TSTP Solution File: REL041-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL041-1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:44:26 EDT 2023
% Result : Unsatisfiable 19.09s 2.95s
% Output : Proof 20.76s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL041-1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n006.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 22:29:37 EDT 2023
% 0.12/0.34 % CPUTime :
% 19.09/2.95 Command-line arguments: --flatten
% 19.09/2.95
% 19.09/2.95 % SZS status Unsatisfiable
% 19.09/2.95
% 20.76/3.03 % SZS output start Proof
% 20.76/3.03 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 20.76/3.03 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 20.76/3.03 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 20.76/3.03 Axiom 4 (def_zero_13): zero = meet(X, complement(X)).
% 20.76/3.03 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 20.76/3.03 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 20.76/3.03 Axiom 7 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 20.76/3.03 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 20.76/3.03 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 20.76/3.03 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 20.76/3.03 Axiom 11 (goals_14): join(composition(converse(sk1), sk1), one) = one.
% 20.76/3.04 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 20.76/3.04 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 20.76/3.04 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 20.76/3.04
% 20.76/3.04 Lemma 15: complement(top) = zero.
% 20.76/3.04 Proof:
% 20.76/3.04 complement(top)
% 20.76/3.04 = { by axiom 5 (def_top_12) }
% 20.76/3.04 complement(join(complement(X), complement(complement(X))))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 20.76/3.04 meet(X, complement(X))
% 20.76/3.04 = { by axiom 4 (def_zero_13) R->L }
% 20.76/3.04 zero
% 20.76/3.04
% 20.76/3.04 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 20.76/3.04 Proof:
% 20.76/3.04 join(X, join(Y, complement(X)))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(X, join(complement(X), Y))
% 20.76/3.04 = { by axiom 7 (maddux2_join_associativity_2) }
% 20.76/3.04 join(join(X, complement(X)), Y)
% 20.76/3.04 = { by axiom 5 (def_top_12) R->L }
% 20.76/3.04 join(top, Y)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.04 join(Y, top)
% 20.76/3.04
% 20.76/3.04 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 20.76/3.04 Proof:
% 20.76/3.04 converse(composition(converse(X), Y))
% 20.76/3.04 = { by axiom 8 (converse_multiplicativity_10) }
% 20.76/3.04 composition(converse(Y), converse(converse(X)))
% 20.76/3.04 = { by axiom 1 (converse_idempotence_8) }
% 20.76/3.04 composition(converse(Y), X)
% 20.76/3.04
% 20.76/3.04 Lemma 18: composition(converse(join(composition(converse(sk1), sk1), one)), X) = X.
% 20.76/3.04 Proof:
% 20.76/3.04 composition(converse(join(composition(converse(sk1), sk1), one)), X)
% 20.76/3.04 = { by axiom 11 (goals_14) }
% 20.76/3.04 composition(converse(one), X)
% 20.76/3.04 = { by lemma 17 R->L }
% 20.76/3.04 converse(composition(converse(X), one))
% 20.76/3.04 = { by axiom 3 (composition_identity_6) }
% 20.76/3.04 converse(converse(X))
% 20.76/3.04 = { by axiom 1 (converse_idempotence_8) }
% 20.76/3.04 X
% 20.76/3.04
% 20.76/3.04 Lemma 19: composition(join(composition(converse(sk1), sk1), one), X) = X.
% 20.76/3.04 Proof:
% 20.76/3.04 composition(join(composition(converse(sk1), sk1), one), X)
% 20.76/3.04 = { by lemma 18 R->L }
% 20.76/3.04 composition(converse(join(composition(converse(sk1), sk1), one)), composition(join(composition(converse(sk1), sk1), one), X))
% 20.76/3.04 = { by axiom 11 (goals_14) }
% 20.76/3.04 composition(converse(join(composition(converse(sk1), sk1), one)), composition(one, X))
% 20.76/3.04 = { by axiom 9 (composition_associativity_5) }
% 20.76/3.04 composition(composition(converse(join(composition(converse(sk1), sk1), one)), one), X)
% 20.76/3.04 = { by axiom 3 (composition_identity_6) }
% 20.76/3.04 composition(converse(join(composition(converse(sk1), sk1), one)), X)
% 20.76/3.04 = { by lemma 18 }
% 20.76/3.04 X
% 20.76/3.04
% 20.76/3.04 Lemma 20: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 20.76/3.04 Proof:
% 20.76/3.04 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 20.76/3.04 = { by axiom 13 (converse_cancellativity_11) }
% 20.76/3.04 complement(X)
% 20.76/3.04
% 20.76/3.04 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 20.76/3.04 Proof:
% 20.76/3.04 join(complement(X), complement(X))
% 20.76/3.04 = { by lemma 18 R->L }
% 20.76/3.04 join(complement(X), composition(converse(join(composition(converse(sk1), sk1), one)), complement(X)))
% 20.76/3.04 = { by lemma 19 R->L }
% 20.76/3.04 join(complement(X), composition(converse(join(composition(converse(sk1), sk1), one)), complement(composition(join(composition(converse(sk1), sk1), one), X))))
% 20.76/3.04 = { by lemma 20 }
% 20.76/3.04 complement(X)
% 20.76/3.04
% 20.76/3.04 Lemma 22: join(top, complement(X)) = top.
% 20.76/3.04 Proof:
% 20.76/3.04 join(top, complement(X))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(complement(X), top)
% 20.76/3.04 = { by lemma 16 R->L }
% 20.76/3.04 join(X, join(complement(X), complement(X)))
% 20.76/3.04 = { by lemma 21 }
% 20.76/3.04 join(X, complement(X))
% 20.76/3.04 = { by axiom 5 (def_top_12) R->L }
% 20.76/3.04 top
% 20.76/3.04
% 20.76/3.04 Lemma 23: join(Y, top) = join(X, top).
% 20.76/3.04 Proof:
% 20.76/3.04 join(Y, top)
% 20.76/3.04 = { by lemma 22 R->L }
% 20.76/3.04 join(Y, join(top, complement(Y)))
% 20.76/3.04 = { by lemma 16 }
% 20.76/3.04 join(top, top)
% 20.76/3.04 = { by lemma 16 R->L }
% 20.76/3.04 join(X, join(top, complement(X)))
% 20.76/3.04 = { by lemma 22 }
% 20.76/3.04 join(X, top)
% 20.76/3.04
% 20.76/3.04 Lemma 24: join(X, top) = top.
