TSTP Solution File: REL035-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL035-2 : 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 : n019.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 13:44:21 EDT 2023
% Result : Unsatisfiable 37.44s 5.15s
% Output : Proof 38.16s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL035-2 : 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 : n019.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 22:00:58 EDT 2023
% 0.12/0.34 % CPUTime :
% 37.44/5.15 Command-line arguments: --flatten
% 37.44/5.15
% 37.44/5.15 % SZS status Unsatisfiable
% 37.44/5.15
% 38.16/5.25 % SZS output start Proof
% 38.16/5.25 Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 38.16/5.25 Axiom 2 (converse_idempotence_8): converse(converse(X)) = X.
% 38.16/5.25 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 38.16/5.25 Axiom 4 (goals_17): composition(sk1, top) = sk1.
% 38.16/5.25 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 38.16/5.25 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 38.16/5.25 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 38.16/5.25 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 38.16/5.25 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 38.16/5.25 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 38.16/5.25 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 38.16/5.25 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 38.16/5.25 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 38.16/5.25 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 38.16/5.25 Axiom 15 (modular_law_1_15): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 38.16/5.25 Axiom 16 (modular_law_2_16): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 38.16/5.25
% 38.16/5.25 Lemma 17: complement(top) = zero.
% 38.16/5.25 Proof:
% 38.16/5.25 complement(top)
% 38.16/5.25 = { by axiom 5 (def_top_12) }
% 38.16/5.25 complement(join(complement(X), complement(complement(X))))
% 38.16/5.25 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.25 meet(X, complement(X))
% 38.16/5.25 = { by axiom 6 (def_zero_13) R->L }
% 38.16/5.25 zero
% 38.16/5.25
% 38.16/5.25 Lemma 18: join(X, join(Y, complement(X))) = join(Y, top).
% 38.16/5.25 Proof:
% 38.16/5.25 join(X, join(Y, complement(X)))
% 38.16/5.25 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.25 join(X, join(complement(X), Y))
% 38.16/5.25 = { by axiom 8 (maddux2_join_associativity_2) }
% 38.16/5.25 join(join(X, complement(X)), Y)
% 38.16/5.25 = { by axiom 5 (def_top_12) R->L }
% 38.16/5.25 join(top, Y)
% 38.16/5.25 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.25 join(Y, top)
% 38.16/5.25
% 38.16/5.25 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 38.16/5.25 Proof:
% 38.16/5.25 converse(composition(converse(X), Y))
% 38.16/5.25 = { by axiom 9 (converse_multiplicativity_10) }
% 38.16/5.25 composition(converse(Y), converse(converse(X)))
% 38.16/5.25 = { by axiom 2 (converse_idempotence_8) }
% 38.16/5.25 composition(converse(Y), X)
% 38.16/5.25
% 38.16/5.25 Lemma 20: composition(converse(one), X) = X.
% 38.16/5.25 Proof:
% 38.16/5.25 composition(converse(one), X)
% 38.16/5.25 = { by lemma 19 R->L }
% 38.16/5.25 converse(composition(converse(X), one))
% 38.16/5.25 = { by axiom 3 (composition_identity_6) }
% 38.16/5.25 converse(converse(X))
% 38.16/5.25 = { by axiom 2 (converse_idempotence_8) }
% 38.16/5.25 X
% 38.16/5.25
% 38.16/5.25 Lemma 21: composition(one, X) = X.
% 38.16/5.25 Proof:
% 38.16/5.25 composition(one, X)
% 38.16/5.25 = { by lemma 20 R->L }
% 38.16/5.25 composition(converse(one), composition(one, X))
% 38.16/5.25 = { by axiom 10 (composition_associativity_5) }
% 38.16/5.25 composition(composition(converse(one), one), X)
% 38.16/5.25 = { by axiom 3 (composition_identity_6) }
% 38.16/5.25 composition(converse(one), X)
% 38.16/5.25 = { by lemma 20 }
% 38.16/5.25 X
% 38.16/5.25
% 38.16/5.25 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 38.16/5.25 Proof:
% 38.16/5.25 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 38.16/5.25 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.25 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 38.16/5.25 = { by axiom 13 (converse_cancellativity_11) }
% 38.16/5.25 complement(X)
% 38.16/5.25
% 38.16/5.25 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 38.16/5.25 Proof:
% 38.16/5.25 join(complement(X), complement(X))
% 38.16/5.25 = { by lemma 20 R->L }
% 38.16/5.25 join(complement(X), composition(converse(one), complement(X)))
% 38.16/5.25 = { by lemma 21 R->L }
% 38.16/5.25 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 38.16/5.25 = { by lemma 22 }
% 38.16/5.25 complement(X)
% 38.16/5.25
% 38.16/5.25 Lemma 24: join(top, complement(X)) = top.
% 38.16/5.25 Proof:
% 38.16/5.25 join(top, complement(X))
% 38.16/5.25 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.25 join(complement(X), top)
% 38.16/5.25 = { by lemma 18 R->L }
% 38.16/5.25 join(X, join(complement(X), complement(X)))
% 38.16/5.25 = { by lemma 23 }
% 38.16/5.25 join(X, complement(X))
% 38.16/5.25 = { by axiom 5 (def_top_12) R->L }
% 38.16/5.25 top
% 38.16/5.25
% 38.16/5.25 Lemma 25: join(Y, top) = join(X, top).
% 38.16/5.25 Proof:
% 38.16/5.25 join(Y, top)
% 38.16/5.25 = { by lemma 24 R->L }
% 38.16/5.25 join(Y, join(top, complement(Y)))
% 38.16/5.25 = { by lemma 18 }
% 38.16/5.25 join(top, top)
% 38.16/5.25 = { by lemma 18 R->L }
% 38.16/5.25 join(X, join(top, complement(X)))
% 38.16/5.25 = { by lemma 24 }
% 38.16/5.25 join(X, top)
% 38.16/5.25
% 38.16/5.25 Lemma 26: join(X, top) = top.
% 38.16/5.25 Proof:
% 38.16/5.25 join(X, top)
% 38.16/5.25 = { by lemma 25 }
% 38.16/5.25 join(zero, top)
% 38.16/5.25 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.25 join(top, zero)
% 38.16/5.25 = { by lemma 17 R->L }
% 38.16/5.25 join(top, complement(top))
% 38.16/5.25 = { by axiom 5 (def_top_12) R->L }
% 38.16/5.25 top
% 38.16/5.25
% 38.16/5.25 Lemma 27: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 38.16/5.25 Proof:
% 38.16/5.25 converse(join(X, converse(Y)))
% 38.16/5.25 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.25 converse(join(converse(Y), X))
% 38.16/5.25 = { by axiom 7 (converse_additivity_9) }
% 38.16/5.25 join(converse(converse(Y)), converse(X))
% 38.16/5.25 = { by axiom 2 (converse_idempotence_8) }
% 38.16/5.25 join(Y, converse(X))
% 38.16/5.25
% 38.16/5.25 Lemma 28: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 38.16/5.25 Proof:
% 38.16/5.25 converse(join(converse(X), Y))
% 38.16/5.25 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.25 converse(join(Y, converse(X)))
% 38.16/5.25 = { by lemma 27 }
% 38.16/5.25 join(X, converse(Y))
% 38.16/5.25
% 38.16/5.25 Lemma 29: join(X, converse(complement(converse(X)))) = converse(top).
% 38.16/5.25 Proof:
% 38.16/5.25 join(X, converse(complement(converse(X))))
% 38.16/5.25 = { by lemma 28 R->L }
% 38.16/5.25 converse(join(converse(X), complement(converse(X))))
% 38.16/5.25 = { by axiom 5 (def_top_12) R->L }
% 38.16/5.25 converse(top)
% 38.16/5.25
% 38.16/5.25 Lemma 30: join(X, join(complement(X), Y)) = top.
