TSTP Solution File: REL030+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL030+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 : n028.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 13:44:14 EDT 2023
% Result : Theorem 37.23s 5.18s
% Output : Proof 38.92s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL030+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n028.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Aug 25 21:45:06 EDT 2023
% 0.12/0.33 % CPUTime :
% 37.23/5.18 Command-line arguments: --ground-connectedness --complete-subsets
% 37.23/5.18
% 37.23/5.18 % SZS status Theorem
% 37.23/5.18
% 38.92/5.31 % SZS output start Proof
% 38.92/5.31 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 38.92/5.31 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 38.92/5.31 Axiom 3 (goals): join(x0, one) = one.
% 38.92/5.31 Axiom 4 (composition_identity): composition(X, one) = X.
% 38.92/5.31 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 38.92/5.31 Axiom 6 (def_top): top = join(X, complement(X)).
% 38.92/5.31 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 38.92/5.31 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 38.92/5.31 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 38.92/5.31 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 38.92/5.31 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 38.92/5.31 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 38.92/5.31 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 38.92/5.31 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 38.92/5.31
% 38.92/5.31 Lemma 15: complement(top) = zero.
% 38.92/5.31 Proof:
% 38.92/5.31 complement(top)
% 38.92/5.31 = { by axiom 6 (def_top) }
% 38.92/5.31 complement(join(complement(X), complement(complement(X))))
% 38.92/5.31 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.31 meet(X, complement(X))
% 38.92/5.31 = { by axiom 5 (def_zero) R->L }
% 38.92/5.31 zero
% 38.92/5.31
% 38.92/5.31 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 38.92/5.31 Proof:
% 38.92/5.31 join(X, join(Y, complement(X)))
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(X, join(complement(X), Y))
% 38.92/5.31 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.31 join(join(X, complement(X)), Y)
% 38.92/5.31 = { by axiom 6 (def_top) R->L }
% 38.92/5.31 join(top, Y)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.31 join(Y, top)
% 38.92/5.31
% 38.92/5.31 Lemma 17: composition(converse(one), X) = X.
% 38.92/5.31 Proof:
% 38.92/5.31 composition(converse(one), X)
% 38.92/5.31 = { by axiom 1 (converse_idempotence) R->L }
% 38.92/5.31 composition(converse(one), converse(converse(X)))
% 38.92/5.31 = { by axiom 9 (converse_multiplicativity) R->L }
% 38.92/5.31 converse(composition(converse(X), one))
% 38.92/5.31 = { by axiom 4 (composition_identity) }
% 38.92/5.31 converse(converse(X))
% 38.92/5.31 = { by axiom 1 (converse_idempotence) }
% 38.92/5.31 X
% 38.92/5.31
% 38.92/5.31 Lemma 18: composition(one, X) = X.
% 38.92/5.31 Proof:
% 38.92/5.31 composition(one, X)
% 38.92/5.31 = { by lemma 17 R->L }
% 38.92/5.31 composition(converse(one), composition(one, X))
% 38.92/5.31 = { by axiom 10 (composition_associativity) }
% 38.92/5.31 composition(composition(converse(one), one), X)
% 38.92/5.31 = { by axiom 4 (composition_identity) }
% 38.92/5.31 composition(converse(one), X)
% 38.92/5.31 = { by lemma 17 }
% 38.92/5.31 X
% 38.92/5.31
% 38.92/5.31 Lemma 19: join(complement(X), complement(X)) = complement(X).
% 38.92/5.31 Proof:
% 38.92/5.31 join(complement(X), complement(X))
% 38.92/5.31 = { by lemma 17 R->L }
% 38.92/5.31 join(complement(X), composition(converse(one), complement(X)))
% 38.92/5.31 = { by lemma 18 R->L }
% 38.92/5.31 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 38.92/5.31 = { by axiom 13 (converse_cancellativity) }
% 38.92/5.31 complement(X)
% 38.92/5.31
% 38.92/5.31 Lemma 20: join(top, complement(X)) = top.
% 38.92/5.31 Proof:
% 38.92/5.31 join(top, complement(X))
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(complement(X), top)
% 38.92/5.31 = { by lemma 16 R->L }
% 38.92/5.31 join(X, join(complement(X), complement(X)))
% 38.92/5.31 = { by lemma 19 }
% 38.92/5.31 join(X, complement(X))
% 38.92/5.31 = { by axiom 6 (def_top) R->L }
% 38.92/5.31 top
% 38.92/5.31
% 38.92/5.31 Lemma 21: join(Y, top) = join(X, top).
% 38.92/5.31 Proof:
% 38.92/5.31 join(Y, top)
% 38.92/5.31 = { by lemma 20 R->L }
% 38.92/5.31 join(Y, join(top, complement(Y)))
% 38.92/5.31 = { by lemma 16 }
% 38.92/5.31 join(top, top)
% 38.92/5.31 = { by lemma 16 R->L }
% 38.92/5.31 join(X, join(top, complement(X)))
% 38.92/5.31 = { by lemma 20 }
% 38.92/5.31 join(X, top)
% 38.92/5.31
% 38.92/5.31 Lemma 22: join(X, top) = top.
% 38.92/5.31 Proof:
% 38.92/5.31 join(X, top)
% 38.92/5.31 = { by lemma 21 }
% 38.92/5.31 join(complement(Y), top)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(top, complement(Y))
% 38.92/5.31 = { by lemma 20 }
% 38.92/5.31 top
% 38.92/5.31
% 38.92/5.31 Lemma 23: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 38.92/5.31 Proof:
% 38.92/5.31 converse(join(X, converse(Y)))
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 converse(join(converse(Y), X))
% 38.92/5.31 = { by axiom 7 (converse_additivity) }
% 38.92/5.31 join(converse(converse(Y)), converse(X))
% 38.92/5.31 = { by axiom 1 (converse_idempotence) }
% 38.92/5.31 join(Y, converse(X))
% 38.92/5.31
% 38.92/5.31 Lemma 24: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 38.92/5.31 Proof:
% 38.92/5.31 converse(join(converse(X), Y))
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 converse(join(Y, converse(X)))
% 38.92/5.31 = { by lemma 23 }
% 38.92/5.31 join(X, converse(Y))
% 38.92/5.31
% 38.92/5.31 Lemma 25: join(X, converse(complement(converse(X)))) = converse(top).
% 38.92/5.31 Proof:
% 38.92/5.31 join(X, converse(complement(converse(X))))
% 38.92/5.31 = { by lemma 24 R->L }
% 38.92/5.31 converse(join(converse(X), complement(converse(X))))
% 38.92/5.31 = { by axiom 6 (def_top) R->L }
% 38.92/5.31 converse(top)
% 38.92/5.31
% 38.92/5.31 Lemma 26: join(X, join(complement(X), Y)) = top.
% 38.92/5.31 Proof:
% 38.92/5.31 join(X, join(complement(X), Y))
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(X, join(Y, complement(X)))
% 38.92/5.31 = { by lemma 16 }
% 38.92/5.31 join(Y, top)
% 38.92/5.31 = { by lemma 21 R->L }
% 38.92/5.31 join(Z, top)
% 38.92/5.31 = { by lemma 22 }
% 38.92/5.31 top
% 38.92/5.31
% 38.92/5.31 Lemma 27: converse(top) = top.
% 38.92/5.31 Proof:
% 38.92/5.31 converse(top)
% 38.92/5.31 = { by lemma 22 R->L }
% 38.92/5.31 converse(join(X, top))
% 38.92/5.31 = { by axiom 7 (converse_additivity) }
% 38.92/5.31 join(converse(X), converse(top))
% 38.92/5.31 = { by lemma 25 R->L }
% 38.92/5.31 join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X)))))))
% 38.92/5.31 = { by lemma 26 }
% 38.92/5.31 top
% 38.92/5.31
% 38.92/5.31 Lemma 28: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 38.92/5.31 Proof:
% 38.92/5.31 composition(join(X, one), Y)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 composition(join(one, X), Y)
% 38.92/5.31 = { by axiom 12 (composition_distributivity) }
% 38.92/5.31 join(composition(one, Y), composition(X, Y))
% 38.92/5.31 = { by lemma 18 }
% 38.92/5.31 join(Y, composition(X, Y))
% 38.92/5.31
% 38.92/5.31 Lemma 29: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 38.92/5.31 Proof:
% 38.92/5.31 join(meet(X, Y), complement(join(complement(X), Y)))
% 38.92/5.31 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.92/5.31 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 38.92/5.31 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 38.92/5.31 X
% 38.92/5.31
% 38.92/5.31 Lemma 30: join(zero, meet(X, X)) = X.