% 20.76/3.04 Proof:
% 20.76/3.04 join(X, top)
% 20.76/3.04 = { by lemma 23 }
% 20.76/3.04 join(complement(Y), top)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(top, complement(Y))
% 20.76/3.04 = { by lemma 22 }
% 20.76/3.04 top
% 20.76/3.04
% 20.76/3.04 Lemma 25: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 20.76/3.04 Proof:
% 20.76/3.04 converse(join(X, converse(Y)))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 converse(join(converse(Y), X))
% 20.76/3.04 = { by axiom 6 (converse_additivity_9) }
% 20.76/3.04 join(converse(converse(Y)), converse(X))
% 20.76/3.04 = { by axiom 1 (converse_idempotence_8) }
% 20.76/3.04 join(Y, converse(X))
% 20.76/3.04
% 20.76/3.04 Lemma 26: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 20.76/3.04 Proof:
% 20.76/3.04 converse(join(converse(X), Y))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 converse(join(Y, converse(X)))
% 20.76/3.04 = { by lemma 25 }
% 20.76/3.04 join(X, converse(Y))
% 20.76/3.04
% 20.76/3.04 Lemma 27: join(X, converse(complement(converse(X)))) = converse(top).
% 20.76/3.04 Proof:
% 20.76/3.04 join(X, converse(complement(converse(X))))
% 20.76/3.04 = { by lemma 26 R->L }
% 20.76/3.04 converse(join(converse(X), complement(converse(X))))
% 20.76/3.04 = { by axiom 5 (def_top_12) R->L }
% 20.76/3.04 converse(top)
% 20.76/3.04
% 20.76/3.04 Lemma 28: join(X, join(complement(X), Y)) = top.
% 20.76/3.04 Proof:
% 20.76/3.04 join(X, join(complement(X), Y))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(X, join(Y, complement(X)))
% 20.76/3.04 = { by lemma 16 }
% 20.76/3.04 join(Y, top)
% 20.76/3.04 = { by lemma 23 R->L }
% 20.76/3.04 join(Z, top)
% 20.76/3.04 = { by lemma 24 }
% 20.76/3.04 top
% 20.76/3.04
% 20.76/3.04 Lemma 29: converse(top) = top.
% 20.76/3.04 Proof:
% 20.76/3.04 converse(top)
% 20.76/3.04 = { by lemma 24 R->L }
% 20.76/3.04 converse(join(X, top))
% 20.76/3.04 = { by axiom 6 (converse_additivity_9) }
% 20.76/3.04 join(converse(X), converse(top))
% 20.76/3.04 = { by lemma 27 R->L }
% 20.76/3.04 join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X)))))))
% 20.76/3.04 = { by lemma 28 }
% 20.76/3.04 top
% 20.76/3.04
% 20.76/3.04 Lemma 30: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 20.76/3.04 Proof:
% 20.76/3.04 join(meet(X, Y), complement(join(complement(X), Y)))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 20.76/3.04 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 20.76/3.04 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 20.76/3.04 X
% 20.76/3.04
% 20.76/3.04 Lemma 31: join(zero, meet(X, X)) = X.
% 20.76/3.04 Proof:
% 20.76/3.04 join(zero, meet(X, X))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 20.76/3.04 join(zero, complement(join(complement(X), complement(X))))
% 20.76/3.04 = { by axiom 4 (def_zero_13) }
% 20.76/3.04 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 20.76/3.04 = { by lemma 30 }
% 20.76/3.04 X
% 20.76/3.04
% 20.76/3.04 Lemma 32: complement(complement(X)) = meet(X, X).
% 20.76/3.04 Proof:
% 20.76/3.04 complement(complement(X))
% 20.76/3.04 = { by lemma 21 R->L }
% 20.76/3.04 complement(join(complement(X), complement(X)))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 20.76/3.04 meet(X, X)
% 20.76/3.04
% 20.76/3.04 Lemma 33: meet(Y, X) = meet(X, Y).
% 20.76/3.04 Proof:
% 20.76/3.04 meet(Y, X)
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 20.76/3.04 complement(join(complement(Y), complement(X)))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 complement(join(complement(X), complement(Y)))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 20.76/3.04 meet(X, Y)
% 20.76/3.04
% 20.76/3.04 Lemma 34: complement(join(zero, complement(X))) = meet(X, top).
% 20.76/3.04 Proof:
% 20.76/3.04 complement(join(zero, complement(X)))
% 20.76/3.04 = { by lemma 15 R->L }
% 20.76/3.04 complement(join(complement(top), complement(X)))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 20.76/3.04 meet(top, X)
% 20.76/3.04 = { by lemma 33 R->L }
% 20.76/3.04 meet(X, top)
% 20.76/3.04
% 20.76/3.04 Lemma 35: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 20.76/3.04 Proof:
% 20.76/3.04 join(zero, join(X, meet(Y, Y)))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(zero, join(meet(Y, Y), X))
% 20.76/3.04 = { by axiom 7 (maddux2_join_associativity_2) }
% 20.76/3.04 join(join(zero, meet(Y, Y)), X)
% 20.76/3.04 = { by lemma 31 }
% 20.76/3.04 join(Y, X)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.04 join(X, Y)
% 20.76/3.04
% 20.76/3.04 Lemma 36: join(X, complement(zero)) = top.
% 20.76/3.04 Proof:
% 20.76/3.04 join(X, complement(zero))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(complement(zero), X)
% 20.76/3.04 = { by lemma 35 R->L }
% 20.76/3.04 join(zero, join(complement(zero), meet(X, X)))
% 20.76/3.04 = { by lemma 28 }
% 20.76/3.04 top
% 20.76/3.04
% 20.76/3.04 Lemma 37: meet(X, zero) = zero.
% 20.76/3.04 Proof:
% 20.76/3.04 meet(X, zero)
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 20.76/3.04 complement(join(complement(X), complement(zero)))
% 20.76/3.04 = { by lemma 36 }
% 20.76/3.04 complement(top)
% 20.76/3.04 = { by lemma 15 }
% 20.76/3.04 zero
% 20.76/3.04
% 20.76/3.04 Lemma 38: join(meet(X, Y), meet(X, complement(Y))) = X.
% 20.76/3.04 Proof:
% 20.76/3.04 join(meet(X, Y), meet(X, complement(Y)))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(meet(X, complement(Y)), meet(X, Y))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 20.76/3.04 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 20.76/3.04 = { by lemma 30 }
% 20.76/3.04 X
% 20.76/3.04
% 20.76/3.04 Lemma 39: join(zero, meet(X, top)) = X.
% 20.76/3.04 Proof:
% 20.76/3.04 join(zero, meet(X, top))
% 20.76/3.04 = { by lemma 36 R->L }
% 20.76/3.04 join(zero, meet(X, join(complement(zero), complement(zero))))
% 20.76/3.04 = { by lemma 21 }
% 20.76/3.04 join(zero, meet(X, complement(zero)))
% 20.76/3.04 = { by lemma 37 R->L }
% 20.76/3.04 join(meet(X, zero), meet(X, complement(zero)))
% 20.76/3.04 = { by lemma 38 }
% 20.76/3.04 X
% 20.76/3.04
% 20.76/3.04 Lemma 40: join(zero, complement(X)) = complement(X).