% 38.16/5.25 Proof:
% 38.16/5.25 join(X, join(complement(X), Y))
% 38.16/5.25 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.25 join(X, join(Y, complement(X)))
% 38.16/5.25 = { by lemma 18 }
% 38.16/5.25 join(Y, top)
% 38.16/5.25 = { by lemma 25 R->L }
% 38.16/5.25 join(Z, top)
% 38.16/5.25 = { by lemma 26 }
% 38.16/5.25 top
% 38.16/5.25
% 38.16/5.25 Lemma 31: join(X, converse(top)) = top.
% 38.16/5.25 Proof:
% 38.16/5.25 join(X, converse(top))
% 38.16/5.25 = { by lemma 29 R->L }
% 38.16/5.25 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 38.16/5.25 = { by lemma 30 }
% 38.16/5.25 top
% 38.16/5.26
% 38.16/5.26 Lemma 32: converse(top) = top.
% 38.16/5.26 Proof:
% 38.16/5.26 converse(top)
% 38.16/5.26 = { by lemma 26 R->L }
% 38.16/5.26 converse(join(X, top))
% 38.16/5.26 = { by axiom 7 (converse_additivity_9) }
% 38.16/5.26 join(converse(X), converse(top))
% 38.16/5.26 = { by lemma 31 }
% 38.16/5.26 top
% 38.16/5.26
% 38.16/5.26 Lemma 33: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 38.16/5.26 Proof:
% 38.16/5.26 join(meet(X, Y), complement(join(complement(X), Y)))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 38.16/5.26 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 38.16/5.26 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 38.16/5.26 X
% 38.16/5.26
% 38.16/5.26 Lemma 34: join(zero, meet(X, X)) = X.
% 38.16/5.26 Proof:
% 38.16/5.26 join(zero, meet(X, X))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 38.16/5.26 join(zero, complement(join(complement(X), complement(X))))
% 38.16/5.26 = { by axiom 6 (def_zero_13) }
% 38.16/5.26 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 38.16/5.26 = { by lemma 33 }
% 38.16/5.26 X
% 38.16/5.26
% 38.16/5.26 Lemma 35: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 38.16/5.26 Proof:
% 38.16/5.26 join(zero, join(X, complement(complement(Y))))
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.26 join(zero, join(complement(complement(Y)), X))
% 38.16/5.26 = { by lemma 23 R->L }
% 38.16/5.26 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.26 join(zero, join(meet(Y, Y), X))
% 38.16/5.26 = { by axiom 8 (maddux2_join_associativity_2) }
% 38.16/5.26 join(join(zero, meet(Y, Y)), X)
% 38.16/5.26 = { by lemma 34 }
% 38.16/5.26 join(Y, X)
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.26 join(X, Y)
% 38.16/5.26
% 38.16/5.26 Lemma 36: join(zero, complement(complement(X))) = X.
% 38.16/5.26 Proof:
% 38.16/5.26 join(zero, complement(complement(X)))
% 38.16/5.26 = { by axiom 6 (def_zero_13) }
% 38.16/5.26 join(meet(X, complement(X)), complement(complement(X)))
% 38.16/5.26 = { by lemma 23 R->L }
% 38.16/5.26 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 38.16/5.26 = { by lemma 33 }
% 38.16/5.26 X
% 38.16/5.26
% 38.16/5.26 Lemma 37: join(X, zero) = join(X, X).
% 38.16/5.26 Proof:
% 38.16/5.26 join(X, zero)
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.26 join(zero, X)
% 38.16/5.26 = { by lemma 36 R->L }
% 38.16/5.26 join(zero, join(zero, complement(complement(X))))
% 38.16/5.26 = { by lemma 23 R->L }
% 38.16/5.26 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 38.16/5.26 = { by lemma 35 }
% 38.16/5.26 join(zero, join(complement(complement(X)), X))
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.26 join(zero, join(X, complement(complement(X))))
% 38.16/5.26 = { by lemma 35 }
% 38.16/5.26 join(X, X)
% 38.16/5.26
% 38.16/5.26 Lemma 38: join(zero, complement(X)) = complement(X).
% 38.16/5.26 Proof:
% 38.16/5.26 join(zero, complement(X))
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.26 join(complement(X), zero)
% 38.16/5.26 = { by lemma 37 }
% 38.16/5.26 join(complement(X), complement(X))
% 38.16/5.26 = { by lemma 23 }
% 38.16/5.26 complement(X)
% 38.16/5.26
% 38.16/5.26 Lemma 39: join(X, zero) = X.
% 38.16/5.26 Proof:
% 38.16/5.26 join(X, zero)
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.26 join(zero, X)
% 38.16/5.26 = { by lemma 35 R->L }
% 38.16/5.26 join(zero, join(zero, complement(complement(X))))
% 38.16/5.26 = { by lemma 38 }
% 38.16/5.26 join(zero, complement(complement(X)))
% 38.16/5.26 = { by lemma 36 }
% 38.16/5.26 X
% 38.16/5.26
% 38.16/5.26 Lemma 40: join(zero, X) = X.
% 38.16/5.26 Proof:
% 38.16/5.26 join(zero, X)
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.26 join(X, zero)
% 38.16/5.26 = { by lemma 39 }
% 38.16/5.26 X
% 38.16/5.26
% 38.16/5.26 Lemma 41: meet(Y, X) = meet(X, Y).
% 38.16/5.26 Proof:
% 38.16/5.26 meet(Y, X)
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 38.16/5.26 complement(join(complement(Y), complement(X)))
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.26 complement(join(complement(X), complement(Y)))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.26 meet(X, Y)
% 38.16/5.26
% 38.16/5.26 Lemma 42: complement(join(zero, complement(X))) = meet(X, top).
% 38.16/5.26 Proof:
% 38.16/5.26 complement(join(zero, complement(X)))
% 38.16/5.26 = { by lemma 17 R->L }
% 38.16/5.26 complement(join(complement(top), complement(X)))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.26 meet(top, X)
% 38.16/5.26 = { by lemma 41 R->L }
% 38.16/5.26 meet(X, top)
% 38.16/5.26
% 38.16/5.26 Lemma 43: join(X, complement(zero)) = top.
% 38.16/5.26 Proof:
% 38.16/5.26 join(X, complement(zero))
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.26 join(complement(zero), X)
% 38.16/5.26 = { by lemma 35 R->L }
% 38.16/5.26 join(zero, join(complement(zero), complement(complement(X))))
% 38.16/5.26 = { by lemma 30 }
% 38.16/5.26 top
% 38.16/5.26
% 38.16/5.26 Lemma 44: meet(X, zero) = zero.
% 38.16/5.26 Proof:
% 38.16/5.26 meet(X, zero)
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 38.16/5.26 complement(join(complement(X), complement(zero)))
% 38.16/5.26 = { by lemma 43 }
% 38.16/5.26 complement(top)
% 38.16/5.26 = { by lemma 17 }
% 38.16/5.26 zero
% 38.16/5.26
% 38.16/5.26 Lemma 45: join(meet(X, Y), meet(X, complement(Y))) = X.
% 38.16/5.26 Proof:
% 38.16/5.26 join(meet(X, Y), meet(X, complement(Y)))
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.26 join(meet(X, complement(Y)), meet(X, Y))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 38.16/5.26 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 38.16/5.26 = { by lemma 33 }
% 38.16/5.26 X
% 38.16/5.26
% 38.16/5.26 Lemma 46: meet(X, top) = X.
