 19 }
% 55.42/7.44 complement(meet(join(X, complement(composition(meet(one, Y), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by lemma 24 R->L }
% 55.42/7.44 complement(meet(join(X, complement(composition(meet(Y, one), X))), join(complement(composition(meet(Y, one), X)), complement(X))))
% 55.42/7.44 = { by lemma 33 R->L }
% 55.42/7.44 complement(meet(join(X, complement(composition(meet(Y, one), X))), join(complement(composition(meet(Y, one), X)), complement(complement(complement(X))))))
% 55.42/7.44 = { by lemma 24 }
% 55.42/7.44 complement(meet(join(complement(composition(meet(Y, one), X)), complement(complement(complement(X)))), join(X, complement(composition(meet(Y, one), X)))))
% 55.42/7.44 = { by lemma 48 R->L }
% 55.42/7.44 join(complement(join(complement(composition(meet(Y, one), X)), complement(complement(complement(X))))), complement(join(X, complement(composition(meet(Y, one), X)))))
% 55.42/7.44 = { by lemma 49 }
% 55.42/7.44 join(meet(complement(complement(X)), complement(complement(composition(meet(Y, one), X)))), complement(join(X, complement(composition(meet(Y, one), X)))))
% 55.42/7.44 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.44 join(complement(join(X, complement(composition(meet(Y, one), X)))), meet(complement(complement(X)), complement(complement(composition(meet(Y, one), X)))))
% 55.42/7.44 = { by lemma 24 R->L }
% 55.42/7.44 join(complement(join(X, complement(composition(meet(Y, one), X)))), meet(complement(complement(composition(meet(Y, one), X))), complement(complement(X))))
% 55.42/7.44 = { by lemma 55 R->L }
% 55.42/7.44 join(meet(complement(X), complement(complement(composition(meet(Y, one), X)))), meet(complement(complement(composition(meet(Y, one), X))), complement(complement(X))))
% 55.42/7.44 = { by lemma 56 }
% 55.42/7.44 complement(complement(composition(meet(Y, one), X)))
% 55.42/7.44 = { by lemma 33 }
% 55.42/7.44 composition(meet(Y, one), X)
% 55.42/7.44
% 55.42/7.44 Lemma 67: meet(X, composition(sk1, composition(sk2, X))) = composition(sk1, composition(sk2, X)).
% 55.42/7.44 Proof:
% 55.42/7.44 meet(X, composition(sk1, composition(sk2, X)))
% 55.42/7.44 = { by lemma 24 }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), X)
% 55.42/7.44 = { by lemma 19 R->L }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), composition(one, X))
% 55.42/7.44 = { by lemma 39 R->L }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), composition(join(one, sk2), X))
% 55.42/7.44 = { by lemma 19 R->L }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), composition(join(one, composition(one, sk2)), X))
% 55.42/7.44 = { by lemma 38 R->L }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), composition(join(one, composition(join(one, sk1), sk2)), X))
% 55.42/7.44 = { by lemma 42 }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), composition(join(one, join(sk2, composition(sk1, sk2))), X))
% 55.42/7.44 = { by axiom 10 (maddux2_join_associativity_2) }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), composition(join(join(one, sk2), composition(sk1, sk2)), X))
% 55.42/7.44 = { by lemma 39 }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), composition(join(one, composition(sk1, sk2)), X))
% 55.42/7.44 = { by lemma 42 }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), join(X, composition(composition(sk1, sk2), X)))
% 55.42/7.44 = { by axiom 11 (composition_associativity_5) R->L }
% 55.42/7.44 meet(composition(sk1, composition(sk2, X)), join(X, composition(sk1, composition(sk2, X))))
% 55.42/7.44 = { by lemma 53 }
% 55.42/7.44 composition(sk1, composition(sk2, X))
% 55.42/7.44
% 55.42/7.44 Lemma 68: composition(meet(sk1, sk2), X) = composition(sk1, composition(sk2, X)).