% 38.92/5.31 Proof:
% 38.92/5.31 join(zero, meet(X, X))
% 38.92/5.31 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.92/5.31 join(zero, complement(join(complement(X), complement(X))))
% 38.92/5.31 = { by axiom 5 (def_zero) }
% 38.92/5.31 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 38.92/5.31 = { by lemma 29 }
% 38.92/5.31 X
% 38.92/5.31
% 38.92/5.31 Lemma 31: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 38.92/5.31 Proof:
% 38.92/5.31 join(zero, join(X, meet(Y, Y)))
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(zero, join(meet(Y, Y), X))
% 38.92/5.31 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.31 join(join(zero, meet(Y, Y)), X)
% 38.92/5.31 = { by lemma 30 }
% 38.92/5.31 join(Y, X)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.31 join(X, Y)
% 38.92/5.31
% 38.92/5.31 Lemma 32: join(X, zero) = join(X, X).
% 38.92/5.31 Proof:
% 38.92/5.31 join(X, zero)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(zero, X)
% 38.92/5.31 = { by lemma 30 R->L }
% 38.92/5.31 join(zero, join(zero, meet(X, X)))
% 38.92/5.31 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.92/5.31 join(zero, join(zero, complement(join(complement(X), complement(X)))))
% 38.92/5.31 = { by lemma 19 R->L }
% 38.92/5.31 join(zero, join(zero, join(complement(join(complement(X), complement(X))), complement(join(complement(X), complement(X))))))
% 38.92/5.31 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.31 join(zero, join(zero, join(meet(X, X), complement(join(complement(X), complement(X))))))
% 38.92/5.31 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.31 join(zero, join(zero, join(meet(X, X), meet(X, X))))
% 38.92/5.31 = { by lemma 31 }
% 38.92/5.31 join(zero, join(meet(X, X), X))
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.31 join(zero, join(X, meet(X, X)))
% 38.92/5.31 = { by lemma 31 }
% 38.92/5.31 join(X, X)
% 38.92/5.31
% 38.92/5.31 Lemma 33: join(one, x0) = one.
% 38.92/5.31 Proof:
% 38.92/5.31 join(one, x0)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(x0, one)
% 38.92/5.31 = { by axiom 3 (goals) }
% 38.92/5.31 one
% 38.92/5.31
% 38.92/5.31 Lemma 34: join(one, join(X, x0)) = join(X, one).
% 38.92/5.31 Proof:
% 38.92/5.31 join(one, join(X, x0))
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(one, join(x0, X))
% 38.92/5.31 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.31 join(join(one, x0), X)
% 38.92/5.31 = { by lemma 33 }
% 38.92/5.31 join(one, X)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.31 join(X, one)
% 38.92/5.31
% 38.92/5.31 Lemma 35: join(X, X) = X.
% 38.92/5.31 Proof:
% 38.92/5.31 join(X, X)
% 38.92/5.31 = { by lemma 17 R->L }
% 38.92/5.31 join(X, composition(converse(one), X))
% 38.92/5.31 = { by lemma 28 R->L }
% 38.92/5.31 composition(join(converse(one), one), X)
% 38.92/5.31 = { by axiom 4 (composition_identity) R->L }
% 38.92/5.31 composition(join(composition(converse(one), one), one), X)
% 38.92/5.31 = { by lemma 17 }
% 38.92/5.31 composition(join(one, one), X)
% 38.92/5.31 = { by lemma 32 R->L }
% 38.92/5.31 composition(join(one, zero), X)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.31 composition(join(zero, one), X)
% 38.92/5.31 = { by lemma 34 R->L }
% 38.92/5.31 composition(join(one, join(zero, x0)), X)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.31 composition(join(one, join(x0, zero)), X)
% 38.92/5.31 = { by lemma 32 }
% 38.92/5.31 composition(join(one, join(x0, x0)), X)
% 38.92/5.31 = { by lemma 34 }
% 38.92/5.31 composition(join(x0, one), X)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.31 composition(join(one, x0), X)
% 38.92/5.31 = { by lemma 33 }
% 38.92/5.31 composition(one, X)
% 38.92/5.31 = { by lemma 18 }
% 38.92/5.31 X
% 38.92/5.31
% 38.92/5.31 Lemma 36: join(X, zero) = X.
% 38.92/5.31 Proof:
% 38.92/5.31 join(X, zero)
% 38.92/5.31 = { by lemma 32 }
% 38.92/5.31 join(X, X)
% 38.92/5.31 = { by lemma 35 }
% 38.92/5.31 X
% 38.92/5.31
% 38.92/5.31 Lemma 37: join(zero, X) = X.
% 38.92/5.31 Proof:
% 38.92/5.31 join(zero, X)
% 38.92/5.31 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.31 join(X, zero)
% 38.92/5.31 = { by lemma 36 }
% 38.92/5.31 X
% 38.92/5.31
% 38.92/5.31 Lemma 38: complement(complement(X)) = meet(X, X).
% 38.92/5.31 Proof:
% 38.92/5.31 complement(complement(X))
% 38.92/5.31 = { by lemma 19 R->L }
% 38.92/5.31 complement(join(complement(X), complement(X)))
% 38.92/5.31 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.31 meet(X, X)
% 38.92/5.31
% 38.92/5.31 Lemma 39: complement(complement(X)) = X.
% 38.92/5.31 Proof:
% 38.92/5.31 complement(complement(X))
% 38.92/5.31 = { by lemma 37 R->L }
% 38.92/5.31 join(zero, complement(complement(X)))
% 38.92/5.32 = { by lemma 38 }
% 38.92/5.32 join(zero, meet(X, X))
% 38.92/5.32 = { by lemma 30 }
% 38.92/5.32 X
% 38.92/5.32
% 38.92/5.32 Lemma 40: meet(Y, X) = meet(X, Y).
% 38.92/5.32 Proof:
% 38.92/5.32 meet(Y, X)
% 38.92/5.32 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.92/5.32 complement(join(complement(Y), complement(X)))
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.32 complement(join(complement(X), complement(Y)))
% 38.92/5.32 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.32 meet(X, Y)
% 38.92/5.32
% 38.92/5.32 Lemma 41: complement(join(zero, complement(X))) = meet(X, top).
% 38.92/5.32 Proof:
% 38.92/5.32 complement(join(zero, complement(X)))
% 38.92/5.32 = { by lemma 15 R->L }
% 38.92/5.32 complement(join(complement(top), complement(X)))
% 38.92/5.32 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.32 meet(top, X)
% 38.92/5.32 = { by lemma 40 R->L }
% 38.92/5.32 meet(X, top)
% 38.92/5.32
% 38.92/5.32 Lemma 42: meet(X, top) = X.
% 38.92/5.32 Proof:
% 38.92/5.32 meet(X, top)
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 meet(top, X)
% 38.92/5.32 = { by lemma 39 R->L }
% 38.92/5.32 meet(top, complement(complement(X)))
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 meet(complement(complement(X)), top)
% 38.92/5.32 = { by lemma 41 R->L }
% 38.92/5.32 complement(join(zero, complement(complement(complement(X)))))
% 38.92/5.32 = { by lemma 38 }
% 38.92/5.32 complement(join(zero, meet(complement(X), complement(X))))
% 38.92/5.32 = { by lemma 30 }
% 38.92/5.32 complement(complement(X))
% 38.92/5.32 = { by lemma 39 }
% 38.92/5.32 X
% 38.92/5.32
% 38.92/5.32 Lemma 43: meet(top, X) = X.
% 38.92/5.32 Proof:
% 38.92/5.32 meet(top, X)
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 meet(X, top)
% 38.92/5.32 = { by lemma 42 }
% 38.92/5.32 X
% 38.92/5.32
% 38.92/5.32 Lemma 44: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 38.92/5.32 Proof:
% 38.92/5.32 composition(join(one, Y), X)
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.32 composition(join(Y, one), X)
% 38.92/5.32 = { by lemma 28 }
% 38.92/5.32 join(X, composition(Y, X))
% 38.92/5.32
% 38.92/5.32 Lemma 45: join(X, composition(x0, X)) = X.