% 38.16/5.26 Proof:
% 38.16/5.26 meet(X, top)
% 38.16/5.26 = { by lemma 42 R->L }
% 38.16/5.26 complement(join(zero, complement(X)))
% 38.16/5.26 = { by lemma 38 R->L }
% 38.16/5.26 join(zero, complement(join(zero, complement(X))))
% 38.16/5.26 = { by lemma 42 }
% 38.16/5.26 join(zero, meet(X, top))
% 38.16/5.26 = { by lemma 43 R->L }
% 38.16/5.26 join(zero, meet(X, join(complement(zero), complement(zero))))
% 38.16/5.26 = { by lemma 23 }
% 38.16/5.26 join(zero, meet(X, complement(zero)))
% 38.16/5.26 = { by lemma 44 R->L }
% 38.16/5.26 join(meet(X, zero), meet(X, complement(zero)))
% 38.16/5.26 = { by lemma 45 }
% 38.16/5.26 X
% 38.16/5.26
% 38.16/5.26 Lemma 47: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 38.16/5.26 Proof:
% 38.16/5.26 join(meet(X, Y), meet(X, Y))
% 38.16/5.26 = { by lemma 41 }
% 38.16/5.26 join(meet(Y, X), meet(X, Y))
% 38.16/5.26 = { by lemma 41 }
% 38.16/5.26 join(meet(Y, X), meet(Y, X))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 38.16/5.26 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 38.16/5.26 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 38.16/5.26 = { by lemma 23 }
% 38.16/5.26 complement(join(complement(Y), complement(X)))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.26 meet(Y, X)
% 38.16/5.26 = { by lemma 41 R->L }
% 38.16/5.26 meet(X, Y)
% 38.16/5.26
% 38.16/5.26 Lemma 48: converse(zero) = zero.
% 38.16/5.26 Proof:
% 38.16/5.26 converse(zero)
% 38.16/5.26 = { by lemma 40 R->L }
% 38.16/5.26 join(zero, converse(zero))
% 38.16/5.26 = { by lemma 28 R->L }
% 38.16/5.26 converse(join(converse(zero), zero))
% 38.16/5.26 = { by lemma 37 }
% 38.16/5.26 converse(join(converse(zero), converse(zero)))
% 38.16/5.26 = { by lemma 27 }
% 38.16/5.26 join(zero, converse(converse(zero)))
% 38.16/5.26 = { by axiom 2 (converse_idempotence_8) }
% 38.16/5.26 join(zero, zero)
% 38.16/5.26 = { by lemma 46 R->L }
% 38.16/5.26 join(zero, meet(zero, top))
% 38.16/5.26 = { by lemma 46 R->L }
% 38.16/5.26 join(meet(zero, top), meet(zero, top))
% 38.16/5.26 = { by lemma 47 }
% 38.16/5.26 meet(zero, top)
% 38.16/5.26 = { by lemma 46 }
% 38.16/5.26 zero
% 38.16/5.26
% 38.16/5.26 Lemma 49: complement(complement(X)) = X.
% 38.16/5.26 Proof:
% 38.16/5.26 complement(complement(X))
% 38.16/5.26 = { by lemma 38 R->L }
% 38.16/5.26 join(zero, complement(complement(X)))
% 38.16/5.26 = { by lemma 36 }
% 38.16/5.26 X
% 38.16/5.26
% 38.16/5.26 Lemma 50: meet(zero, X) = zero.
% 38.16/5.26 Proof:
% 38.16/5.26 meet(zero, X)
% 38.16/5.26 = { by lemma 41 }
% 38.16/5.26 meet(X, zero)
% 38.16/5.26 = { by lemma 44 }
% 38.16/5.26 zero
% 38.16/5.26
% 38.16/5.26 Lemma 51: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 38.16/5.26 Proof:
% 38.16/5.26 meet(X, join(complement(Y), complement(Z)))
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.26 meet(X, join(complement(Z), complement(Y)))
% 38.16/5.26 = { by lemma 41 }
% 38.16/5.26 meet(join(complement(Z), complement(Y)), X)
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 38.16/5.26 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 38.16/5.26 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.26 complement(join(meet(Z, Y), complement(X)))
% 38.16/5.26 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.26 complement(join(complement(X), meet(Z, Y)))
% 38.16/5.26 = { by lemma 41 R->L }
% 38.16/5.27 complement(join(complement(X), meet(Y, Z)))
% 38.16/5.27
% 38.16/5.27 Lemma 52: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 38.16/5.27 Proof:
% 38.16/5.27 complement(join(X, complement(Y)))
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.27 complement(join(complement(Y), X))
% 38.16/5.27 = { by lemma 46 R->L }
% 38.16/5.27 complement(join(complement(Y), meet(X, top)))
% 38.16/5.27 = { by lemma 41 R->L }
% 38.16/5.27 complement(join(complement(Y), meet(top, X)))
% 38.16/5.27 = { by lemma 51 R->L }
% 38.16/5.27 meet(Y, join(complement(top), complement(X)))
% 38.16/5.27 = { by lemma 17 }
% 38.16/5.27 meet(Y, join(zero, complement(X)))
% 38.16/5.27 = { by lemma 38 }
% 38.16/5.27 meet(Y, complement(X))
% 38.16/5.27
% 38.16/5.27 Lemma 53: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 38.16/5.27 Proof:
% 38.16/5.27 complement(meet(X, complement(Y)))
% 38.16/5.27 = { by lemma 40 R->L }
% 38.16/5.27 complement(join(zero, meet(X, complement(Y))))
% 38.16/5.27 = { by lemma 52 R->L }
% 38.16/5.27 complement(join(zero, complement(join(Y, complement(X)))))
% 38.16/5.27 = { by lemma 42 }
% 38.16/5.27 meet(join(Y, complement(X)), top)
% 38.16/5.27 = { by lemma 46 }
% 38.16/5.27 join(Y, complement(X))
% 38.16/5.27
% 38.16/5.27 Lemma 54: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 38.16/5.27 Proof:
% 38.16/5.27 complement(join(complement(X), Y))
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.27 complement(join(Y, complement(X)))
% 38.16/5.27 = { by lemma 52 }
% 38.16/5.27 meet(X, complement(Y))
% 38.16/5.27
% 38.16/5.27 Lemma 55: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 38.16/5.27 Proof:
% 38.16/5.27 join(complement(X), complement(Y))
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.27 join(complement(Y), complement(X))
% 38.16/5.27 = { by lemma 34 R->L }
% 38.16/5.27 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 38.16/5.27 = { by lemma 51 }
% 38.16/5.27 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 38.16/5.27 = { by lemma 38 }
% 38.16/5.27 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 38.16/5.27 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.27 complement(join(meet(Y, X), meet(Y, X)))
% 38.16/5.27 = { by lemma 47 }
% 38.16/5.27 complement(meet(Y, X))
% 38.16/5.27 = { by lemma 41 R->L }
% 38.16/5.27 complement(meet(X, Y))
% 38.16/5.27
% 38.16/5.27 Lemma 56: join(X, complement(meet(X, Y))) = top.
% 38.16/5.27 Proof:
% 38.16/5.27 join(X, complement(meet(X, Y)))
% 38.16/5.27 = { by lemma 41 }
% 38.16/5.27 join(X, complement(meet(Y, X)))
% 38.16/5.27 = { by lemma 55 R->L }
% 38.16/5.27 join(X, join(complement(Y), complement(X)))
% 38.16/5.27 = { by lemma 18 }
% 38.16/5.27 join(complement(Y), top)
% 38.16/5.27 = { by lemma 26 }
% 38.16/5.27 top
% 38.16/5.27
% 38.16/5.27 Lemma 57: meet(X, meet(Y, complement(X))) = zero.