% 20.76/3.04 Proof:
% 20.76/3.04 join(zero, complement(X))
% 20.76/3.04 = { by lemma 31 R->L }
% 20.76/3.04 join(zero, complement(join(zero, meet(X, X))))
% 20.76/3.04 = { by lemma 32 R->L }
% 20.76/3.04 join(zero, complement(join(zero, complement(complement(X)))))
% 20.76/3.04 = { by lemma 34 }
% 20.76/3.04 join(zero, meet(complement(X), top))
% 20.76/3.04 = { by lemma 39 }
% 20.76/3.04 complement(X)
% 20.76/3.04
% 20.76/3.04 Lemma 41: complement(complement(X)) = X.
% 20.76/3.04 Proof:
% 20.76/3.04 complement(complement(X))
% 20.76/3.04 = { by lemma 40 R->L }
% 20.76/3.04 join(zero, complement(complement(X)))
% 20.76/3.04 = { by lemma 32 }
% 20.76/3.04 join(zero, meet(X, X))
% 20.76/3.04 = { by lemma 31 }
% 20.76/3.04 X
% 20.76/3.04
% 20.76/3.04 Lemma 42: join(X, zero) = X.
% 20.76/3.04 Proof:
% 20.76/3.04 join(X, zero)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(zero, X)
% 20.76/3.04 = { by lemma 41 R->L }
% 20.76/3.04 join(zero, complement(complement(X)))
% 20.76/3.04 = { by lemma 32 }
% 20.76/3.04 join(zero, meet(X, X))
% 20.76/3.04 = { by lemma 31 }
% 20.76/3.04 X
% 20.76/3.04
% 20.76/3.04 Lemma 43: join(zero, X) = X.
% 20.76/3.04 Proof:
% 20.76/3.04 join(zero, X)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(X, zero)
% 20.76/3.04 = { by lemma 42 }
% 20.76/3.04 X
% 20.76/3.04
% 20.76/3.04 Lemma 44: join(X, zero) = join(X, X).
% 20.76/3.04 Proof:
% 20.76/3.04 join(X, zero)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(zero, X)
% 20.76/3.04 = { by lemma 31 R->L }
% 20.76/3.04 join(zero, join(zero, meet(X, X)))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 20.76/3.04 join(zero, join(zero, complement(join(complement(X), complement(X)))))
% 20.76/3.04 = { by lemma 21 R->L }
% 20.76/3.04 join(zero, join(zero, join(complement(join(complement(X), complement(X))), complement(join(complement(X), complement(X))))))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 20.76/3.04 join(zero, join(zero, join(meet(X, X), complement(join(complement(X), complement(X))))))
% 20.76/3.04 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 20.76/3.04 join(zero, join(zero, join(meet(X, X), meet(X, X))))
% 20.76/3.04 = { by lemma 35 }
% 20.76/3.04 join(zero, join(meet(X, X), X))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.04 join(zero, join(X, meet(X, X)))
% 20.76/3.04 = { by lemma 35 }
% 20.76/3.04 join(X, X)
% 20.76/3.04
% 20.76/3.04 Lemma 45: composition(join(X, join(composition(converse(sk1), sk1), one)), Y) = join(Y, composition(X, Y)).
% 20.76/3.04 Proof:
% 20.76/3.04 composition(join(X, join(composition(converse(sk1), sk1), one)), Y)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 composition(join(join(composition(converse(sk1), sk1), one), X), Y)
% 20.76/3.04 = { by axiom 12 (composition_distributivity_7) }
% 20.76/3.04 join(composition(join(composition(converse(sk1), sk1), one), Y), composition(X, Y))
% 20.76/3.04 = { by lemma 19 }
% 20.76/3.04 join(Y, composition(X, Y))
% 20.76/3.04
% 20.76/3.04 Lemma 46: join(join(composition(converse(sk1), sk1), one), X) = join(X, one).
% 20.76/3.04 Proof:
% 20.76/3.04 join(join(composition(converse(sk1), sk1), one), X)
% 20.76/3.04 = { by axiom 11 (goals_14) }
% 20.76/3.04 join(one, X)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(X, one)
% 20.76/3.04
% 20.76/3.04 Lemma 47: join(join(composition(converse(sk1), sk1), one), join(X, Y)) = join(X, join(Y, one)).
% 20.76/3.04 Proof:
% 20.76/3.04 join(join(composition(converse(sk1), sk1), one), join(X, Y))
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 join(join(composition(converse(sk1), sk1), one), join(Y, X))
% 20.76/3.04 = { by axiom 7 (maddux2_join_associativity_2) }
% 20.76/3.04 join(join(join(composition(converse(sk1), sk1), one), Y), X)
% 20.76/3.04 = { by lemma 46 }
% 20.76/3.04 join(join(Y, one), X)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.04 join(X, join(Y, one))
% 20.76/3.04
% 20.76/3.04 Lemma 48: converse(zero) = zero.
% 20.76/3.04 Proof:
% 20.76/3.04 converse(zero)
% 20.76/3.04 = { by lemma 43 R->L }
% 20.76/3.04 join(zero, converse(zero))
% 20.76/3.04 = { by lemma 26 R->L }
% 20.76/3.04 converse(join(converse(zero), zero))
% 20.76/3.04 = { by lemma 44 }
% 20.76/3.04 converse(join(converse(zero), converse(zero)))
% 20.76/3.04 = { by lemma 25 }
% 20.76/3.04 join(zero, converse(converse(zero)))
% 20.76/3.04 = { by axiom 1 (converse_idempotence_8) }
% 20.76/3.04 join(zero, zero)
% 20.76/3.04 = { by lemma 19 R->L }
% 20.76/3.04 join(zero, composition(join(composition(converse(sk1), sk1), one), zero))
% 20.76/3.04 = { by lemma 45 R->L }
% 20.76/3.04 composition(join(join(composition(converse(sk1), sk1), one), join(composition(converse(sk1), sk1), one)), zero)
% 20.76/3.04 = { by lemma 44 R->L }
% 20.76/3.04 composition(join(join(composition(converse(sk1), sk1), one), zero), zero)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.04 composition(join(zero, join(composition(converse(sk1), sk1), one)), zero)
% 20.76/3.04 = { by lemma 47 R->L }
% 20.76/3.04 composition(join(join(composition(converse(sk1), sk1), one), join(zero, composition(converse(sk1), sk1))), zero)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.04 composition(join(join(composition(converse(sk1), sk1), one), join(composition(converse(sk1), sk1), zero)), zero)
% 20.76/3.04 = { by lemma 44 }
% 20.76/3.04 composition(join(join(composition(converse(sk1), sk1), one), join(composition(converse(sk1), sk1), composition(converse(sk1), sk1))), zero)
% 20.76/3.04 = { by lemma 47 }
% 20.76/3.04 composition(join(composition(converse(sk1), sk1), join(composition(converse(sk1), sk1), one)), zero)
% 20.76/3.04 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.05 composition(join(join(composition(converse(sk1), sk1), one), composition(converse(sk1), sk1)), zero)
% 20.76/3.05 = { by lemma 46 }
% 20.76/3.05 composition(join(composition(converse(sk1), sk1), one), zero)
% 20.76/3.05 = { by lemma 19 }
% 20.76/3.05 zero
% 20.76/3.05
% 20.76/3.05 Lemma 49: meet(X, X) = X.