% 55.42/7.44 Proof:
% 55.42/7.44 composition(meet(sk1, sk2), X)
% 55.42/7.44 = { by lemma 19 R->L }
% 55.42/7.44 composition(meet(sk1, composition(one, sk2)), X)
% 55.42/7.44 = { by lemma 61 R->L }
% 55.42/7.44 composition(meet(sk1, meet(one, composition(top, sk2))), X)
% 55.42/7.44 = { by lemma 63 R->L }
% 55.42/7.44 composition(meet(one, meet(sk1, composition(top, sk2))), X)
% 55.42/7.44 = { by lemma 61 }
% 55.42/7.44 composition(meet(one, composition(sk1, sk2)), X)
% 55.42/7.44 = { by lemma 24 R->L }
% 55.42/7.44 composition(meet(composition(sk1, sk2), one), X)
% 55.42/7.44 = { by axiom 5 (composition_identity_6) R->L }
% 55.42/7.44 composition(meet(composition(sk1, composition(sk2, one)), one), X)
% 55.42/7.44 = { by lemma 66 R->L }
% 55.42/7.44 meet(X, composition(meet(one, composition(sk1, composition(sk2, one))), X))
% 55.42/7.44 = { by lemma 67 }
% 55.42/7.44 meet(X, composition(composition(sk1, composition(sk2, one)), X))
% 55.42/7.44 = { by axiom 11 (composition_associativity_5) R->L }
% 55.42/7.44 meet(X, composition(sk1, composition(composition(sk2, one), X)))
% 55.42/7.44 = { by axiom 11 (composition_associativity_5) R->L }
% 55.42/7.44 meet(X, composition(sk1, composition(sk2, composition(one, X))))
% 55.42/7.44 = { by lemma 19 }
% 55.42/7.44 meet(X, composition(sk1, composition(sk2, X)))
% 55.42/7.44 = { by lemma 67 }
% 55.42/7.44 composition(sk1, composition(sk2, X))
% 55.42/7.44
% 55.42/7.44 Lemma 69: composition(sk2, composition(sk1, X)) = composition(sk1, composition(sk2, X)).
% 55.42/7.44 Proof:
% 55.42/7.44 composition(sk2, composition(sk1, X))
% 55.42/7.44 = { by lemma 65 R->L }
% 55.42/7.44 meet(X, composition(sk2, composition(sk1, X)))
% 55.42/7.44 = { by lemma 19 R->L }
% 55.42/7.44 meet(X, composition(sk2, composition(sk1, composition(one, X))))
% 55.42/7.44 = { by axiom 11 (composition_associativity_5) }
% 55.42/7.44 meet(X, composition(sk2, composition(composition(sk1, one), X)))
% 55.42/7.44 = { by axiom 11 (composition_associativity_5) }
% 55.42/7.44 meet(X, composition(composition(sk2, composition(sk1, one)), X))
% 55.42/7.44 = { by lemma 65 R->L }
% 55.42/7.44 meet(X, composition(meet(one, composition(sk2, composition(sk1, one))), X))
% 55.42/7.44 = { by lemma 66 }
% 55.42/7.44 composition(meet(composition(sk2, composition(sk1, one)), one), X)
% 55.42/7.44 = { by axiom 5 (composition_identity_6) }
% 55.42/7.44 composition(meet(composition(sk2, sk1), one), X)
% 55.42/7.44 = { by lemma 24 }
% 55.42/7.44 composition(meet(one, composition(sk2, sk1)), X)
% 55.42/7.44 = { by lemma 60 R->L }
% 55.42/7.44 composition(meet(one, meet(sk2, composition(top, sk1))), X)
% 55.42/7.44 = { by lemma 63 }
% 55.42/7.44 composition(meet(sk2, meet(one, composition(top, sk1))), X)
% 55.42/7.44 = { by lemma 60 }
% 55.42/7.44 composition(meet(sk2, composition(one, sk1)), X)
% 55.42/7.44 = { by lemma 19 }
% 55.42/7.44 composition(meet(sk2, sk1), X)
% 55.42/7.44 = { by lemma 24 R->L }
% 55.42/7.44 composition(meet(sk1, sk2), X)
% 55.42/7.44 = { by lemma 68 }
% 55.42/7.44 composition(sk1, composition(sk2, X))
% 55.42/7.44
% 55.42/7.44 Goal 1 (goals_19): tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))) = tuple(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))).