% 38.16/5.27 Proof:
% 38.16/5.27 meet(X, meet(Y, complement(X)))
% 38.16/5.27 = { by lemma 41 }
% 38.16/5.27 meet(X, meet(complement(X), Y))
% 38.16/5.27 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 38.16/5.27 complement(join(complement(X), complement(meet(complement(X), Y))))
% 38.16/5.27 = { by lemma 56 }
% 38.16/5.27 complement(top)
% 38.16/5.27 = { by lemma 17 }
% 38.16/5.27 zero
% 38.16/5.27
% 38.16/5.27 Lemma 58: meet(X, join(X, complement(Y))) = X.
% 38.16/5.27 Proof:
% 38.16/5.27 meet(X, join(X, complement(Y)))
% 38.16/5.27 = { by lemma 53 R->L }
% 38.16/5.27 meet(X, complement(meet(Y, complement(X))))
% 38.16/5.27 = { by lemma 54 R->L }
% 38.16/5.27 complement(join(complement(X), meet(Y, complement(X))))
% 38.16/5.27 = { by lemma 38 R->L }
% 38.16/5.27 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 38.16/5.27 = { by lemma 57 R->L }
% 38.16/5.27 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 38.16/5.27 = { by lemma 33 }
% 38.16/5.27 X
% 38.16/5.27
% 38.16/5.27 Lemma 59: meet(X, join(X, Y)) = X.
% 38.16/5.27 Proof:
% 38.16/5.27 meet(X, join(X, Y))
% 38.16/5.27 = { by lemma 46 R->L }
% 38.16/5.27 meet(X, join(X, meet(Y, top)))
% 38.16/5.27 = { by lemma 42 R->L }
% 38.16/5.27 meet(X, join(X, complement(join(zero, complement(Y)))))
% 38.16/5.27 = { by lemma 58 }
% 38.16/5.27 X
% 38.16/5.27
% 38.16/5.27 Lemma 60: join(zero, composition(converse(sk1), complement(composition(sk1, top)))) = zero.
% 38.16/5.27 Proof:
% 38.16/5.27 join(zero, composition(converse(sk1), complement(composition(sk1, top))))
% 38.16/5.27 = { by axiom 4 (goals_17) R->L }
% 38.16/5.27 join(zero, composition(converse(composition(sk1, top)), complement(composition(sk1, top))))
% 38.16/5.27 = { by lemma 17 R->L }
% 38.16/5.27 join(complement(top), composition(converse(composition(sk1, top)), complement(composition(sk1, top))))
% 38.16/5.27 = { by axiom 4 (goals_17) R->L }
% 38.16/5.27 join(complement(top), composition(converse(composition(sk1, top)), complement(composition(composition(sk1, top), top))))
% 38.16/5.27 = { by lemma 22 }
% 38.16/5.27 complement(top)
% 38.16/5.27 = { by lemma 17 }
% 38.16/5.27 zero
% 38.16/5.27
% 38.16/5.27 Lemma 61: composition(converse(sk1), complement(composition(sk1, top))) = zero.
% 38.16/5.27 Proof:
% 38.16/5.27 composition(converse(sk1), complement(composition(sk1, top)))
% 38.16/5.27 = { by lemma 59 R->L }
% 38.16/5.27 meet(composition(converse(sk1), complement(composition(sk1, top))), join(composition(converse(sk1), complement(composition(sk1, top))), zero))
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.27 meet(composition(converse(sk1), complement(composition(sk1, top))), join(zero, composition(converse(sk1), complement(composition(sk1, top)))))
% 38.16/5.27 = { by lemma 60 }
% 38.16/5.27 meet(composition(converse(sk1), complement(composition(sk1, top))), zero)
% 38.16/5.27 = { by lemma 44 }
% 38.16/5.27 zero
% 38.16/5.27
% 38.16/5.27 Lemma 62: composition(top, zero) = zero.
% 38.16/5.27 Proof:
% 38.16/5.27 composition(top, zero)
% 38.16/5.27 = { by lemma 61 R->L }
% 38.16/5.27 composition(top, composition(converse(sk1), complement(composition(sk1, top))))
% 38.16/5.27 = { by lemma 32 R->L }
% 38.16/5.27 composition(converse(top), composition(converse(sk1), complement(composition(sk1, top))))
% 38.16/5.27 = { by axiom 10 (composition_associativity_5) }
% 38.16/5.27 composition(composition(converse(top), converse(sk1)), complement(composition(sk1, top)))
% 38.16/5.27 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 38.16/5.27 composition(converse(composition(sk1, top)), complement(composition(sk1, top)))
% 38.16/5.27 = { by axiom 4 (goals_17) }
% 38.16/5.27 composition(converse(sk1), complement(composition(sk1, top)))
% 38.16/5.27 = { by lemma 61 }
% 38.16/5.27 zero
% 38.16/5.27
% 38.16/5.27 Lemma 63: composition(X, zero) = zero.
% 38.16/5.27 Proof:
% 38.16/5.27 composition(X, zero)
% 38.16/5.27 = { by lemma 40 R->L }
% 38.16/5.27 join(zero, composition(X, zero))
% 38.16/5.27 = { by lemma 62 R->L }
% 38.16/5.27 join(composition(top, zero), composition(X, zero))
% 38.16/5.27 = { by axiom 12 (composition_distributivity_7) R->L }
% 38.16/5.27 composition(join(top, X), zero)
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.27 composition(join(X, top), zero)
% 38.16/5.27 = { by lemma 25 R->L }
% 38.16/5.27 composition(join(Y, top), zero)
% 38.16/5.27 = { by lemma 26 }
% 38.16/5.27 composition(top, zero)
% 38.16/5.27 = { by lemma 62 }
% 38.16/5.27 zero
% 38.16/5.27
% 38.16/5.27 Lemma 64: composition(zero, X) = zero.
% 38.16/5.27 Proof:
% 38.16/5.27 composition(zero, X)
% 38.16/5.27 = { by lemma 48 R->L }
% 38.16/5.27 composition(converse(zero), X)
% 38.16/5.27 = { by lemma 19 R->L }
% 38.16/5.27 converse(composition(converse(X), zero))
% 38.16/5.27 = { by lemma 63 }
% 38.16/5.27 converse(zero)
% 38.16/5.27 = { by lemma 48 }
% 38.16/5.27 zero
% 38.16/5.27
% 38.16/5.27 Lemma 65: meet(one, composition(converse(complement(X)), X)) = zero.
% 38.16/5.27 Proof:
% 38.16/5.27 meet(one, composition(converse(complement(X)), X))
% 38.16/5.27 = { by lemma 41 }
% 38.16/5.27 meet(composition(converse(complement(X)), X), one)
% 38.16/5.27 = { by lemma 49 R->L }
% 38.16/5.27 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 38.16/5.27 = { by lemma 22 R->L }
% 38.16/5.27 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), one))))))
% 38.16/5.27 = { by axiom 3 (composition_identity_6) }
% 38.16/5.27 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 38.16/5.27 = { by lemma 42 }
% 38.16/5.27 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 38.16/5.27 = { by lemma 38 }
% 38.16/5.27 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), meet(X, top)))))
% 38.16/5.27 = { by lemma 46 }
% 38.16/5.27 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 38.16/5.27 = { by lemma 54 }
% 38.16/5.27 meet(composition(converse(complement(X)), X), meet(one, complement(composition(converse(complement(X)), X))))
% 38.16/5.27 = { by lemma 57 }
% 38.16/5.27 zero
% 38.16/5.27
% 38.16/5.27 Lemma 66: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 38.16/5.27 Proof:
% 38.16/5.27 join(meet(X, Y), meet(Y, complement(X)))
% 38.16/5.27 = { by lemma 41 }
% 38.16/5.27 join(meet(Y, X), meet(Y, complement(X)))
% 38.16/5.27 = { by lemma 45 }
% 38.16/5.27 Y
% 38.16/5.27
% 38.16/5.27 Lemma 67: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 38.16/5.27 Proof:
% 38.16/5.27 join(meet(X, Y), meet(complement(X), Y))
% 38.16/5.27 = { by lemma 41 }
% 38.16/5.27 join(meet(X, Y), meet(Y, complement(X)))
% 38.16/5.27 = { by lemma 66 }
% 38.16/5.27 Y
% 38.16/5.27
% 38.16/5.27 Lemma 68: converse(complement(X)) = complement(converse(X)).