% 20.76/3.05 Proof:
% 20.76/3.05 meet(X, X)
% 20.76/3.05 = { by lemma 32 R->L }
% 20.76/3.05 complement(complement(X))
% 20.76/3.05 = { by lemma 41 }
% 20.76/3.05 X
% 20.76/3.05
% 20.76/3.05 Lemma 50: meet(X, top) = X.
% 20.76/3.05 Proof:
% 20.76/3.05 meet(X, top)
% 20.76/3.05 = { by lemma 34 R->L }
% 20.76/3.05 complement(join(zero, complement(X)))
% 20.76/3.05 = { by lemma 40 R->L }
% 20.76/3.05 join(zero, complement(join(zero, complement(X))))
% 20.76/3.05 = { by lemma 34 }
% 20.76/3.05 join(zero, meet(X, top))
% 20.76/3.05 = { by lemma 39 }
% 20.76/3.05 X
% 20.76/3.05
% 20.76/3.05 Lemma 51: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 20.76/3.05 Proof:
% 20.76/3.05 complement(join(complement(X), meet(Y, Z)))
% 20.76/3.05 = { by lemma 33 }
% 20.76/3.05 complement(join(complement(X), meet(Z, Y)))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.05 complement(join(meet(Z, Y), complement(X)))
% 20.76/3.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 20.76/3.05 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 20.76/3.05 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 20.76/3.05 meet(join(complement(Z), complement(Y)), X)
% 20.76/3.05 = { by lemma 33 R->L }
% 20.76/3.05 meet(X, join(complement(Z), complement(Y)))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.05 meet(X, join(complement(Y), complement(Z)))
% 20.76/3.05
% 20.76/3.05 Lemma 52: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 20.76/3.05 Proof:
% 20.76/3.05 join(complement(X), complement(Y))
% 20.76/3.05 = { by lemma 50 R->L }
% 20.76/3.05 meet(join(complement(X), complement(Y)), top)
% 20.76/3.05 = { by lemma 33 R->L }
% 20.76/3.05 meet(top, join(complement(X), complement(Y)))
% 20.76/3.05 = { by lemma 51 R->L }
% 20.76/3.05 complement(join(complement(top), meet(X, Y)))
% 20.76/3.05 = { by lemma 15 }
% 20.76/3.05 complement(join(zero, meet(X, Y)))
% 20.76/3.05 = { by lemma 33 R->L }
% 20.76/3.05 complement(join(zero, meet(Y, X)))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.05 complement(join(meet(Y, X), zero))
% 20.76/3.05 = { by lemma 42 }
% 20.76/3.05 complement(meet(Y, X))
% 20.76/3.05 = { by lemma 33 R->L }
% 20.76/3.05 complement(meet(X, Y))
% 20.76/3.05
% 20.76/3.05 Lemma 53: join(complement(X), complement(Y)) = complement(meet(Y, X)).
% 20.76/3.05 Proof:
% 20.76/3.05 join(complement(X), complement(Y))
% 20.76/3.05 = { by lemma 52 }
% 20.76/3.05 complement(meet(X, Y))
% 20.76/3.05 = { by lemma 33 R->L }
% 20.76/3.05 complement(meet(Y, X))
% 20.76/3.05
% 20.76/3.05 Lemma 54: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 20.76/3.05 Proof:
% 20.76/3.05 complement(meet(X, complement(Y)))
% 20.76/3.05 = { by lemma 33 }
% 20.76/3.05 complement(meet(complement(Y), X))
% 20.76/3.05 = { by lemma 40 R->L }
% 20.76/3.05 complement(meet(join(zero, complement(Y)), X))
% 20.76/3.05 = { by lemma 52 R->L }
% 20.76/3.05 join(complement(join(zero, complement(Y))), complement(X))
% 20.76/3.05 = { by lemma 34 }
% 20.76/3.05 join(meet(Y, top), complement(X))
% 20.76/3.05 = { by lemma 50 }
% 20.76/3.05 join(Y, complement(X))
% 20.76/3.05
% 20.76/3.05 Lemma 55: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 20.76/3.05 Proof:
% 20.76/3.05 complement(join(X, complement(Y)))
% 20.76/3.05 = { by lemma 54 R->L }
% 20.76/3.05 complement(complement(meet(Y, complement(X))))
% 20.76/3.05 = { by lemma 32 }
% 20.76/3.05 meet(meet(Y, complement(X)), meet(Y, complement(X)))
% 20.76/3.05 = { by lemma 49 }
% 20.76/3.05 meet(Y, complement(X))
% 20.76/3.05
% 20.76/3.05 Lemma 56: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 20.76/3.05 Proof:
% 20.76/3.05 complement(join(complement(X), Y))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.05 complement(join(Y, complement(X)))
% 20.76/3.05 = { by lemma 55 }
% 20.76/3.05 meet(X, complement(Y))
% 20.76/3.05
% 20.76/3.05 Lemma 57: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 20.76/3.05 Proof:
% 20.76/3.05 complement(meet(complement(X), Y))
% 20.76/3.05 = { by lemma 33 }
% 20.76/3.05 complement(meet(Y, complement(X)))
% 20.76/3.05 = { by lemma 54 }
% 20.76/3.05 join(X, complement(Y))
% 20.76/3.05
% 20.76/3.05 Lemma 58: join(X, complement(meet(X, Y))) = top.
% 20.76/3.05 Proof:
% 20.76/3.05 join(X, complement(meet(X, Y)))
% 20.76/3.05 = { by lemma 33 }
% 20.76/3.05 join(X, complement(meet(Y, X)))
% 20.76/3.05 = { by lemma 52 R->L }
% 20.76/3.05 join(X, join(complement(Y), complement(X)))
% 20.76/3.05 = { by lemma 16 }
% 20.76/3.05 join(complement(Y), top)
% 20.76/3.05 = { by lemma 24 }
% 20.76/3.05 top
% 20.76/3.05
% 20.76/3.05 Lemma 59: meet(X, meet(Y, complement(X))) = zero.