% 38.92/5.32 Proof:
% 38.92/5.32 join(X, composition(x0, X))
% 38.92/5.32 = { by lemma 44 R->L }
% 38.92/5.32 composition(join(one, x0), X)
% 38.92/5.32 = { by lemma 33 }
% 38.92/5.32 composition(one, X)
% 38.92/5.32 = { by lemma 18 }
% 38.92/5.32 X
% 38.92/5.32
% 38.92/5.32 Lemma 46: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 38.92/5.32 Proof:
% 38.92/5.32 join(Y, join(X, Z))
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.32 join(join(X, Z), Y)
% 38.92/5.32 = { by axiom 8 (maddux2_join_associativity) R->L }
% 38.92/5.32 join(X, join(Z, Y))
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.32 join(X, join(Y, Z))
% 38.92/5.32
% 38.92/5.32 Lemma 47: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 38.92/5.32 Proof:
% 38.92/5.32 complement(join(complement(X), meet(Y, Z)))
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 complement(join(complement(X), meet(Z, Y)))
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.32 complement(join(meet(Z, Y), complement(X)))
% 38.92/5.32 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.92/5.32 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 38.92/5.32 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.32 meet(join(complement(Z), complement(Y)), X)
% 38.92/5.32 = { by lemma 40 R->L }
% 38.92/5.32 meet(X, join(complement(Z), complement(Y)))
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.32 meet(X, join(complement(Y), complement(Z)))
% 38.92/5.32
% 38.92/5.32 Lemma 48: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 38.92/5.32 Proof:
% 38.92/5.32 join(complement(X), complement(Y))
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.32 join(complement(Y), complement(X))
% 38.92/5.32 = { by lemma 43 R->L }
% 38.92/5.32 meet(top, join(complement(Y), complement(X)))
% 38.92/5.32 = { by lemma 47 R->L }
% 38.92/5.32 complement(join(complement(top), meet(Y, X)))
% 38.92/5.32 = { by lemma 15 }
% 38.92/5.32 complement(join(zero, meet(Y, X)))
% 38.92/5.32 = { by lemma 37 }
% 38.92/5.32 complement(meet(Y, X))
% 38.92/5.32 = { by lemma 40 R->L }
% 38.92/5.32 complement(meet(X, Y))
% 38.92/5.32
% 38.92/5.32 Lemma 49: join(composition(x0, X), X) = X.
% 38.92/5.32 Proof:
% 38.92/5.32 join(composition(x0, X), X)
% 38.92/5.32 = { by lemma 18 R->L }
% 38.92/5.32 join(composition(x0, X), composition(one, X))
% 38.92/5.32 = { by axiom 12 (composition_distributivity) R->L }
% 38.92/5.32 composition(join(x0, one), X)
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.32 composition(join(one, x0), X)
% 38.92/5.32 = { by lemma 33 }
% 38.92/5.32 composition(one, X)
% 38.92/5.32 = { by lemma 18 }
% 38.92/5.32 X
% 38.92/5.32
% 38.92/5.32 Lemma 50: meet(X, meet(X, X)) = X.
% 38.92/5.32 Proof:
% 38.92/5.32 meet(X, meet(X, X))
% 38.92/5.32 = { by lemma 38 R->L }
% 38.92/5.32 meet(X, complement(complement(X)))
% 38.92/5.32 = { by lemma 36 R->L }
% 38.92/5.32 join(meet(X, complement(complement(X))), zero)
% 38.92/5.32 = { by lemma 15 R->L }
% 38.92/5.32 join(meet(X, complement(complement(X))), complement(top))
% 38.92/5.32 = { by axiom 6 (def_top) }
% 38.92/5.32 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 38.92/5.32 = { by lemma 29 }
% 38.92/5.32 X
% 38.92/5.32
% 38.92/5.32 Lemma 51: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 38.92/5.32 Proof:
% 38.92/5.32 complement(join(X, complement(Y)))
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.32 complement(join(complement(Y), X))
% 38.92/5.32 = { by lemma 50 R->L }
% 38.92/5.32 complement(join(complement(Y), meet(X, meet(X, X))))
% 38.92/5.32 = { by lemma 47 }
% 38.92/5.32 meet(Y, join(complement(X), complement(meet(X, X))))
% 38.92/5.32 = { by lemma 48 }
% 38.92/5.32 meet(Y, complement(meet(X, meet(X, X))))
% 38.92/5.32 = { by lemma 50 }
% 38.92/5.32 meet(Y, complement(X))
% 38.92/5.32
% 38.92/5.32 Lemma 52: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 38.92/5.32 Proof:
% 38.92/5.32 complement(meet(X, complement(Y)))
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 complement(meet(complement(Y), X))
% 38.92/5.32 = { by lemma 37 R->L }
% 38.92/5.32 complement(meet(join(zero, complement(Y)), X))
% 38.92/5.32 = { by lemma 48 R->L }
% 38.92/5.32 join(complement(join(zero, complement(Y))), complement(X))
% 38.92/5.32 = { by lemma 41 }
% 38.92/5.32 join(meet(Y, top), complement(X))
% 38.92/5.32 = { by lemma 42 }
% 38.92/5.32 join(Y, complement(X))
% 38.92/5.32
% 38.92/5.32 Lemma 53: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 38.92/5.32 Proof:
% 38.92/5.32 complement(meet(complement(X), Y))
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 complement(meet(Y, complement(X)))
% 38.92/5.32 = { by lemma 52 }
% 38.92/5.32 join(X, complement(Y))
% 38.92/5.32
% 38.92/5.32 Lemma 54: meet(X, join(X, complement(Y))) = X.
% 38.92/5.32 Proof:
% 38.92/5.32 meet(X, join(X, complement(Y)))
% 38.92/5.32 = { by lemma 52 R->L }
% 38.92/5.32 meet(X, complement(meet(Y, complement(X))))
% 38.92/5.32 = { by lemma 48 R->L }
% 38.92/5.32 meet(X, join(complement(Y), complement(complement(X))))
% 38.92/5.32 = { by lemma 47 R->L }
% 38.92/5.32 complement(join(complement(X), meet(Y, complement(X))))
% 38.92/5.32 = { by lemma 37 R->L }
% 38.92/5.32 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 38.92/5.32 = { by lemma 15 R->L }
% 38.92/5.32 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 38.92/5.32 = { by lemma 22 R->L }
% 38.92/5.32 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 38.92/5.32 = { by lemma 16 R->L }
% 38.92/5.32 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 38.92/5.32 = { by lemma 48 }
% 38.92/5.32 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 38.92/5.32 = { by lemma 40 R->L }
% 38.92/5.32 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 38.92/5.32 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.32 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 38.92/5.32 = { by lemma 40 R->L }
% 38.92/5.32 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 38.92/5.32 = { by lemma 29 }
% 38.92/5.32 X
% 38.92/5.32
% 38.92/5.32 Lemma 55: meet(X, join(Y, X)) = X.
% 38.92/5.32 Proof:
% 38.92/5.32 meet(X, join(Y, X))
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.32 meet(X, join(X, Y))
% 38.92/5.32 = { by lemma 39 R->L }
% 38.92/5.32 meet(X, join(X, complement(complement(Y))))
% 38.92/5.32 = { by lemma 54 }
% 38.92/5.32 X
% 38.92/5.32
% 38.92/5.32 Lemma 56: complement(meet(Y, meet(Z, X))) = complement(meet(X, meet(Y, Z))).