% 38.16/5.27 Proof:
% 38.16/5.27 converse(complement(X))
% 38.16/5.27 = { by lemma 38 R->L }
% 38.16/5.27 converse(join(zero, complement(X)))
% 38.16/5.27 = { by lemma 33 R->L }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), complement(converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.16/5.27 = { by lemma 54 R->L }
% 38.16/5.27 converse(join(complement(join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.16/5.27 = { by lemma 29 }
% 38.16/5.27 converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.16/5.27 = { by lemma 32 }
% 38.16/5.27 converse(join(complement(top), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.16/5.27 = { by lemma 17 }
% 38.16/5.27 converse(join(zero, complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 38.16/5.27 = { by lemma 38 }
% 38.16/5.27 converse(complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.16/5.27 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.27 converse(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))))
% 38.16/5.27 = { by lemma 39 R->L }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), zero))
% 38.16/5.27 = { by lemma 50 R->L }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(X)))))))))
% 38.16/5.27 = { by lemma 64 R->L }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 38.16/5.27 = { by lemma 65 R->L }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(X)))))), converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 38.16/5.27 = { by axiom 16 (modular_law_2_16) R->L }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(one, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(X)))))), converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.16/5.27 = { by lemma 65 }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(one, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.16/5.27 = { by lemma 21 }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.16/5.27 = { by lemma 64 }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(X))))))))))
% 38.16/5.27 = { by lemma 50 }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), zero)))
% 38.16/5.27 = { by lemma 39 }
% 38.16/5.27 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 38.16/5.27 = { by lemma 67 }
% 38.16/5.27 converse(converse(complement(converse(complement(join(zero, complement(X)))))))
% 38.16/5.27 = { by axiom 2 (converse_idempotence_8) }
% 38.16/5.27 complement(converse(complement(join(zero, complement(X)))))
% 38.16/5.27 = { by lemma 42 }
% 38.16/5.27 complement(converse(meet(X, top)))
% 38.16/5.27 = { by lemma 46 }
% 38.16/5.27 complement(converse(X))
% 38.16/5.27
% 38.16/5.27 Lemma 69: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 38.16/5.27 Proof:
% 38.16/5.27 complement(meet(complement(X), Y))
% 38.16/5.27 = { by lemma 41 }
% 38.16/5.27 complement(meet(Y, complement(X)))
% 38.16/5.27 = { by lemma 53 }
% 38.16/5.27 join(X, complement(Y))
% 38.16/5.27
% 38.16/5.27 Lemma 70: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 38.16/5.27 Proof:
% 38.16/5.27 meet(complement(X), complement(Y))
% 38.16/5.27 = { by lemma 41 }
% 38.16/5.27 meet(complement(Y), complement(X))
% 38.16/5.27 = { by lemma 38 R->L }
% 38.16/5.27 meet(join(zero, complement(Y)), complement(X))
% 38.16/5.27 = { by lemma 52 R->L }
% 38.16/5.27 complement(join(X, complement(join(zero, complement(Y)))))
% 38.16/5.27 = { by lemma 42 }
% 38.16/5.27 complement(join(X, meet(Y, top)))
% 38.16/5.27 = { by lemma 46 }
% 38.16/5.27 complement(join(X, Y))
% 38.16/5.27
% 38.16/5.27 Lemma 71: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 38.16/5.27 Proof:
% 38.16/5.27 converse(composition(X, converse(Y)))
% 38.16/5.27 = { by axiom 9 (converse_multiplicativity_10) }
% 38.16/5.27 composition(converse(converse(Y)), converse(X))
% 38.16/5.27 = { by axiom 2 (converse_idempotence_8) }
% 38.16/5.27 composition(Y, converse(X))
% 38.16/5.27
% 38.16/5.27 Lemma 72: join(complement(X), meet(Y, complement(Z))) = complement(meet(X, join(Z, complement(Y)))).
% 38.16/5.27 Proof:
% 38.16/5.27 join(complement(X), meet(Y, complement(Z)))
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.27 join(meet(Y, complement(Z)), complement(X))
% 38.16/5.27 = { by lemma 52 R->L }
% 38.16/5.27 join(complement(join(Z, complement(Y))), complement(X))
% 38.16/5.27 = { by lemma 55 }
% 38.16/5.27 complement(meet(join(Z, complement(Y)), X))
% 38.16/5.27 = { by lemma 41 R->L }
% 38.16/5.27 complement(meet(X, join(Z, complement(Y))))
% 38.16/5.27
% 38.16/5.27 Lemma 73: complement(meet(Y, join(X, complement(Y)))) = complement(meet(X, join(Y, complement(X)))).
% 38.16/5.27 Proof:
% 38.16/5.27 complement(meet(Y, join(X, complement(Y))))
% 38.16/5.27 = { by lemma 72 R->L }
% 38.16/5.27 join(complement(Y), meet(Y, complement(X)))
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.27 join(meet(Y, complement(X)), complement(Y))
% 38.16/5.27 = { by lemma 67 R->L }
% 38.16/5.27 join(meet(Y, complement(X)), join(meet(X, complement(Y)), meet(complement(X), complement(Y))))
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.27 join(meet(Y, complement(X)), join(meet(complement(X), complement(Y)), meet(X, complement(Y))))
% 38.16/5.27 = { by axiom 8 (maddux2_join_associativity_2) }
% 38.16/5.27 join(join(meet(Y, complement(X)), meet(complement(X), complement(Y))), meet(X, complement(Y)))
% 38.16/5.27 = { by lemma 66 }
% 38.16/5.27 join(complement(X), meet(X, complement(Y)))
% 38.16/5.27 = { by lemma 72 }
% 38.16/5.27 complement(meet(X, join(Y, complement(X))))
% 38.16/5.27
% 38.16/5.27 Lemma 74: join(complement(X), meet(Y, X)) = join(Y, complement(join(X, Y))).
% 38.16/5.27 Proof:
% 38.16/5.27 join(complement(X), meet(Y, X))
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.27 join(meet(Y, X), complement(X))
% 38.16/5.27 = { by lemma 53 R->L }
% 38.16/5.27 complement(meet(X, complement(meet(Y, X))))
% 38.16/5.27 = { by lemma 55 R->L }
% 38.16/5.27 complement(meet(X, join(complement(Y), complement(X))))
% 38.16/5.27 = { by lemma 73 }
% 38.16/5.27 complement(meet(complement(Y), join(X, complement(complement(Y)))))
% 38.16/5.27 = { by lemma 69 }
% 38.16/5.27 join(Y, complement(join(X, complement(complement(Y)))))
% 38.16/5.27 = { by lemma 52 }
% 38.16/5.27 join(Y, meet(complement(Y), complement(X)))
% 38.16/5.27 = { by lemma 70 }
% 38.16/5.27 join(Y, complement(join(Y, X)))
% 38.16/5.27 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.28 join(Y, complement(join(X, Y)))
% 38.16/5.28
% 38.16/5.28 Lemma 75: join(X, meet(Y, complement(X))) = join(X, Y).