% 20.76/3.05 Proof:
% 20.76/3.05 meet(X, meet(Y, complement(X)))
% 20.76/3.05 = { by lemma 33 }
% 20.76/3.05 meet(X, meet(complement(X), Y))
% 20.76/3.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 20.76/3.05 complement(join(complement(X), complement(meet(complement(X), Y))))
% 20.76/3.05 = { by lemma 58 }
% 20.76/3.05 complement(top)
% 20.76/3.05 = { by lemma 15 }
% 20.76/3.05 zero
% 20.76/3.05
% 20.76/3.05 Lemma 60: meet(X, join(Y, X)) = X.
% 20.76/3.05 Proof:
% 20.76/3.05 meet(X, join(Y, X))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.05 meet(X, join(X, Y))
% 20.76/3.05 = { by lemma 49 R->L }
% 20.76/3.05 meet(X, join(X, meet(Y, Y)))
% 20.76/3.05 = { by lemma 32 R->L }
% 20.76/3.05 meet(X, join(X, complement(complement(Y))))
% 20.76/3.05 = { by lemma 54 R->L }
% 20.76/3.05 meet(X, complement(meet(complement(Y), complement(X))))
% 20.76/3.05 = { by lemma 52 R->L }
% 20.76/3.05 meet(X, join(complement(complement(Y)), complement(complement(X))))
% 20.76/3.05 = { by lemma 51 R->L }
% 20.76/3.05 complement(join(complement(X), meet(complement(Y), complement(X))))
% 20.76/3.05 = { by lemma 40 R->L }
% 20.76/3.05 join(zero, complement(join(complement(X), meet(complement(Y), complement(X)))))
% 20.76/3.05 = { by lemma 59 R->L }
% 20.76/3.05 join(meet(X, meet(complement(Y), complement(X))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 20.76/3.05 = { by lemma 30 }
% 20.76/3.05 X
% 20.76/3.05
% 20.76/3.05 Lemma 61: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 20.76/3.05 Proof:
% 20.76/3.05 meet(Y, meet(X, Z))
% 20.76/3.05 = { by lemma 33 }
% 20.76/3.05 meet(Y, meet(Z, X))
% 20.76/3.05 = { by lemma 41 R->L }
% 20.76/3.05 complement(complement(meet(Y, meet(Z, X))))
% 20.76/3.05 = { by lemma 33 }
% 20.76/3.05 complement(complement(meet(Y, meet(X, Z))))
% 20.76/3.05 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 20.76/3.05 complement(complement(meet(Y, complement(join(complement(X), complement(Z))))))
% 20.76/3.05 = { by lemma 54 }
% 20.76/3.05 complement(join(join(complement(X), complement(Z)), complement(Y)))
% 20.76/3.05 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 20.76/3.05 complement(join(complement(X), join(complement(Z), complement(Y))))
% 20.76/3.05 = { by lemma 52 }
% 20.76/3.05 complement(join(complement(X), complement(meet(Z, Y))))
% 20.76/3.05 = { by lemma 52 }
% 20.76/3.05 complement(complement(meet(X, meet(Z, Y))))
% 20.76/3.05 = { by lemma 33 R->L }
% 20.76/3.05 complement(complement(meet(X, meet(Y, Z))))
% 20.76/3.05 = { by lemma 32 }
% 20.76/3.05 meet(meet(X, meet(Y, Z)), meet(X, meet(Y, Z)))
% 20.76/3.05 = { by lemma 49 }
% 20.76/3.05 meet(X, meet(Y, Z))
% 20.76/3.05
% 20.76/3.05 Lemma 62: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 20.76/3.05 Proof:
% 20.76/3.05 converse(composition(X, converse(Y)))
% 20.76/3.05 = { by axiom 8 (converse_multiplicativity_10) }
% 20.76/3.05 composition(converse(converse(Y)), converse(X))
% 20.76/3.05 = { by axiom 1 (converse_idempotence_8) }
% 20.76/3.05 composition(Y, converse(X))
% 20.76/3.05
% 20.76/3.05 Lemma 63: complement(join(complement(X), complement(Y))) = meet(Y, X).
% 20.76/3.05 Proof:
% 20.76/3.05 complement(join(complement(X), complement(Y)))
% 20.76/3.05 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 20.76/3.05 meet(X, Y)
% 20.76/3.05 = { by lemma 33 R->L }
% 20.76/3.05 meet(Y, X)
% 20.76/3.05
% 20.76/3.05 Lemma 64: join(composition(X, complement(Y)), composition(X, Y)) = converse(composition(top, converse(X))).
% 20.76/3.05 Proof:
% 20.76/3.05 join(composition(X, complement(Y)), composition(X, Y))
% 20.76/3.05 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.76/3.05 converse(converse(join(composition(X, complement(Y)), composition(X, Y))))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.05 converse(converse(join(composition(X, Y), composition(X, complement(Y)))))
% 20.76/3.05 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.76/3.05 converse(converse(join(composition(X, Y), composition(X, complement(converse(converse(Y)))))))
% 20.76/3.05 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.76/3.05 converse(converse(join(composition(X, Y), composition(converse(converse(X)), complement(converse(converse(Y)))))))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.05 converse(converse(join(composition(converse(converse(X)), complement(converse(converse(Y)))), composition(X, Y))))
% 20.76/3.05 = { by axiom 6 (converse_additivity_9) }
% 20.76/3.05 converse(join(converse(composition(converse(converse(X)), complement(converse(converse(Y))))), converse(composition(X, Y))))
% 20.76/3.05 = { by lemma 17 }
% 20.76/3.05 converse(join(composition(converse(complement(converse(converse(Y)))), converse(X)), converse(composition(X, Y))))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.05 converse(join(converse(composition(X, Y)), composition(converse(complement(converse(converse(Y)))), converse(X))))
% 20.76/3.05 = { by axiom 8 (converse_multiplicativity_10) }
% 20.76/3.05 converse(join(composition(converse(Y), converse(X)), composition(converse(complement(converse(converse(Y)))), converse(X))))
% 20.76/3.05 = { by axiom 12 (composition_distributivity_7) R->L }
% 20.76/3.05 converse(composition(join(converse(Y), converse(complement(converse(converse(Y))))), converse(X)))
% 20.76/3.05 = { by lemma 27 }
% 20.76/3.05 converse(composition(converse(top), converse(X)))
% 20.76/3.05 = { by lemma 29 }
% 20.76/3.05 converse(composition(top, converse(X)))
% 20.76/3.05
% 20.76/3.05 Lemma 65: composition(converse(X), complement(composition(X, top))) = zero.