% 38.92/5.32 Proof:
% 38.92/5.32 complement(meet(Y, meet(Z, X)))
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 complement(meet(Y, meet(X, Z)))
% 38.92/5.32 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.92/5.32 complement(meet(Y, complement(join(complement(X), complement(Z)))))
% 38.92/5.32 = { by lemma 52 }
% 38.92/5.32 join(join(complement(X), complement(Z)), complement(Y))
% 38.92/5.32 = { by axiom 8 (maddux2_join_associativity) R->L }
% 38.92/5.32 join(complement(X), join(complement(Z), complement(Y)))
% 38.92/5.32 = { by lemma 48 }
% 38.92/5.32 join(complement(X), complement(meet(Z, Y)))
% 38.92/5.32 = { by lemma 48 }
% 38.92/5.32 complement(meet(X, meet(Z, Y)))
% 38.92/5.32 = { by lemma 40 R->L }
% 38.92/5.32 complement(meet(X, meet(Y, Z)))
% 38.92/5.32
% 38.92/5.32 Lemma 57: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 38.92/5.32 Proof:
% 38.92/5.32 meet(Y, meet(X, Z))
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 meet(Y, meet(Z, X))
% 38.92/5.32 = { by lemma 42 R->L }
% 38.92/5.32 meet(meet(Y, meet(Z, X)), top)
% 38.92/5.32 = { by lemma 41 R->L }
% 38.92/5.32 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 38.92/5.32 = { by lemma 56 }
% 38.92/5.32 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 38.92/5.32 = { by lemma 41 }
% 38.92/5.32 meet(meet(X, meet(Y, Z)), top)
% 38.92/5.32 = { by lemma 42 }
% 38.92/5.32 meet(X, meet(Y, Z))
% 38.92/5.32
% 38.92/5.32 Lemma 58: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 38.92/5.32 Proof:
% 38.92/5.32 meet(complement(X), complement(Y))
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 meet(complement(Y), complement(X))
% 38.92/5.32 = { by lemma 37 R->L }
% 38.92/5.32 meet(join(zero, complement(Y)), complement(X))
% 38.92/5.32 = { by lemma 51 R->L }
% 38.92/5.32 complement(join(X, complement(join(zero, complement(Y)))))
% 38.92/5.32 = { by lemma 41 }
% 38.92/5.32 complement(join(X, meet(Y, top)))
% 38.92/5.32 = { by lemma 42 }
% 38.92/5.32 complement(join(X, Y))
% 38.92/5.32
% 38.92/5.32 Lemma 59: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 38.92/5.32 Proof:
% 38.92/5.32 meet(meet(X, Y), Z)
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 meet(Z, meet(X, Y))
% 38.92/5.32 = { by lemma 57 R->L }
% 38.92/5.32 meet(X, meet(Z, Y))
% 38.92/5.32
% 38.92/5.32 Lemma 60: complement(join(complement(X), complement(Y))) = meet(Y, X).
% 38.92/5.32 Proof:
% 38.92/5.32 complement(join(complement(X), complement(Y)))
% 38.92/5.32 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.32 meet(X, Y)
% 38.92/5.32 = { by lemma 40 R->L }
% 38.92/5.32 meet(Y, X)
% 38.92/5.32
% 38.92/5.32 Lemma 61: join(complement(composition(x0, X)), X) = top.
% 38.92/5.32 Proof:
% 38.92/5.32 join(complement(composition(x0, X)), X)
% 38.92/5.32 = { by lemma 49 R->L }
% 38.92/5.32 join(complement(composition(x0, X)), join(composition(x0, X), X))
% 38.92/5.32 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.32 join(join(complement(composition(x0, X)), composition(x0, X)), X)
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.32 join(join(composition(x0, X), complement(composition(x0, X))), X)
% 38.92/5.32 = { by axiom 6 (def_top) R->L }
% 38.92/5.32 join(top, X)
% 38.92/5.32 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.32 join(X, top)
% 38.92/5.32 = { by lemma 21 R->L }
% 38.92/5.32 join(Y, top)
% 38.92/5.32 = { by lemma 22 }
% 38.92/5.32 top
% 38.92/5.32
% 38.92/5.32 Lemma 62: meet(complement(X), composition(x0, X)) = zero.
% 38.92/5.32 Proof:
% 38.92/5.32 meet(complement(X), composition(x0, X))
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 meet(composition(x0, X), complement(X))
% 38.92/5.32 = { by lemma 36 R->L }
% 38.92/5.32 meet(composition(x0, X), join(complement(X), zero))
% 38.92/5.32 = { by lemma 15 R->L }
% 38.92/5.32 meet(composition(x0, X), join(complement(X), complement(top)))
% 38.92/5.32 = { by lemma 40 }
% 38.92/5.32 meet(join(complement(X), complement(top)), composition(x0, X))
% 38.92/5.32 = { by lemma 60 R->L }
% 38.92/5.32 complement(join(complement(composition(x0, X)), complement(join(complement(X), complement(top)))))
% 38.92/5.32 = { by lemma 60 }
% 38.92/5.32 complement(join(complement(composition(x0, X)), meet(top, X)))
% 38.92/5.32 = { by lemma 43 }
% 38.92/5.32 complement(join(complement(composition(x0, X)), X))
% 38.92/5.32 = { by lemma 61 }
% 38.92/5.32 complement(top)
% 38.92/5.32 = { by lemma 15 }
% 38.92/5.33 zero
% 38.92/5.33
% 38.92/5.33 Lemma 63: join(meet(X, Y), complement(join(X, complement(Y)))) = Y.
% 38.92/5.33 Proof:
% 38.92/5.33 join(meet(X, Y), complement(join(X, complement(Y))))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.33 join(meet(X, Y), complement(join(complement(Y), X)))
% 38.92/5.33 = { by lemma 40 }
% 38.92/5.33 join(meet(Y, X), complement(join(complement(Y), X)))
% 38.92/5.33 = { by lemma 29 }
% 38.92/5.33 Y
% 38.92/5.33
% 38.92/5.33 Lemma 64: meet(composition(x0, X), X) = composition(x0, X).
% 38.92/5.33 Proof:
% 38.92/5.33 meet(composition(x0, X), X)
% 38.92/5.33 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.92/5.33 complement(join(complement(composition(x0, X)), complement(X)))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.33 complement(join(complement(X), complement(composition(x0, X))))
% 38.92/5.33 = { by lemma 37 R->L }
% 38.92/5.33 join(zero, complement(join(complement(X), complement(composition(x0, X)))))
% 38.92/5.33 = { by lemma 62 R->L }
% 38.92/5.33 join(meet(complement(X), composition(x0, X)), complement(join(complement(X), complement(composition(x0, X)))))
% 38.92/5.33 = { by lemma 63 }
% 38.92/5.33 composition(x0, X)
% 38.92/5.33
% 38.92/5.33 Lemma 65: join(meet(X, Y), meet(X, complement(Y))) = X.
% 38.92/5.33 Proof:
% 38.92/5.33 join(meet(X, Y), meet(X, complement(Y)))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.33 join(meet(X, complement(Y)), meet(X, Y))
% 38.92/5.33 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.92/5.33 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 38.92/5.33 = { by lemma 29 }
% 38.92/5.33 X
% 38.92/5.33
% 38.92/5.33 Lemma 66: join(meet(X, Y), meet(complement(Y), X)) = X.
% 38.92/5.33 Proof:
% 38.92/5.33 join(meet(X, Y), meet(complement(Y), X))
% 38.92/5.33 = { by lemma 40 }
% 38.92/5.33 join(meet(X, Y), meet(X, complement(Y)))
% 38.92/5.33 = { by lemma 65 }
% 38.92/5.33 X
% 38.92/5.33
% 38.92/5.33 Lemma 67: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 38.92/5.33 Proof:
% 38.92/5.33 meet(complement(X), join(X, Y))
% 38.92/5.33 = { by lemma 39 R->L }
% 38.92/5.33 meet(complement(X), join(X, complement(complement(Y))))
% 38.92/5.33 = { by lemma 66 R->L }
% 38.92/5.33 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(complement(X), join(X, complement(complement(Y))))))
% 38.92/5.33 = { by lemma 40 }
% 38.92/5.33 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(join(X, complement(complement(Y))), complement(X))))
% 38.92/5.33 = { by lemma 57 }
% 38.92/5.33 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), meet(complement(Y), complement(X))))
% 38.92/5.33 = { by lemma 51 R->L }
% 38.92/5.33 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), complement(join(X, complement(complement(Y))))))
% 38.92/5.33 = { by axiom 5 (def_zero) R->L }
% 38.92/5.33 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), zero)
% 38.92/5.33 = { by lemma 36 }
% 38.92/5.33 meet(meet(complement(X), join(X, complement(complement(Y)))), Y)
% 38.92/5.33 = { by lemma 59 }
% 38.92/5.33 meet(complement(X), meet(Y, join(X, complement(complement(Y)))))
% 38.92/5.33 = { by lemma 39 }
% 38.92/5.33 meet(complement(X), meet(Y, join(X, Y)))
% 38.92/5.33 = { by lemma 55 }
% 38.92/5.33 meet(complement(X), Y)
% 38.92/5.33 = { by lemma 40 R->L }
% 38.92/5.33 meet(Y, complement(X))
% 38.92/5.33
% 38.92/5.33 Lemma 68: meet(X, join(Y, complement(X))) = meet(X, Y).