% 38.16/5.28 Proof:
% 38.16/5.28 join(X, meet(Y, complement(X)))
% 38.16/5.28 = { by lemma 52 R->L }
% 38.16/5.28 join(X, complement(join(X, complement(Y))))
% 38.16/5.28 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.28 join(X, complement(join(complement(Y), X)))
% 38.16/5.28 = { by lemma 74 R->L }
% 38.16/5.28 join(complement(complement(Y)), meet(X, complement(Y)))
% 38.16/5.28 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.28 join(meet(X, complement(Y)), complement(complement(Y)))
% 38.16/5.28 = { by lemma 67 R->L }
% 38.16/5.28 join(meet(X, complement(Y)), complement(join(meet(X, complement(Y)), meet(complement(X), complement(Y)))))
% 38.16/5.28 = { by lemma 70 R->L }
% 38.16/5.28 join(meet(X, complement(Y)), meet(complement(meet(X, complement(Y))), complement(meet(complement(X), complement(Y)))))
% 38.16/5.28 = { by lemma 52 R->L }
% 38.16/5.28 join(meet(X, complement(Y)), complement(join(meet(complement(X), complement(Y)), complement(complement(meet(X, complement(Y)))))))
% 38.16/5.28 = { by lemma 69 R->L }
% 38.16/5.28 complement(meet(complement(meet(X, complement(Y))), join(meet(complement(X), complement(Y)), complement(complement(meet(X, complement(Y)))))))
% 38.16/5.28 = { by lemma 73 }
% 38.16/5.28 complement(meet(meet(complement(X), complement(Y)), join(complement(meet(X, complement(Y))), complement(meet(complement(X), complement(Y))))))
% 38.16/5.28 = { by lemma 55 }
% 38.16/5.28 complement(meet(meet(complement(X), complement(Y)), complement(meet(meet(X, complement(Y)), meet(complement(X), complement(Y))))))
% 38.16/5.28 = { by lemma 53 }
% 38.16/5.28 join(meet(meet(X, complement(Y)), meet(complement(X), complement(Y))), complement(meet(complement(X), complement(Y))))
% 38.16/5.28 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.28 join(complement(meet(complement(X), complement(Y))), meet(meet(X, complement(Y)), meet(complement(X), complement(Y))))
% 38.16/5.28 = { by lemma 41 R->L }
% 38.16/5.28 join(complement(meet(complement(X), complement(Y))), meet(meet(complement(X), complement(Y)), meet(X, complement(Y))))
% 38.16/5.28 = { by lemma 69 }
% 38.16/5.28 join(join(X, complement(complement(Y))), meet(meet(complement(X), complement(Y)), meet(X, complement(Y))))
% 38.16/5.28 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(meet(complement(X), complement(Y)), meet(X, complement(Y)))))
% 38.16/5.28 = { by lemma 41 }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(meet(X, complement(Y)), meet(complement(X), complement(Y)))))
% 38.16/5.28 = { by lemma 41 }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(meet(X, complement(Y)), meet(complement(Y), complement(X)))))
% 38.16/5.28 = { by lemma 46 R->L }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(meet(X, complement(Y)), meet(complement(Y), meet(complement(X), top)))))
% 38.16/5.28 = { by lemma 42 R->L }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(meet(X, complement(Y)), meet(complement(Y), complement(join(zero, complement(complement(X))))))))
% 38.16/5.28 = { by lemma 54 R->L }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(meet(X, complement(Y)), complement(join(complement(complement(Y)), join(zero, complement(complement(X))))))))
% 38.16/5.28 = { by lemma 54 R->L }
% 38.16/5.28 join(X, join(complement(complement(Y)), complement(join(complement(meet(X, complement(Y))), join(complement(complement(Y)), join(zero, complement(complement(X))))))))
% 38.16/5.28 = { by axiom 8 (maddux2_join_associativity_2) }
% 38.16/5.28 join(X, join(complement(complement(Y)), complement(join(join(complement(meet(X, complement(Y))), complement(complement(Y))), join(zero, complement(complement(X)))))))
% 38.16/5.28 = { by lemma 70 R->L }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(complement(join(complement(meet(X, complement(Y))), complement(complement(Y)))), complement(join(zero, complement(complement(X)))))))
% 38.16/5.28 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(meet(meet(X, complement(Y)), complement(Y)), complement(join(zero, complement(complement(X)))))))
% 38.16/5.28 = { by lemma 41 R->L }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(complement(join(zero, complement(complement(X)))), meet(meet(X, complement(Y)), complement(Y)))))
% 38.16/5.28 = { by lemma 41 R->L }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(complement(join(zero, complement(complement(X)))), meet(complement(Y), meet(X, complement(Y))))))
% 38.16/5.28 = { by lemma 42 }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(meet(complement(X), top), meet(complement(Y), meet(X, complement(Y))))))
% 38.16/5.28 = { by lemma 46 }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(complement(X), meet(complement(Y), meet(X, complement(Y))))))
% 38.16/5.28 = { by lemma 41 }
% 38.16/5.28 join(X, join(complement(complement(Y)), meet(meet(complement(Y), meet(X, complement(Y))), complement(X))))
% 38.16/5.28 = { by lemma 72 }
% 38.16/5.28 join(X, complement(meet(complement(Y), join(X, complement(meet(complement(Y), meet(X, complement(Y))))))))
% 38.16/5.28 = { by lemma 41 }
% 38.16/5.28 join(X, complement(meet(complement(Y), join(X, complement(meet(complement(Y), meet(complement(Y), X)))))))
% 38.16/5.28 = { by lemma 41 }
% 38.16/5.28 join(X, complement(meet(complement(Y), join(X, complement(meet(meet(complement(Y), X), complement(Y)))))))
% 38.16/5.28 = { by lemma 33 R->L }
% 38.16/5.28 join(X, complement(meet(complement(Y), join(X, complement(meet(meet(complement(Y), X), join(meet(complement(Y), X), complement(join(complement(complement(Y)), X)))))))))
% 38.16/5.28 = { by lemma 58 }
% 38.16/5.28 join(X, complement(meet(complement(Y), join(X, complement(meet(complement(Y), X))))))
% 38.16/5.28 = { by lemma 41 }
% 38.16/5.28 join(X, complement(meet(complement(Y), join(X, complement(meet(X, complement(Y)))))))
% 38.16/5.28 = { by lemma 56 }
% 38.16/5.28 join(X, complement(meet(complement(Y), top)))
% 38.16/5.28 = { by lemma 46 }
% 38.16/5.28 join(X, complement(complement(Y)))
% 38.16/5.28 = { by lemma 49 }
% 38.16/5.28 join(X, Y)
% 38.16/5.28
% 38.16/5.28 Lemma 76: join(Y, meet(X, complement(Y))) = join(X, meet(Y, complement(X))).
% 38.16/5.28 Proof:
% 38.16/5.28 join(Y, meet(X, complement(Y)))
% 38.16/5.28 = { by lemma 49 R->L }
% 38.16/5.28 join(Y, meet(X, complement(complement(complement(Y)))))
% 38.16/5.28 = { by lemma 49 R->L }
% 38.16/5.28 complement(complement(join(Y, meet(X, complement(complement(complement(Y)))))))
% 38.16/5.28 = { by lemma 52 R->L }
% 38.16/5.28 complement(complement(join(Y, complement(join(complement(complement(Y)), complement(X))))))
% 38.16/5.28 = { by lemma 52 }
% 38.16/5.28 complement(meet(join(complement(complement(Y)), complement(X)), complement(Y)))
% 38.16/5.28 = { by lemma 41 R->L }
% 38.16/5.28 complement(meet(complement(Y), join(complement(complement(Y)), complement(X))))
% 38.16/5.28 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.28 complement(meet(complement(Y), join(complement(X), complement(complement(Y)))))
% 38.16/5.28 = { by lemma 73 R->L }
% 38.16/5.28 complement(meet(complement(X), join(complement(Y), complement(complement(X)))))
% 38.16/5.28 = { by lemma 69 }
% 38.16/5.28 join(X, complement(join(complement(Y), complement(complement(X)))))
% 38.16/5.28 = { by lemma 54 }
% 38.16/5.28 join(X, meet(Y, complement(complement(complement(X)))))
% 38.16/5.28 = { by lemma 49 }
% 38.16/5.28 join(X, meet(Y, complement(X)))
% 38.16/5.28
% 38.16/5.28 Lemma 77: composition(converse(X), complement(composition(X, top))) = zero.