% 20.76/3.05 Proof:
% 20.76/3.05 composition(converse(X), complement(composition(X, top)))
% 20.76/3.05 = { by lemma 43 R->L }
% 20.76/3.05 join(zero, composition(converse(X), complement(composition(X, top))))
% 20.76/3.05 = { by lemma 15 R->L }
% 20.76/3.05 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 20.76/3.05 = { by lemma 20 }
% 20.76/3.05 complement(top)
% 20.76/3.05 = { by lemma 15 }
% 20.76/3.05 zero
% 20.76/3.05
% 20.76/3.05 Lemma 66: join(composition(X, Y), composition(X, converse(Z))) = composition(X, join(Y, converse(Z))).
% 20.76/3.05 Proof:
% 20.76/3.05 join(composition(X, Y), composition(X, converse(Z)))
% 20.76/3.05 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.76/3.05 converse(converse(join(composition(X, Y), composition(X, converse(Z)))))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.05 converse(converse(join(composition(X, converse(Z)), composition(X, Y))))
% 20.76/3.05 = { by axiom 6 (converse_additivity_9) }
% 20.76/3.05 converse(join(converse(composition(X, converse(Z))), converse(composition(X, Y))))
% 20.76/3.05 = { by lemma 62 }
% 20.76/3.05 converse(join(composition(Z, converse(X)), converse(composition(X, Y))))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.05 converse(join(converse(composition(X, Y)), composition(Z, converse(X))))
% 20.76/3.05 = { by axiom 8 (converse_multiplicativity_10) }
% 20.76/3.05 converse(join(composition(converse(Y), converse(X)), composition(Z, converse(X))))
% 20.76/3.05 = { by axiom 12 (composition_distributivity_7) R->L }
% 20.76/3.05 converse(composition(join(converse(Y), Z), converse(X)))
% 20.76/3.05 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.05 converse(composition(join(Z, converse(Y)), converse(X)))
% 20.76/3.05 = { by lemma 25 R->L }
% 20.76/3.05 converse(composition(converse(join(Y, converse(Z))), converse(X)))
% 20.76/3.05 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 20.76/3.05 converse(converse(composition(X, join(Y, converse(Z)))))
% 20.76/3.05 = { by axiom 1 (converse_idempotence_8) }
% 20.76/3.05 composition(X, join(Y, converse(Z)))
% 20.76/3.05
% 20.76/3.05 Lemma 67: composition(composition(converse(X), X), top) = composition(converse(X), top).
% 20.76/3.05 Proof:
% 20.76/3.05 composition(composition(converse(X), X), top)
% 20.76/3.05 = { by lemma 29 R->L }
% 20.76/3.05 composition(composition(converse(X), X), converse(top))
% 20.76/3.05 = { by lemma 17 R->L }
% 20.76/3.05 composition(converse(composition(converse(X), X)), converse(top))
% 20.76/3.05 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 20.76/3.05 converse(composition(top, composition(converse(X), X)))
% 20.76/3.05 = { by axiom 9 (composition_associativity_5) }
% 20.76/3.05 converse(composition(composition(top, converse(X)), X))
% 20.76/3.05 = { by axiom 8 (converse_multiplicativity_10) }
% 20.76/3.05 composition(converse(X), converse(composition(top, converse(X))))
% 20.76/3.05 = { by lemma 64 R->L }
% 20.76/3.05 composition(converse(X), join(composition(X, complement(Y)), composition(X, Y)))
% 20.76/3.05 = { by lemma 42 R->L }
% 20.76/3.05 join(composition(converse(X), join(composition(X, complement(Y)), composition(X, Y))), zero)
% 20.76/3.05 = { by lemma 48 R->L }
% 20.76/3.05 join(composition(converse(X), join(composition(X, complement(Y)), composition(X, Y))), converse(zero))
% 20.76/3.05 = { by lemma 48 R->L }
% 20.76/3.05 join(composition(converse(X), join(composition(X, complement(Y)), composition(X, Y))), converse(converse(zero)))
% 20.76/3.05 = { by lemma 65 R->L }
% 20.76/3.05 join(composition(converse(X), join(composition(X, complement(Y)), composition(X, Y))), converse(converse(composition(converse(X), complement(composition(X, top))))))
% 20.76/3.05 = { by lemma 17 }
% 20.76/3.05 join(composition(converse(X), join(composition(X, complement(Y)), composition(X, Y))), converse(composition(converse(complement(composition(X, top))), X)))
% 20.76/3.05 = { by axiom 8 (converse_multiplicativity_10) }
% 20.76/3.05 join(composition(converse(X), join(composition(X, complement(Y)), composition(X, Y))), composition(converse(X), converse(converse(complement(composition(X, top))))))
% 20.76/3.05 = { by lemma 66 }
% 20.76/3.05 composition(converse(X), join(join(composition(X, complement(Y)), composition(X, Y)), converse(converse(complement(composition(X, top))))))
% 20.76/3.05 = { by axiom 1 (converse_idempotence_8) }
% 20.76/3.05 composition(converse(X), join(join(composition(X, complement(Y)), composition(X, Y)), complement(composition(X, top))))
% 20.76/3.05 = { by lemma 29 R->L }
% 20.76/3.05 composition(converse(X), join(join(composition(X, complement(Y)), composition(X, Y)), complement(composition(X, converse(top)))))
% 20.76/3.05 = { by lemma 62 R->L }
% 20.76/3.05 composition(converse(X), join(join(composition(X, complement(Y)), composition(X, Y)), complement(converse(composition(top, converse(X))))))
% 20.76/3.06 = { by lemma 64 R->L }
% 20.76/3.06 composition(converse(X), join(join(composition(X, complement(Y)), composition(X, Y)), complement(join(composition(X, complement(Y)), composition(X, Y)))))
% 20.76/3.06 = { by axiom 5 (def_top_12) R->L }
% 20.76/3.06 composition(converse(X), top)
% 20.76/3.06
% 20.76/3.06 Lemma 68: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 20.76/3.06 Proof:
% 20.76/3.06 join(composition(X, Y), composition(X, Z))
% 20.76/3.06 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.76/3.06 join(composition(X, Y), composition(X, converse(converse(Z))))
% 20.76/3.06 = { by lemma 66 }
% 20.76/3.06 composition(X, join(Y, converse(converse(Z))))
% 20.76/3.06 = { by axiom 1 (converse_idempotence_8) }
% 20.76/3.06 composition(X, join(Y, Z))
% 20.76/3.06
% 20.76/3.06 Goal 1 (goals_15): meet(composition(sk1, sk2), composition(sk1, complement(sk2))) = zero.