% 38.92/5.33 Proof:
% 38.92/5.33 meet(X, join(Y, complement(X)))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.33 meet(X, join(complement(X), Y))
% 38.92/5.33 = { by lemma 37 R->L }
% 38.92/5.33 meet(X, join(zero, join(complement(X), Y)))
% 38.92/5.33 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.33 meet(X, join(join(zero, complement(X)), Y))
% 38.92/5.33 = { by lemma 42 R->L }
% 38.92/5.33 meet(X, meet(join(join(zero, complement(X)), Y), top))
% 38.92/5.33 = { by lemma 59 R->L }
% 38.92/5.33 meet(meet(X, top), join(join(zero, complement(X)), Y))
% 38.92/5.33 = { by lemma 41 R->L }
% 38.92/5.33 meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), Y))
% 38.92/5.33 = { by lemma 67 }
% 38.92/5.33 meet(Y, complement(join(zero, complement(X))))
% 38.92/5.33 = { by lemma 41 }
% 38.92/5.33 meet(Y, meet(X, top))
% 38.92/5.33 = { by lemma 42 }
% 38.92/5.33 meet(Y, X)
% 38.92/5.33 = { by lemma 40 R->L }
% 38.92/5.33 meet(X, Y)
% 38.92/5.33
% 38.92/5.33 Lemma 69: join(composition(X, Y), composition(X, converse(Z))) = composition(X, join(Y, converse(Z))).
% 38.92/5.33 Proof:
% 38.92/5.33 join(composition(X, Y), composition(X, converse(Z)))
% 38.92/5.33 = { by axiom 1 (converse_idempotence) R->L }
% 38.92/5.33 converse(converse(join(composition(X, Y), composition(X, converse(Z)))))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.33 converse(converse(join(composition(X, converse(Z)), composition(X, Y))))
% 38.92/5.33 = { by axiom 7 (converse_additivity) }
% 38.92/5.33 converse(join(converse(composition(X, converse(Z))), converse(composition(X, Y))))
% 38.92/5.33 = { by axiom 9 (converse_multiplicativity) }
% 38.92/5.33 converse(join(composition(converse(converse(Z)), converse(X)), converse(composition(X, Y))))
% 38.92/5.33 = { by axiom 1 (converse_idempotence) }
% 38.92/5.33 converse(join(composition(Z, converse(X)), converse(composition(X, Y))))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.33 converse(join(converse(composition(X, Y)), composition(Z, converse(X))))
% 38.92/5.33 = { by axiom 9 (converse_multiplicativity) }
% 38.92/5.33 converse(join(composition(converse(Y), converse(X)), composition(Z, converse(X))))
% 38.92/5.33 = { by axiom 12 (composition_distributivity) R->L }
% 38.92/5.33 converse(composition(join(converse(Y), Z), converse(X)))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.33 converse(composition(join(Z, converse(Y)), converse(X)))
% 38.92/5.33 = { by lemma 23 R->L }
% 38.92/5.33 converse(composition(converse(join(Y, converse(Z))), converse(X)))
% 38.92/5.33 = { by axiom 9 (converse_multiplicativity) R->L }
% 38.92/5.33 converse(converse(composition(X, join(Y, converse(Z)))))
% 38.92/5.33 = { by axiom 1 (converse_idempotence) }
% 38.92/5.33 composition(X, join(Y, converse(Z)))
% 38.92/5.33
% 38.92/5.33 Lemma 70: join(composition(X, Y), composition(X, complement(composition(x0, Y)))) = composition(X, top).
% 38.92/5.33 Proof:
% 38.92/5.33 join(composition(X, Y), composition(X, complement(composition(x0, Y))))
% 38.92/5.33 = { by axiom 1 (converse_idempotence) R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(complement(composition(x0, Y))))))
% 38.92/5.33 = { by lemma 69 }
% 38.92/5.33 composition(X, join(Y, converse(converse(complement(composition(x0, Y))))))
% 38.92/5.33 = { by axiom 1 (converse_idempotence) }
% 38.92/5.33 composition(X, join(Y, complement(composition(x0, Y))))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.33 composition(X, join(complement(composition(x0, Y)), Y))
% 38.92/5.33 = { by lemma 61 }
% 38.92/5.33 composition(X, top)
% 38.92/5.33
% 38.92/5.33 Lemma 71: complement(join(X, meet(Y, complement(Z)))) = meet(complement(X), join(Z, complement(Y))).
% 38.92/5.33 Proof:
% 38.92/5.33 complement(join(X, meet(Y, complement(Z))))
% 38.92/5.33 = { by lemma 51 R->L }
% 38.92/5.33 complement(join(X, complement(join(Z, complement(Y)))))
% 38.92/5.33 = { by lemma 51 }
% 38.92/5.33 meet(join(Z, complement(Y)), complement(X))
% 38.92/5.33 = { by lemma 40 R->L }
% 38.92/5.33 meet(complement(X), join(Z, complement(Y)))
% 38.92/5.33
% 38.92/5.33 Lemma 72: join(meet(X, complement(Y)), meet(X, Y)) = X.
% 38.92/5.33 Proof:
% 38.92/5.33 join(meet(X, complement(Y)), meet(X, Y))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.33 join(meet(X, Y), meet(X, complement(Y)))
% 38.92/5.33 = { by lemma 65 }
% 38.92/5.33 X
% 38.92/5.33
% 38.92/5.33 Lemma 73: join(composition(X, Y), composition(X, complement(Y))) = composition(X, top).
% 38.92/5.33 Proof:
% 38.92/5.33 join(composition(X, Y), composition(X, complement(Y)))
% 38.92/5.33 = { by axiom 1 (converse_idempotence) R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(complement(Y)))))
% 38.92/5.33 = { by lemma 42 R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(complement(Y), top)))))
% 38.92/5.33 = { by lemma 27 R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(complement(Y), converse(top))))))
% 38.92/5.33 = { by lemma 25 R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(complement(Y), join(Y, converse(complement(converse(Y)))))))))
% 38.92/5.33 = { by lemma 39 R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(complement(Y), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by axiom 1 (converse_idempotence) R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(converse(complement(Y))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 39 R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(complement(complement(converse(complement(Y))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 66 R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(complement(join(meet(complement(converse(complement(Y))), converse(complement(converse(converse(complement(Y)))))), meet(complement(converse(complement(converse(converse(complement(Y)))))), complement(converse(complement(Y))))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 71 }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(meet(complement(meet(complement(converse(complement(Y))), converse(complement(converse(converse(complement(Y))))))), join(converse(complement(Y)), complement(complement(converse(complement(converse(converse(complement(Y)))))))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 53 }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(meet(join(converse(complement(Y)), complement(converse(complement(converse(converse(complement(Y))))))), join(converse(complement(Y)), complement(complement(converse(complement(converse(converse(complement(Y)))))))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 39 }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(meet(join(converse(complement(Y)), complement(converse(complement(converse(converse(complement(Y))))))), join(converse(complement(Y)), converse(complement(converse(converse(complement(Y)))))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 40 R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(meet(join(converse(complement(Y)), converse(complement(converse(converse(complement(Y)))))), join(converse(complement(Y)), complement(converse(complement(converse(converse(complement(Y))))))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 25 }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(meet(converse(top), join(converse(complement(Y)), complement(converse(complement(converse(converse(complement(Y))))))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 27 }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(meet(top, join(converse(complement(Y)), complement(converse(complement(converse(converse(complement(Y))))))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 43 }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(converse(join(converse(complement(Y)), complement(converse(complement(converse(converse(complement(Y)))))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by lemma 24 }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(join(complement(Y), converse(complement(converse(complement(converse(converse(complement(Y)))))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by axiom 1 (converse_idempotence) }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(join(complement(Y), converse(complement(converse(complement(complement(Y)))))), join(Y, converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(join(complement(Y), converse(complement(converse(complement(complement(Y)))))), join(converse(complement(converse(complement(complement(Y))))), Y))))))
% 38.92/5.33 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.33 join(composition(X, Y), composition(X, converse(converse(meet(join(converse(complement(converse(complement(complement(Y))))), complement(Y)), join(converse(complement(converse(complement(complement(Y))))), Y))))))
% 38.92/5.33 = { by lemma 40 }
% 38.92/5.34 join(composition(X, Y), composition(X, converse(converse(meet(join(converse(complement(converse(complement(complement(Y))))), Y), join(converse(complement(converse(complement(complement(Y))))), complement(Y)))))))
% 38.92/5.34 = { by lemma 39 R->L }
% 38.92/5.34 join(composition(X, Y), composition(X, converse(converse(meet(join(converse(complement(converse(complement(complement(Y))))), complement(complement(Y))), join(converse(complement(converse(complement(complement(Y))))), complement(Y)))))))
% 38.92/5.34 = { by lemma 53 R->L }
% 38.92/5.34 join(composition(X, Y), composition(X, converse(converse(meet(complement(meet(complement(converse(complement(converse(complement(complement(Y)))))), complement(Y))), join(converse(complement(converse(complement(complement(Y))))), complement(Y)))))))
% 38.92/5.34 = { by lemma 71 R->L }
% 38.92/5.34 join(composition(X, Y), composition(X, converse(converse(complement(join(meet(complement(converse(complement(converse(complement(complement(Y)))))), complement(Y)), meet(Y, complement(converse(complement(converse(complement(complement(Y)))))))))))))