% 38.16/5.28 Proof:
% 38.16/5.28 composition(converse(X), complement(composition(X, top)))
% 38.16/5.28 = { by lemma 40 R->L }
% 38.16/5.28 join(zero, composition(converse(X), complement(composition(X, top))))
% 38.16/5.28 = { by lemma 17 R->L }
% 38.16/5.28 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 38.16/5.28 = { by lemma 22 }
% 38.16/5.28 complement(top)
% 38.16/5.28 = { by lemma 17 }
% 38.16/5.28 zero
% 38.16/5.28
% 38.16/5.28 Lemma 78: meet(complement(composition(sk1, top)), composition(composition(sk1, top), X)) = zero.
% 38.16/5.28 Proof:
% 38.16/5.28 meet(complement(composition(sk1, top)), composition(composition(sk1, top), X))
% 38.16/5.28 = { by lemma 41 }
% 38.16/5.28 meet(composition(composition(sk1, top), X), complement(composition(sk1, top)))
% 38.16/5.28 = { by axiom 4 (goals_17) }
% 38.16/5.28 meet(composition(sk1, X), complement(composition(sk1, top)))
% 38.16/5.28 = { by lemma 39 R->L }
% 38.16/5.28 join(meet(composition(sk1, X), complement(composition(sk1, top))), zero)
% 38.16/5.28 = { by lemma 50 R->L }
% 38.16/5.28 join(meet(composition(sk1, X), complement(composition(sk1, top))), meet(zero, complement(composition(sk1, top))))
% 38.16/5.28 = { by lemma 63 R->L }
% 38.16/5.28 join(meet(composition(sk1, X), complement(composition(sk1, top))), meet(composition(sk1, zero), complement(composition(sk1, top))))
% 38.16/5.28 = { by lemma 44 R->L }
% 38.16/5.28 join(meet(composition(sk1, X), complement(composition(sk1, top))), meet(composition(sk1, meet(X, zero)), complement(composition(sk1, top))))
% 38.16/5.28 = { by lemma 77 R->L }
% 38.16/5.28 join(meet(composition(sk1, X), complement(composition(sk1, top))), meet(composition(sk1, meet(X, composition(converse(sk1), complement(composition(sk1, top))))), complement(composition(sk1, top))))
% 38.16/5.28 = { by axiom 15 (modular_law_1_15) }
% 38.16/5.28 meet(composition(sk1, meet(X, composition(converse(sk1), complement(composition(sk1, top))))), complement(composition(sk1, top)))
% 38.16/5.28 = { by lemma 77 }
% 38.16/5.28 meet(composition(sk1, meet(X, zero)), complement(composition(sk1, top)))
% 38.16/5.28 = { by lemma 44 }
% 38.16/5.28 meet(composition(sk1, zero), complement(composition(sk1, top)))
% 38.16/5.28 = { by lemma 63 }
% 38.16/5.28 meet(zero, complement(composition(sk1, top)))
% 38.16/5.28 = { by lemma 50 }
% 38.16/5.28 zero
% 38.16/5.28
% 38.16/5.28 Goal 1 (goals_18): composition(meet(sk2, converse(sk1)), meet(sk1, sk3)) = composition(sk2, meet(sk1, sk3)).
% 38.16/5.28 Proof:
% 38.16/5.28 composition(meet(sk2, converse(sk1)), meet(sk1, sk3))
% 38.16/5.28 = { by lemma 59 R->L }
% 38.16/5.28 meet(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3))), composition(meet(sk2, complement(converse(sk1))), meet(sk1, sk3)))))
% 38.16/5.28 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.28 meet(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(composition(meet(sk2, complement(converse(sk1))), meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3))))))
% 38.16/5.28 = { by axiom 8 (maddux2_join_associativity_2) }
% 38.16/5.28 meet(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(meet(sk2, complement(converse(sk1))), meet(sk1, sk3))), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.28 meet(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3))), join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(meet(sk2, complement(converse(sk1))), meet(sk1, sk3)))))
% 38.16/5.28 = { by axiom 12 (composition_distributivity_7) R->L }
% 38.16/5.28 meet(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3))), composition(join(meet(sk2, converse(sk1)), meet(sk2, complement(converse(sk1)))), meet(sk1, sk3))))
% 38.16/5.28 = { by lemma 45 }
% 38.16/5.28 meet(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3))), composition(sk2, meet(sk1, sk3))))
% 38.16/5.28 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.28 meet(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 39 R->L }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), zero), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 17 R->L }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), complement(top)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 22 R->L }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(composition(complement(composition(sk1, top)), top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 66 R->L }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(meet(composition(sk1, top), composition(complement(composition(sk1, top)), top)), meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 41 }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(meet(composition(complement(composition(sk1, top)), top), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 39 R->L }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(join(meet(composition(complement(composition(sk1, top)), top), composition(sk1, top)), zero), meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 50 R->L }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(join(meet(composition(complement(composition(sk1, top)), top), composition(sk1, top)), meet(zero, composition(sk1, top))), meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 64 R->L }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(join(meet(composition(complement(composition(sk1, top)), top), composition(sk1, top)), meet(composition(zero, top), composition(sk1, top))), meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 78 R->L }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(join(meet(composition(complement(composition(sk1, top)), top), composition(sk1, top)), meet(composition(meet(complement(composition(sk1, top)), composition(composition(sk1, top), converse(top))), top), composition(sk1, top))), meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by axiom 16 (modular_law_2_16) }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(meet(composition(meet(complement(composition(sk1, top)), composition(composition(sk1, top), converse(top))), top), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 78 }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(meet(composition(zero, top), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 64 }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(meet(zero, composition(sk1, top)), meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 50 }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(join(zero, meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.28 = { by lemma 40 }
% 38.16/5.28 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(composition(complement(composition(sk1, top)), top), complement(composition(sk1, top))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 41 R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), composition(complement(composition(sk1, top)), top)))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 32 R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), composition(complement(composition(sk1, top)), converse(top))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 71 R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), converse(composition(top, converse(complement(composition(sk1, top)))))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 31 R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), converse(composition(join(one, converse(top)), converse(complement(composition(sk1, top)))))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 12 (composition_distributivity_7) }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), converse(join(composition(one, converse(complement(composition(sk1, top)))), composition(converse(top), converse(complement(composition(sk1, top))))))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 21 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), converse(join(converse(complement(composition(sk1, top))), composition(converse(top), converse(complement(composition(sk1, top))))))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 32 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), converse(join(converse(complement(composition(sk1, top))), composition(top, converse(complement(composition(sk1, top))))))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 28 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), join(complement(composition(sk1, top)), converse(composition(top, converse(complement(composition(sk1, top))))))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 71 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), join(complement(composition(sk1, top)), composition(complement(composition(sk1, top)), converse(top)))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 32 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(meet(complement(composition(sk1, top)), join(complement(composition(sk1, top)), composition(complement(composition(sk1, top)), top))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 59 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(complement(top), composition(converse(complement(composition(sk1, top))), complement(complement(composition(sk1, top)))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 17 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), join(zero, composition(converse(complement(composition(sk1, top))), complement(complement(composition(sk1, top)))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 40 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(converse(complement(composition(sk1, top))), complement(complement(composition(sk1, top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 68 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(composition(sk1, top))), complement(complement(composition(sk1, top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 4 (goals_17) }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), complement(complement(composition(sk1, top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 