% 20.76/3.06 Proof:
% 20.76/3.06 meet(composition(sk1, sk2), composition(sk1, complement(sk2)))
% 20.76/3.06 = { by lemma 33 }
% 20.76/3.06 meet(composition(sk1, complement(sk2)), composition(sk1, sk2))
% 20.76/3.06 = { by lemma 41 R->L }
% 20.76/3.06 meet(composition(sk1, complement(sk2)), complement(complement(composition(sk1, sk2))))
% 20.76/3.06 = { by lemma 38 R->L }
% 20.76/3.06 join(meet(meet(composition(sk1, complement(sk2)), complement(complement(composition(sk1, sk2)))), composition(sk1, sk2)), meet(meet(composition(sk1, complement(sk2)), complement(complement(composition(sk1, sk2)))), complement(composition(sk1, sk2))))
% 20.76/3.06 = { by lemma 33 R->L }
% 20.76/3.06 join(meet(meet(composition(sk1, complement(sk2)), complement(complement(composition(sk1, sk2)))), composition(sk1, sk2)), meet(complement(composition(sk1, sk2)), meet(composition(sk1, complement(sk2)), complement(complement(composition(sk1, sk2))))))
% 20.76/3.06 = { by lemma 59 }
% 20.76/3.06 join(meet(meet(composition(sk1, complement(sk2)), complement(complement(composition(sk1, sk2)))), composition(sk1, sk2)), zero)
% 20.76/3.06 = { by lemma 42 }
% 20.76/3.06 meet(meet(composition(sk1, complement(sk2)), complement(complement(composition(sk1, sk2)))), composition(sk1, sk2))
% 20.76/3.06 = { by lemma 41 }
% 20.76/3.06 meet(meet(composition(sk1, complement(sk2)), composition(sk1, sk2)), composition(sk1, sk2))
% 20.76/3.06 = { by lemma 33 R->L }
% 20.76/3.06 meet(composition(sk1, sk2), meet(composition(sk1, complement(sk2)), composition(sk1, sk2)))
% 20.76/3.06 = { by lemma 33 R->L }
% 20.76/3.06 meet(composition(sk1, sk2), meet(composition(sk1, sk2), composition(sk1, complement(sk2))))
% 20.76/3.06 = { by lemma 33 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), composition(sk1, sk2))
% 20.76/3.06 = { by lemma 41 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(complement(composition(sk1, sk2))))
% 20.76/3.06 = { by lemma 20 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(converse(converse(sk1)), complement(composition(converse(sk1), composition(sk1, sk2)))))))
% 20.76/3.06 = { by axiom 9 (composition_associativity_5) }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(converse(converse(sk1)), complement(composition(composition(converse(sk1), sk1), sk2))))))
% 20.76/3.06 = { by axiom 1 (converse_idempotence_8) }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(composition(composition(converse(sk1), sk1), sk2))))))
% 20.76/3.06 = { by lemma 38 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(meet(complement(composition(composition(converse(sk1), sk1), sk2)), sk2), meet(complement(composition(composition(converse(sk1), sk1), sk2)), complement(sk2)))))))
% 20.76/3.06 = { by lemma 33 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, complement(composition(composition(converse(sk1), sk1), sk2))), meet(complement(composition(composition(converse(sk1), sk1), sk2)), complement(sk2)))))))
% 20.76/3.06 = { by lemma 33 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, complement(composition(composition(converse(sk1), sk1), sk2))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by lemma 56 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(join(complement(sk2), composition(composition(converse(sk1), sk1), sk2))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by axiom 9 (composition_associativity_5) R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(join(complement(sk2), composition(converse(sk1), composition(sk1, sk2)))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by lemma 20 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(join(join(complement(sk2), composition(converse(sk1), complement(composition(sk1, sk2)))), composition(converse(sk1), composition(sk1, sk2)))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(join(complement(sk2), join(composition(converse(sk1), complement(composition(sk1, sk2))), composition(converse(sk1), composition(sk1, sk2))))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(join(complement(sk2), join(composition(converse(sk1), composition(sk1, sk2)), composition(converse(sk1), complement(composition(sk1, sk2)))))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by lemma 68 }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(join(complement(sk2), composition(converse(sk1), join(composition(sk1, sk2), complement(composition(sk1, sk2)))))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by axiom 5 (def_top_12) R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(join(complement(sk2), composition(converse(sk1), top))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by lemma 67 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(join(complement(sk2), composition(composition(converse(sk1), sk1), top))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by lemma 56 }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, complement(composition(composition(converse(sk1), sk1), top))), meet(complement(sk2), complement(composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by lemma 33 }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, complement(composition(composition(converse(sk1), sk1), top))), meet(complement(composition(composition(converse(sk1), sk1), sk2)), complement(sk2)))))))
% 20.76/3.06 = { by lemma 55 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, complement(composition(composition(converse(sk1), sk1), top))), complement(join(sk2, complement(complement(composition(composition(converse(sk1), sk1), sk2))))))))))
% 20.76/3.06 = { by lemma 32 }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, complement(composition(composition(converse(sk1), sk1), top))), complement(join(sk2, meet(composition(composition(converse(sk1), sk1), sk2), composition(composition(converse(sk1), sk1), sk2)))))))))
% 20.76/3.06 = { by lemma 49 }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, complement(composition(composition(converse(sk1), sk1), top))), complement(join(sk2, composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by lemma 55 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(join(composition(composition(converse(sk1), sk1), top), complement(sk2))), complement(join(sk2, composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by lemma 52 }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(join(composition(composition(converse(sk1), sk1), top), complement(sk2)), join(sk2, composition(composition(converse(sk1), sk1), sk2))))))))
% 20.76/3.06 = { by lemma 33 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(join(sk2, composition(composition(converse(sk1), sk1), sk2)), join(composition(composition(converse(sk1), sk1), top), complement(sk2))))))))
% 20.76/3.06 = { by lemma 45 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(composition(join(composition(converse(sk1), sk1), join(composition(converse(sk1), sk1), one)), sk2), join(composition(composition(converse(sk1), sk1), top), complement(sk2))))))))
% 20.76/3.06 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(composition(join(join(composition(converse(sk1), sk1), one), composition(converse(sk1), sk1)), sk2), join(composition(composition(converse(sk1), sk1), top), complement(sk2))))))))
% 20.76/3.06 = { by lemma 46 }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(composition(join(composition(converse(sk1), sk1), one), sk2), join(composition(composition(converse(sk1), sk1), top), complement(sk2))))))))
% 20.76/3.06 = { by lemma 19 }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(sk2, join(composition(composition(converse(sk1), sk1), top), complement(sk2))))))))