% 38.92/5.34 = { by lemma 40 }
% 38.92/5.34 join(composition(X, Y), composition(X, converse(converse(complement(join(meet(complement(converse(complement(converse(complement(complement(Y)))))), complement(Y)), meet(complement(converse(complement(converse(complement(complement(Y)))))), Y)))))))
% 38.92/5.34 = { by lemma 72 }
% 38.92/5.34 join(composition(X, Y), composition(X, converse(converse(complement(complement(converse(complement(converse(complement(complement(Y)))))))))))
% 38.92/5.34 = { by lemma 39 }
% 38.92/5.34 join(composition(X, Y), composition(X, converse(converse(converse(complement(converse(complement(complement(Y)))))))))
% 38.92/5.34 = { by lemma 39 }
% 38.92/5.34 join(composition(X, Y), composition(X, converse(converse(converse(complement(converse(Y)))))))
% 38.92/5.34 = { by axiom 1 (converse_idempotence) }
% 38.92/5.34 join(composition(X, Y), composition(X, converse(complement(converse(Y)))))
% 38.92/5.34 = { by lemma 69 }
% 38.92/5.34 composition(X, join(Y, converse(complement(converse(Y)))))
% 38.92/5.34 = { by lemma 24 R->L }
% 38.92/5.34 composition(X, converse(join(converse(Y), complement(converse(Y)))))
% 38.92/5.34 = { by axiom 6 (def_top) R->L }
% 38.92/5.34 composition(X, converse(top))
% 38.92/5.34 = { by lemma 27 }
% 38.92/5.34 composition(X, top)
% 38.92/5.34
% 38.92/5.34 Lemma 74: composition(x0, complement(composition(x0, X))) = composition(x0, complement(X)).
% 38.92/5.34 Proof:
% 38.92/5.34 composition(x0, complement(composition(x0, X)))
% 38.92/5.34 = { by lemma 55 R->L }
% 38.92/5.34 meet(composition(x0, complement(composition(x0, X))), join(complement(composition(x0, X)), composition(x0, complement(composition(x0, X)))))
% 38.92/5.34 = { by lemma 45 }
% 38.92/5.34 meet(composition(x0, complement(composition(x0, X))), complement(composition(x0, X)))
% 38.92/5.34 = { by lemma 67 R->L }
% 38.92/5.34 meet(complement(composition(x0, X)), join(composition(x0, X), composition(x0, complement(composition(x0, X)))))
% 38.92/5.34 = { by lemma 70 }
% 38.92/5.34 meet(complement(composition(x0, X)), composition(x0, top))
% 38.92/5.34 = { by lemma 36 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), zero)
% 38.92/5.34 = { by lemma 15 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(top))
% 38.92/5.34 = { by lemma 26 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(join(zero, join(complement(zero), meet(complement(composition(x0, X)), complement(composition(x0, X)))))))
% 38.92/5.34 = { by lemma 31 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(join(complement(zero), complement(composition(x0, X)))))
% 38.92/5.34 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(join(complement(composition(x0, X)), complement(zero))))
% 38.92/5.34 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), zero))
% 38.92/5.34 = { by axiom 5 (def_zero) }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), meet(X, complement(X))))
% 38.92/5.34 = { by lemma 45 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), meet(X, join(complement(X), composition(x0, complement(X))))))
% 38.92/5.34 = { by lemma 39 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), meet(X, join(complement(X), complement(complement(composition(x0, complement(X))))))))
% 38.92/5.34 = { by lemma 66 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), join(meet(meet(X, join(complement(X), complement(complement(composition(x0, complement(X)))))), composition(x0, complement(X))), meet(complement(composition(x0, complement(X))), meet(X, join(complement(X), complement(complement(composition(x0, complement(X))))))))))
% 38.92/5.34 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), join(meet(meet(X, join(complement(X), complement(complement(composition(x0, complement(X)))))), composition(x0, complement(X))), meet(complement(composition(x0, complement(X))), meet(X, join(complement(complement(composition(x0, complement(X)))), complement(X)))))))
% 38.92/5.34 = { by lemma 40 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), join(meet(meet(X, join(complement(X), complement(complement(composition(x0, complement(X)))))), composition(x0, complement(X))), meet(complement(composition(x0, complement(X))), meet(join(complement(complement(composition(x0, complement(X)))), complement(X)), X)))))
% 38.92/5.34 = { by lemma 59 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), join(meet(meet(X, join(complement(X), complement(complement(composition(x0, complement(X)))))), composition(x0, complement(X))), meet(meet(complement(composition(x0, complement(X))), X), join(complement(complement(composition(x0, complement(X)))), complement(X))))))
% 38.92/5.34 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), join(meet(meet(X, join(complement(X), complement(complement(composition(x0, complement(X)))))), composition(x0, complement(X))), meet(meet(complement(composition(x0, complement(X))), X), join(complement(X), complement(complement(composition(x0, complement(X)))))))))
% 38.92/5.34 = { by lemma 40 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), join(meet(meet(X, join(complement(X), complement(complement(composition(x0, complement(X)))))), composition(x0, complement(X))), meet(join(complement(X), complement(complement(composition(x0, complement(X))))), meet(complement(composition(x0, complement(X))), X)))))
% 38.92/5.34 = { by lemma 60 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), join(meet(meet(X, join(complement(X), complement(complement(composition(x0, complement(X)))))), composition(x0, complement(X))), meet(join(complement(X), complement(complement(composition(x0, complement(X))))), complement(join(complement(X), complement(complement(composition(x0, complement(X))))))))))
% 38.92/5.34 = { by axiom 5 (def_zero) R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), join(meet(meet(X, join(complement(X), complement(complement(composition(x0, complement(X)))))), composition(x0, complement(X))), zero)))
% 38.92/5.34 = { by lemma 36 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), meet(meet(X, join(complement(X), complement(complement(composition(x0, complement(X)))))), composition(x0, complement(X)))))
% 38.92/5.34 = { by lemma 59 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), meet(X, meet(composition(x0, complement(X)), join(complement(X), complement(complement(composition(x0, complement(X)))))))))
% 38.92/5.34 = { by lemma 39 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), meet(X, meet(composition(x0, complement(X)), join(complement(X), composition(x0, complement(X)))))))
% 38.92/5.34 = { by lemma 55 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), meet(X, composition(x0, complement(X)))))
% 38.92/5.34 = { by lemma 40 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), meet(composition(x0, complement(X)), X)))
% 38.92/5.34 = { by lemma 60 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), complement(join(complement(X), complement(composition(x0, complement(X)))))))
% 38.92/5.34 = { by lemma 51 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(join(join(complement(X), complement(composition(x0, complement(X)))), complement(composition(x0, X)))))
% 38.92/5.34 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(join(complement(composition(x0, X)), join(complement(X), complement(composition(x0, complement(X)))))))
% 38.92/5.34 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(join(join(complement(composition(x0, X)), complement(X)), complement(composition(x0, complement(X))))))
% 38.92/5.34 = { by lemma 52 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(complement(meet(composition(x0, complement(X)), complement(join(complement(composition(x0, X)), complement(X)))))))
% 38.92/5.34 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(complement(meet(composition(x0, complement(X)), meet(composition(x0, X), X)))))
% 38.92/5.34 = { by lemma 64 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(complement(meet(composition(x0, complement(X)), composition(x0, X)))))
% 38.92/5.34 = { by lemma 40 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(complement(meet(composition(x0, X), composition(x0, complement(X))))))
% 38.92/5.34 = { by lemma 48 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), complement(join(complement(composition(x0, X)), complement(composition(x0, complement(X))))))
% 38.92/5.34 = { by lemma 60 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, complement(X)), composition(x0, X)))
% 38.92/5.34 = { by lemma 40 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, top)), meet(composition(x0, X), composition(x0, complement(X))))
% 38.92/5.34 = { by lemma 73 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), join(composition(x0, X), composition(x0, complement(X)))), meet(composition(x0, X), composition(x0, complement(X))))
% 38.92/5.34 = { by lemma 67 }
% 38.92/5.34 join(meet(composition(x0, complement(X)), complement(composition(x0, X))), meet(composition(x0, X), composition(x0, complement(X))))
% 38.92/5.34 = { by lemma 40 R->L }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, complement(X))), meet(composition(x0, X), composition(x0, complement(X))))
% 38.92/5.34 = { by lemma 40 }
% 38.92/5.34 join(meet(complement(composition(x0, X)), composition(x0, complement(X))), meet(composition(x0, complement(X)), composition(x0, X)))
% 38.92/5.34 = { by lemma 40 }
% 38.92/5.34 join(meet(composition(x0, complement(X)), complement(composition(x0, X))), meet(composition(x0, complement(X)), composition(x0, X)))
% 38.92/5.34 = { by lemma 72 }
% 38.92/5.34 composition(x0, complement(X))
% 38.92/5.34
% 38.92/5.34 Lemma 75: complement(meet(X, meet(complement(Y), Z))) = join(Y, complement(meet(X, Z))).