49 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), composition(sk1, top))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 4 (goals_17) }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), sk1)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 49 R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), complement(complement(sk1)))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 58 R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), complement(meet(complement(sk1), join(complement(sk1), complement(sk3)))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 69 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), join(sk1, complement(join(complement(sk1), complement(sk3)))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), join(sk1, meet(sk1, sk3)))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 4 (goals_17) R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), join(composition(sk1, top), meet(sk1, sk3)))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), join(meet(sk1, sk3), composition(sk1, top)))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 4 (goals_17) R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(composition(sk1, top))), join(meet(sk1, sk3), composition(sk1, top)))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 68 R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(converse(complement(composition(sk1, top))), join(meet(sk1, sk3), composition(sk1, top)))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 19 R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(composition(converse(join(meet(sk1, sk3), composition(sk1, top))), complement(composition(sk1, top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(composition(converse(join(composition(sk1, top), meet(sk1, sk3))), complement(composition(sk1, top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 4 (goals_17) }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(composition(converse(join(sk1, meet(sk1, sk3))), complement(composition(sk1, top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 7 (converse_additivity_9) }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(composition(join(converse(sk1), converse(meet(sk1, sk3))), complement(composition(sk1, top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(composition(join(converse(meet(sk1, sk3)), converse(sk1)), complement(composition(sk1, top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 40 R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(join(zero, composition(join(converse(meet(sk1, sk3)), converse(sk1)), complement(composition(sk1, top)))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(join(composition(join(converse(meet(sk1, sk3)), converse(sk1)), complement(composition(sk1, top))), zero))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 12 (composition_distributivity_7) }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(join(join(composition(converse(meet(sk1, sk3)), complement(composition(sk1, top))), composition(converse(sk1), complement(composition(sk1, top)))), zero))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(join(composition(converse(meet(sk1, sk3)), complement(composition(sk1, top))), join(composition(converse(sk1), complement(composition(sk1, top))), zero)))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(join(composition(converse(meet(sk1, sk3)), complement(composition(sk1, top))), join(zero, composition(converse(sk1), complement(composition(sk1, top))))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 60 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(join(composition(converse(meet(sk1, sk3)), complement(composition(sk1, top))), zero))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 39 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), converse(composition(converse(meet(sk1, sk3)), complement(composition(sk1, top))))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 19 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(converse(complement(composition(sk1, top))), meet(sk1, sk3))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 68 }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(composition(sk1, top))), meet(sk1, sk3))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 4 (goals_17) }
% 38.16/5.29 meet(join(composition(meet(sk2, converse(sk1)), meet(sk1, sk3)), composition(complement(converse(sk1)), meet(sk1, sk3))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 12 (composition_distributivity_7) R->L }
% 38.16/5.29 meet(composition(join(meet(sk2, converse(sk1)), complement(converse(sk1))), meet(sk1, sk3)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 66 R->L }
% 38.16/5.29 meet(composition(join(meet(sk2, converse(sk1)), join(meet(sk2, complement(converse(sk1))), meet(complement(converse(sk1)), complement(sk2)))), meet(sk1, sk3)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 8 (maddux2_join_associativity_2) }
% 38.16/5.29 meet(composition(join(join(meet(sk2, converse(sk1)), meet(sk2, complement(converse(sk1)))), meet(complement(converse(sk1)), complement(sk2))), meet(sk1, sk3)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 45 }
% 38.16/5.29 meet(composition(join(sk2, meet(complement(converse(sk1)), complement(sk2))), meet(sk1, sk3)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.29 meet(composition(join(meet(complement(converse(sk1)), complement(sk2)), sk2), meet(sk1, sk3)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 70 }
% 38.16/5.29 meet(composition(join(complement(join(converse(sk1), sk2)), sk2), meet(sk1, sk3)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 1 (maddux1_join_commutativity_1) }
% 38.16/5.29 meet(composition(join(sk2, complement(join(converse(sk1), sk2))), meet(sk1, sk3)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 12 (composition_distributivity_7) }
% 38.16/5.29 meet(join(composition(sk2, meet(sk1, sk3)), composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 38.16/5.29 meet(join(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), composition(sk2, meet(sk1, sk3))), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.29 = { by lemma 41 }
% 38.16/5.30 meet(join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))), join(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), composition(sk2, meet(sk1, sk3))))
% 38.16/5.30 = { by lemma 75 R->L }
% 38.16/5.30 meet(join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))), join(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), meet(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3))))))
% 38.16/5.30 = { by lemma 53 R->L }
% 38.16/5.30 meet(complement(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3))))), join(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), meet(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3))))))
% 38.16/5.30 = { by lemma 76 }
% 38.16/5.30 meet(complement(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3))))), join(composition(sk2, meet(sk1, sk3)), meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3))))))
% 38.16/5.30 = { by lemma 41 }
% 38.16/5.30 meet(join(composition(sk2, meet(sk1, sk3)), meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3))))), complement(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3))))))
% 38.16/5.30 = { by lemma 52 R->L }
% 38.16/5.30 complement(join(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3)))), complement(join(composition(sk2, meet(sk1, sk3)), meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3))))))))
% 38.16/5.30 = { by lemma 74 R->L }
% 38.16/5.30 complement(join(complement(composition(sk2, meet(sk1, sk3))), meet(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3)))), composition(sk2, meet(sk1, sk3)))))
% 38.16/5.30 = { by lemma 54 }
% 38.16/5.30 meet(composition(sk2, meet(sk1, sk3)), complement(meet(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3)))), composition(sk2, meet(sk1, sk3)))))
% 38.16/5.30 = { by lemma 41 }
% 38.16/5.30 meet(composition(sk2, meet(sk1, sk3)), complement(meet(composition(sk2, meet(sk1, sk3)), meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3)))))))
% 38.16/5.30 = { by lemma 55 R->L }
% 38.16/5.30 meet(composition(sk2, meet(sk1, sk3)), join(complement(composition(sk2, meet(sk1, sk3))), complement(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3)))))))
% 38.16/5.30 = { by lemma 53 R->L }
% 38.16/5.30 meet(composition(sk2, meet(sk1, sk3)), complement(meet(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3)))), complement(complement(composition(sk2, meet(sk1, sk3)))))))
% 38.16/5.30 = { by lemma 54 R->L }
% 38.16/5.30 complement(join(complement(composition(sk2, meet(sk1, sk3))), meet(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3)))), complement(complement(composition(sk2, meet(sk1, sk3)))))))
% 38.16/5.30 = { by lemma 76 R->L }
% 38.16/5.30 complement(join(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3)))), meet(complement(composition(sk2, meet(sk1, sk3))), complement(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3))))))))
% 38.16/5.30 = { by lemma 75 }
% 38.16/5.30 complement(join(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3)))), complement(composition(sk2, meet(sk1, sk3)))))
% 38.16/5.30 = { by lemma 52 }
% 38.16/5.30 meet(composition(sk2, meet(sk1, sk3)), complement(meet(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)), complement(composition(sk2, meet(sk1, sk3))))))
% 38.16/5.30 = { by lemma 53 }
% 38.16/5.30 meet(composition(sk2, meet(sk1, sk3)), join(composition(sk2, meet(sk1, sk3)), complement(composition(complement(join(converse(sk1), sk2)), meet(sk1, sk3)))))
% 38.16/5.30 = { by lemma 58 }
% 38.16/5.30 composition(sk2, meet(sk1, sk3))
% 38.16/5.30 % SZS output end Proof
% 38.16/5.30
% 38.16/5.30 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------