% 20.76/3.06 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))))))))
% 20.76/3.06 = { by lemma 60 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), join(composition(composition(converse(sk1), sk1), top), meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))))))))))
% 20.76/3.06 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), join(composition(composition(converse(sk1), sk1), top), meet(sk2, join(composition(composition(converse(sk1), sk1), top), complement(sk2))))))))))
% 20.76/3.06 = { by lemma 57 R->L }
% 20.76/3.06 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), join(composition(composition(converse(sk1), sk1), top), meet(sk2, complement(meet(complement(composition(composition(converse(sk1), sk1), top)), sk2))))))))))
% 20.76/3.06 = { by lemma 53 R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), join(composition(composition(converse(sk1), sk1), top), meet(sk2, join(complement(sk2), complement(complement(composition(composition(converse(sk1), sk1), top))))))))))))
% 20.76/3.07 = { by lemma 33 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), join(composition(composition(converse(sk1), sk1), top), meet(join(complement(sk2), complement(complement(composition(composition(converse(sk1), sk1), top)))), sk2))))))))
% 20.76/3.07 = { by lemma 63 R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), join(composition(composition(converse(sk1), sk1), top), complement(join(complement(sk2), complement(join(complement(sk2), complement(complement(composition(composition(converse(sk1), sk1), top))))))))))))))
% 20.76/3.07 = { by lemma 63 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), join(composition(composition(converse(sk1), sk1), top), complement(join(complement(sk2), meet(complement(composition(composition(converse(sk1), sk1), top)), sk2))))))))))
% 20.76/3.07 = { by lemma 57 R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), complement(meet(complement(composition(composition(converse(sk1), sk1), top)), join(complement(sk2), meet(complement(composition(composition(converse(sk1), sk1), top)), sk2))))))))))
% 20.76/3.07 = { by lemma 42 R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), complement(join(meet(complement(composition(composition(converse(sk1), sk1), top)), join(complement(sk2), meet(complement(composition(composition(converse(sk1), sk1), top)), sk2))), zero))))))))
% 20.76/3.07 = { by lemma 15 R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), complement(join(meet(complement(composition(composition(converse(sk1), sk1), top)), join(complement(sk2), meet(complement(composition(composition(converse(sk1), sk1), top)), sk2))), complement(top)))))))))
% 20.76/3.07 = { by axiom 5 (def_top_12) }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), complement(join(meet(complement(composition(composition(converse(sk1), sk1), top)), join(complement(sk2), meet(complement(composition(composition(converse(sk1), sk1), top)), sk2))), complement(join(join(complement(complement(composition(composition(converse(sk1), sk1), top))), complement(sk2)), complement(join(complement(complement(composition(composition(converse(sk1), sk1), top))), complement(sk2)))))))))))))
% 20.76/3.07 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), complement(join(meet(complement(composition(composition(converse(sk1), sk1), top)), join(complement(sk2), meet(complement(composition(composition(converse(sk1), sk1), top)), sk2))), complement(join(join(complement(complement(composition(composition(converse(sk1), sk1), top))), complement(sk2)), meet(complement(composition(composition(converse(sk1), sk1), top)), sk2)))))))))))
% 20.76/3.07 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), complement(join(meet(complement(composition(composition(converse(sk1), sk1), top)), join(complement(sk2), meet(complement(composition(composition(converse(sk1), sk1), top)), sk2))), complement(join(complement(complement(composition(composition(converse(sk1), sk1), top))), join(complement(sk2), meet(complement(composition(composition(converse(sk1), sk1), top)), sk2))))))))))))
% 20.76/3.07 = { by lemma 30 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), complement(complement(composition(composition(converse(sk1), sk1), top)))))))))
% 20.76/3.07 = { by lemma 41 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top))), composition(composition(converse(sk1), sk1), top)))))))
% 20.76/3.07 = { by lemma 33 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(composition(composition(converse(sk1), sk1), top), meet(sk2, join(complement(sk2), composition(composition(converse(sk1), sk1), top)))))))))
% 20.76/3.07 = { by lemma 61 R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(sk2, meet(composition(composition(converse(sk1), sk1), top), join(complement(sk2), composition(composition(converse(sk1), sk1), top)))))))))
% 20.76/3.07 = { by lemma 60 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(sk2, composition(composition(converse(sk1), sk1), top)))))))
% 20.76/3.07 = { by lemma 33 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(composition(composition(converse(sk1), sk1), top), sk2))))))
% 20.76/3.07 = { by lemma 67 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(meet(composition(converse(sk1), top), sk2))))))
% 20.76/3.07 = { by lemma 53 R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(sk2), complement(composition(converse(sk1), top)))))))
% 20.76/3.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, join(complement(composition(converse(sk1), top)), complement(sk2))))))
% 20.76/3.07 = { by lemma 68 R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), join(composition(sk1, complement(composition(converse(sk1), top))), composition(sk1, complement(sk2))))))
% 20.76/3.07 = { by axiom 1 (converse_idempotence_8) R->L }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), join(composition(converse(converse(sk1)), complement(composition(converse(sk1), top))), composition(sk1, complement(sk2))))))
% 20.76/3.07 = { by lemma 65 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), join(zero, composition(sk1, complement(sk2))))))
% 20.76/3.07 = { by lemma 43 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(complement(composition(sk1, sk2)), composition(sk1, complement(sk2)))))
% 20.76/3.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, sk2)))))
% 20.76/3.07 = { by lemma 55 }
% 20.76/3.07 meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), meet(composition(sk1, sk2), complement(composition(sk1, complement(sk2)))))
% 20.76/3.07 = { by lemma 61 }
% 20.76/3.07 meet(composition(sk1, sk2), meet(meet(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(composition(sk1, complement(sk2)))))
% 20.76/3.07 = { by lemma 55 R->L }
% 20.76/3.07 meet(composition(sk1, sk2), complement(join(composition(sk1, complement(sk2)), complement(meet(composition(sk1, sk2), composition(sk1, complement(sk2)))))))
% 20.76/3.07 = { by lemma 33 R->L }
% 20.76/3.07 meet(composition(sk1, sk2), complement(join(composition(sk1, complement(sk2)), complement(meet(composition(sk1, complement(sk2)), composition(sk1, sk2))))))
% 20.76/3.07 = { by lemma 58 }
% 20.76/3.07 meet(composition(sk1, sk2), complement(top))
% 20.76/3.07 = { by lemma 15 }
% 20.76/3.07 meet(composition(sk1, sk2), zero)
% 20.76/3.07 = { by lemma 37 }
% 20.76/3.07 zero
% 20.76/3.07 % SZS output end Proof
% 20.76/3.07
% 20.76/3.07 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------