% 38.92/5.34 Proof:
% 38.92/5.34 complement(meet(X, meet(complement(Y), Z)))
% 38.92/5.34 = { by lemma 56 R->L }
% 38.92/5.34 complement(meet(complement(Y), meet(Z, X)))
% 38.92/5.34 = { by lemma 53 }
% 38.92/5.35 join(Y, complement(meet(Z, X)))
% 38.92/5.35 = { by lemma 40 R->L }
% 38.92/5.35 join(Y, complement(meet(X, Z)))
% 38.92/5.35
% 38.92/5.35 Lemma 76: join(composition(x0, X), complement(composition(x0, top))) = complement(composition(x0, complement(composition(x0, X)))).
% 38.92/5.35 Proof:
% 38.92/5.35 join(composition(x0, X), complement(composition(x0, top)))
% 38.92/5.35 = { by lemma 53 R->L }
% 38.92/5.35 complement(meet(complement(composition(x0, X)), composition(x0, top)))
% 38.92/5.35 = { by lemma 40 }
% 38.92/5.35 complement(meet(composition(x0, top), complement(composition(x0, X))))
% 38.92/5.35 = { by lemma 45 R->L }
% 38.92/5.35 complement(meet(composition(x0, top), join(complement(composition(x0, X)), composition(x0, complement(composition(x0, X))))))
% 38.92/5.35 = { by lemma 48 R->L }
% 38.92/5.35 join(complement(composition(x0, top)), complement(join(complement(composition(x0, X)), composition(x0, complement(composition(x0, X))))))
% 38.92/5.35 = { by lemma 58 R->L }
% 38.92/5.35 join(complement(composition(x0, top)), meet(complement(complement(composition(x0, X))), complement(composition(x0, complement(composition(x0, X))))))
% 38.92/5.35 = { by lemma 73 R->L }
% 38.92/5.35 join(complement(join(composition(x0, X), composition(x0, complement(X)))), meet(complement(complement(composition(x0, X))), complement(composition(x0, complement(composition(x0, X))))))
% 38.92/5.35 = { by lemma 58 R->L }
% 38.92/5.35 join(meet(complement(composition(x0, X)), complement(composition(x0, complement(X)))), meet(complement(complement(composition(x0, X))), complement(composition(x0, complement(composition(x0, X))))))
% 38.92/5.35 = { by lemma 74 R->L }
% 38.92/5.35 join(meet(complement(composition(x0, X)), complement(composition(x0, complement(composition(x0, X))))), meet(complement(complement(composition(x0, X))), complement(composition(x0, complement(composition(x0, X))))))
% 38.92/5.35 = { by lemma 40 }
% 38.92/5.35 join(meet(complement(composition(x0, X)), complement(composition(x0, complement(composition(x0, X))))), meet(complement(composition(x0, complement(composition(x0, X)))), complement(complement(composition(x0, X)))))
% 38.92/5.35 = { by lemma 40 }
% 38.92/5.35 join(meet(complement(composition(x0, complement(composition(x0, X)))), complement(composition(x0, X))), meet(complement(composition(x0, complement(composition(x0, X)))), complement(complement(composition(x0, X)))))
% 38.92/5.35 = { by lemma 65 }
% 38.92/5.35 complement(composition(x0, complement(composition(x0, X))))
% 38.92/5.35
% 38.92/5.35 Lemma 77: join(complement(X), meet(complement(composition(x0, X)), X)) = complement(composition(x0, X)).
% 38.92/5.35 Proof:
% 38.92/5.35 join(complement(X), meet(complement(composition(x0, X)), X))
% 38.92/5.35 = { by lemma 39 R->L }
% 38.92/5.35 join(complement(X), meet(complement(composition(x0, X)), complement(complement(X))))
% 38.92/5.35 = { by lemma 65 R->L }
% 38.92/5.35 join(join(meet(complement(X), composition(x0, X)), meet(complement(X), complement(composition(x0, X)))), meet(complement(composition(x0, X)), complement(complement(X))))
% 38.92/5.35 = { by lemma 62 }
% 38.92/5.35 join(join(zero, meet(complement(X), complement(composition(x0, X)))), meet(complement(composition(x0, X)), complement(complement(X))))
% 38.92/5.35 = { by lemma 37 }
% 38.92/5.35 join(meet(complement(X), complement(composition(x0, X))), meet(complement(composition(x0, X)), complement(complement(X))))
% 38.92/5.35 = { by lemma 40 R->L }
% 38.92/5.35 join(meet(complement(composition(x0, X)), complement(X)), meet(complement(composition(x0, X)), complement(complement(X))))
% 38.92/5.35 = { by lemma 65 }
% 38.92/5.35 complement(composition(x0, X))
% 38.92/5.35
% 38.92/5.35 Goal 1 (goals_1): meet(composition(x0, x1), complement(x2)) = meet(composition(x0, x1), complement(composition(x0, x2))).