% 55.42/7.44 Proof:
% 55.42/7.44 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))))
% 55.42/7.44 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.42/7.44 tuple(join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))))
% 55.42/7.44 = { by lemma 68 }
% 55.42/7.44 tuple(join(composition(sk1, composition(sk2, sk3)), meet(composition(sk1, sk3), composition(sk2, sk3))), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))))
% 55.42/7.44 = { by lemma 68 }
% 55.42/7.44 tuple(join(composition(sk1, composition(sk2, sk3)), meet(composition(sk1, sk3), composition(sk2, sk3))), join(composition(sk1, composition(sk2, sk3)), meet(composition(sk1, sk3), composition(sk2, sk3))))
% 55.42/7.44 = { by lemma 64 R->L }
% 55.42/7.44 tuple(join(composition(sk1, composition(sk2, sk3)), meet(sk3, composition(sk2, composition(sk1, sk3)))), join(composition(sk1, composition(sk2, sk3)), meet(composition(sk1, sk3), composition(sk2, sk3))))
% 55.42/7.44 = { by lemma 64 R->L }
% 55.42/7.44 tuple(join(composition(sk1, composition(sk2, sk3)), meet(sk3, composition(sk2, composition(sk1, sk3)))), join(composition(sk1, composition(sk2, sk3)), meet(sk3, composition(sk2, composition(sk1, sk3)))))
% 55.42/7.44 = { by lemma 65 }
% 55.42/7.44 tuple(join(composition(sk1, composition(sk2, sk3)), composition(sk2, composition(sk1, sk3))), join(composition(sk1, composition(sk2, sk3)), meet(sk3, composition(sk2, composition(sk1, sk3)))))
% 55.42/7.44 = { by lemma 65 }
% 55.42/7.44 tuple(join(composition(sk1, composition(sk2, sk3)), composition(sk2, composition(sk1, sk3))), join(composition(sk1, composition(sk2, sk3)), composition(sk2, composition(sk1, sk3))))
% 55.42/7.44 = { by lemma 69 }
% 55.42/7.44 tuple(join(composition(sk1, composition(sk2, sk3)), composition(sk1, composition(sk2, sk3))), join(composition(sk1, composition(sk2, sk3)), composition(sk2, composition(sk1, sk3))))
% 55.42/7.44 = { by lemma 69 }
% 55.42/7.44 tuple(join(composition(sk1, composition(sk2, sk3)), composition(sk1, composition(sk2, sk3))), join(composition(sk1, composition(sk2, sk3)), composition(sk1, composition(sk2, sk3))))
% 55.42/7.44 = { by lemma 36 }
% 55.42/7.44 tuple(composition(sk1, composition(sk2, sk3)), join(composition(sk1, composition(sk2, sk3)), composition(sk1, composition(sk2, sk3))))
% 55.42/7.44 = { by lemma 36 }
% 55.42/7.44 tuple(composition(sk1, composition(sk2, sk3)), composition(sk1, composition(sk2, sk3)))
% 55.42/7.44 = { by lemma 69 R->L }
% 55.42/7.44 tuple(composition(sk1, composition(sk2, sk3)), composition(sk2, composition(sk1, sk3)))
% 55.42/7.44 = { by lemma 65 R->L }
% 55.42/7.44 tuple(composition(sk1, composition(sk2, sk3)), meet(sk3, composition(sk2, composition(sk1, sk3))))
% 55.42/7.44 = { by lemma 64 }
% 55.42/7.44 tuple(composition(sk1, composition(sk2, sk3)), meet(composition(sk1, sk3), composition(sk2, sk3)))
% 55.42/7.44 = { by lemma 68 R->L }
% 55.42/7.44 tuple(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))
% 55.42/7.44 % SZS output end Proof
% 55.42/7.44
% 55.42/7.44 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------