% 38.92/5.35 Proof:
% 38.92/5.35 meet(composition(x0, x1), complement(x2))
% 38.92/5.35 = { by lemma 39 R->L }
% 38.92/5.35 complement(complement(meet(composition(x0, x1), complement(x2))))
% 38.92/5.35 = { by lemma 52 }
% 38.92/5.35 complement(join(x2, complement(composition(x0, x1))))
% 38.92/5.35 = { by lemma 49 R->L }
% 38.92/5.35 complement(join(join(composition(x0, x2), x2), complement(composition(x0, x1))))
% 38.92/5.35 = { by axiom 8 (maddux2_join_associativity) R->L }
% 38.92/5.35 complement(join(composition(x0, x2), join(x2, complement(composition(x0, x1)))))
% 38.92/5.35 = { by lemma 52 R->L }
% 38.92/5.35 complement(join(composition(x0, x2), complement(meet(composition(x0, x1), complement(x2)))))
% 38.92/5.35 = { by lemma 51 }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(x2)), complement(composition(x0, x2)))
% 38.92/5.35 = { by lemma 40 R->L }
% 38.92/5.35 meet(complement(composition(x0, x2)), meet(composition(x0, x1), complement(x2)))
% 38.92/5.35 = { by lemma 40 R->L }
% 38.92/5.35 meet(complement(composition(x0, x2)), meet(complement(x2), composition(x0, x1)))
% 38.92/5.35 = { by lemma 57 R->L }
% 38.92/5.35 meet(complement(x2), meet(complement(composition(x0, x2)), composition(x0, x1)))
% 38.92/5.35 = { by lemma 40 }
% 38.92/5.35 meet(complement(x2), meet(composition(x0, x1), complement(composition(x0, x2))))
% 38.92/5.35 = { by lemma 40 }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), complement(x2))
% 38.92/5.35 = { by lemma 18 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), composition(one, complement(x2)))
% 38.92/5.35 = { by lemma 33 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), composition(join(one, x0), complement(x2)))
% 38.92/5.35 = { by lemma 44 }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), composition(x0, complement(x2))))
% 38.92/5.35 = { by lemma 74 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), composition(x0, complement(composition(x0, x2)))))
% 38.92/5.35 = { by lemma 39 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), complement(complement(composition(x0, complement(composition(x0, x2)))))))
% 38.92/5.35 = { by lemma 54 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), complement(meet(complement(composition(x0, complement(composition(x0, x2)))), join(complement(composition(x0, complement(composition(x0, x2)))), complement(composition(x0, x1)))))))
% 38.92/5.35 = { by lemma 53 }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), complement(join(complement(composition(x0, complement(composition(x0, x2)))), complement(composition(x0, x1)))))))
% 38.92/5.35 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), composition(x0, x1)))))
% 38.92/5.35 = { by lemma 68 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), join(composition(x0, x1), complement(composition(x0, complement(composition(x0, x2)))))))))
% 38.92/5.35 = { by lemma 76 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), join(composition(x0, x1), join(composition(x0, x2), complement(composition(x0, top))))))))
% 38.92/5.35 = { by lemma 46 }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), join(composition(x0, x2), join(composition(x0, x1), complement(composition(x0, top))))))))
% 38.92/5.35 = { by lemma 52 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), join(composition(x0, x2), complement(meet(composition(x0, top), complement(composition(x0, x1)))))))))
% 38.92/5.35 = { by lemma 75 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), meet(complement(composition(x0, x2)), complement(composition(x0, x1)))))))))
% 38.92/5.35 = { by lemma 67 R->L }
% 38.92/5.35 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), meet(complement(composition(x0, x1)), join(composition(x0, x1), complement(composition(x0, x2))))))))))
% 38.92/5.36 = { by lemma 71 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), complement(join(composition(x0, x1), meet(composition(x0, x2), complement(composition(x0, x1)))))))))))
% 38.92/5.36 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), complement(join(meet(composition(x0, x2), complement(composition(x0, x1))), composition(x0, x1)))))))))
% 38.92/5.36 = { by lemma 51 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), complement(join(complement(join(composition(x0, x1), complement(composition(x0, x2)))), composition(x0, x1)))))))))
% 38.92/5.36 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), complement(join(composition(x0, x1), complement(join(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 38.92/5.36 = { by lemma 72 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), complement(join(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x1), composition(x0, x2))), complement(join(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 38.92/5.36 = { by axiom 8 (maddux2_join_associativity) R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), join(meet(composition(x0, x1), composition(x0, x2)), complement(join(composition(x0, x1), complement(composition(x0, x2)))))))))))))
% 38.92/5.36 = { by lemma 63 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2)))))))))
% 38.92/5.36 = { by lemma 64 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, x2), x2)))))))))
% 38.92/5.36 = { by axiom 11 (maddux4_definiton_of_meet) }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), complement(join(complement(composition(x0, x2)), complement(x2)))))))))))
% 38.92/5.36 = { by lemma 51 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), meet(join(complement(composition(x0, x2)), complement(x2)), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))))
% 38.92/5.36 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), meet(join(complement(x2), complement(composition(x0, x2))), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))))
% 38.92/5.36 = { by lemma 77 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), meet(join(complement(x2), join(complement(x2), meet(complement(composition(x0, x2)), x2))), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))))
% 38.92/5.36 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), meet(join(join(complement(x2), complement(x2)), meet(complement(composition(x0, x2)), x2)), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))))
% 38.92/5.36 = { by lemma 35 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), meet(join(complement(x2), meet(complement(composition(x0, x2)), x2)), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))))
% 38.92/5.36 = { by lemma 77 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(meet(composition(x0, top), meet(complement(composition(x0, x2)), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))))
% 38.92/5.36 = { by lemma 75 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), join(composition(x0, x2), complement(meet(composition(x0, top), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))))
% 38.92/5.36 = { by lemma 52 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), join(composition(x0, x2), join(meet(composition(x0, x1), complement(composition(x0, x2))), complement(composition(x0, top))))))))
% 38.92/5.36 = { by lemma 46 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), join(meet(composition(x0, x1), complement(composition(x0, x2))), join(composition(x0, x2), complement(composition(x0, top))))))))
% 38.92/5.36 = { by lemma 76 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), join(meet(composition(x0, x1), complement(composition(x0, x2))), complement(composition(x0, complement(composition(x0, x2)))))))))
% 38.92/5.36 = { by lemma 68 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))
% 38.92/5.36 = { by lemma 39 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, complement(composition(x0, x2))), complement(complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 38.92/5.36 = { by lemma 67 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), composition(x0, complement(composition(x0, x2))))))))
% 38.92/5.36 = { by lemma 52 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(join(composition(x0, x2), complement(composition(x0, x1))), composition(x0, complement(composition(x0, x2))))))))
% 38.92/5.36 = { by axiom 8 (maddux2_join_associativity) R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(composition(x0, x2), join(complement(composition(x0, x1)), composition(x0, complement(composition(x0, x2)))))))))
% 38.92/5.36 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(composition(x0, x2), join(composition(x0, complement(composition(x0, x2))), complement(composition(x0, x1))))))))
% 38.92/5.36 = { by lemma 74 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(composition(x0, x2), join(composition(x0, complement(x2)), complement(composition(x0, x1))))))))
% 38.92/5.36 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(join(composition(x0, x2), composition(x0, complement(x2))), complement(composition(x0, x1)))))))
% 38.92/5.36 = { by lemma 73 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(composition(x0, top), complement(composition(x0, x1)))))))
% 38.92/5.36 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(complement(composition(x0, x1)), composition(x0, top))))))
% 38.92/5.36 = { by lemma 70 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(complement(composition(x0, x1)), join(composition(x0, x1), composition(x0, complement(composition(x0, x1)))))))))
% 38.92/5.36 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(complement(composition(x0, x1)), join(composition(x0, complement(composition(x0, x1))), composition(x0, x1)))))))
% 38.92/5.36 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(join(complement(composition(x0, x1)), composition(x0, complement(composition(x0, x1)))), composition(x0, x1))))))
% 38.92/5.36 = { by lemma 45 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(complement(composition(x0, x1)), composition(x0, x1))))))
% 38.92/5.36 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(composition(x0, x1), complement(composition(x0, x1)))))))
% 38.92/5.36 = { by axiom 6 (def_top) R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(complement(complement(meet(composition(x0, x1), complement(composition(x0, x2))))), top))))
% 38.92/5.36 = { by lemma 39 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(meet(composition(x0, x1), complement(composition(x0, x2))), top))))
% 38.92/5.36 = { by lemma 40 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(top, meet(composition(x0, x1), complement(composition(x0, x2)))))))
% 38.92/5.36 = { by lemma 43 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))
% 38.92/5.36 = { by axiom 2 (maddux1_join_commutativity) }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, complement(composition(x0, x2))))))
% 38.92/5.36 = { by lemma 74 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, complement(x2)))))
% 38.92/5.36 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), join(composition(x0, complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2))))))
% 38.92/5.36 = { by axiom 8 (maddux2_join_associativity) }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(join(complement(x2), composition(x0, complement(x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 38.92/5.36 = { by lemma 44 R->L }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(composition(join(one, x0), complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 38.92/5.36 = { by lemma 33 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(composition(one, complement(x2)), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 38.92/5.36 = { by lemma 18 }
% 38.92/5.36 meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(complement(x2), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 38.92/5.36 = { by lemma 55 }
% 38.92/5.36 meet(composition(x0, x1), complement(composition(x0, x2)))
% 38.92/5.36 % SZS output end Proof
% 38.92/5.36
% 38.92/5.36 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------