TSTP Solution File: REL045+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL045+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 : n001.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 13:44:31 EDT 2023
% Result : Theorem 29.25s 4.09s
% Output : Proof 29.51s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : REL045+2 : 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 : n001.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 19:57:14 EDT 2023
% 0.12/0.33 % CPUTime :
% 29.25/4.09 Command-line arguments: --flatten
% 29.25/4.09
% 29.25/4.09 % SZS status Theorem
% 29.25/4.09
% 29.51/4.17 % SZS output start Proof
% 29.51/4.17 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 29.51/4.17 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 29.51/4.17 Axiom 3 (composition_identity): composition(X, one) = X.
% 29.51/4.17 Axiom 4 (def_zero): zero = meet(X, complement(X)).
% 29.51/4.17 Axiom 5 (def_top): top = join(X, complement(X)).
% 29.51/4.17 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 29.51/4.17 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 29.51/4.17 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 29.51/4.17 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 29.51/4.17 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 29.51/4.17 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 29.51/4.17 Axiom 12 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 29.51/4.17 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 29.51/4.17 Axiom 14 (modular_law_2): 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).
% 29.51/4.17
% 29.51/4.17 Lemma 15: complement(top) = zero.
% 29.51/4.17 Proof:
% 29.51/4.17 complement(top)
% 29.51/4.17 = { by axiom 5 (def_top) }
% 29.51/4.17 complement(join(complement(X), complement(complement(X))))
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 29.51/4.17 meet(X, complement(X))
% 29.51/4.17 = { by axiom 4 (def_zero) R->L }
% 29.51/4.17 zero
% 29.51/4.17
% 29.51/4.17 Lemma 16: converse(composition(converse(Y), X)) = composition(converse(X), Y).
% 29.51/4.17 Proof:
% 29.51/4.17 converse(composition(converse(Y), X))
% 29.51/4.17 = { by axiom 8 (converse_multiplicativity) }
% 29.51/4.17 composition(converse(X), converse(converse(Y)))
% 29.51/4.17 = { by axiom 2 (converse_idempotence) }
% 29.51/4.17 composition(converse(X), Y)
% 29.51/4.17
% 29.51/4.17 Lemma 17: composition(converse(one), X) = X.
% 29.51/4.17 Proof:
% 29.51/4.17 composition(converse(one), X)
% 29.51/4.17 = { by lemma 16 R->L }
% 29.51/4.17 converse(composition(converse(X), one))
% 29.51/4.17 = { by axiom 3 (composition_identity) }
% 29.51/4.17 converse(converse(X))
% 29.51/4.17 = { by axiom 2 (converse_idempotence) }
% 29.51/4.17 X
% 29.51/4.17
% 29.51/4.17 Lemma 18: composition(one, X) = X.
% 29.51/4.17 Proof:
% 29.51/4.17 composition(one, X)
% 29.51/4.17 = { by lemma 17 R->L }
% 29.51/4.17 composition(converse(one), composition(one, X))
% 29.51/4.17 = { by axiom 9 (composition_associativity) }
% 29.51/4.17 composition(composition(converse(one), one), X)
% 29.51/4.17 = { by axiom 3 (composition_identity) }
% 29.51/4.17 composition(converse(one), X)
% 29.51/4.17 = { by lemma 17 }
% 29.51/4.17 X
% 29.51/4.17
% 29.51/4.17 Lemma 19: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 29.51/4.17 Proof:
% 29.51/4.17 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 29.51/4.17 = { by axiom 12 (converse_cancellativity) }
% 29.51/4.17 complement(X)
% 29.51/4.17
% 29.51/4.17 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 29.51/4.17 Proof:
% 29.51/4.17 join(complement(X), complement(X))
% 29.51/4.17 = { by lemma 17 R->L }
% 29.51/4.17 join(complement(X), composition(converse(one), complement(X)))
% 29.51/4.17 = { by lemma 18 R->L }
% 29.51/4.17 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 29.51/4.17 = { by lemma 19 }
% 29.51/4.17 complement(X)
% 29.51/4.17
% 29.51/4.17 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 29.51/4.17 Proof:
% 29.51/4.17 join(meet(X, Y), complement(join(complement(X), Y)))
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.17 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 29.51/4.17 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 29.51/4.17 X
% 29.51/4.17
% 29.51/4.17 Lemma 22: join(zero, meet(X, X)) = X.
% 29.51/4.17 Proof:
% 29.51/4.17 join(zero, meet(X, X))
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.17 join(zero, complement(join(complement(X), complement(X))))
% 29.51/4.17 = { by axiom 4 (def_zero) }
% 29.51/4.17 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 29.51/4.17 = { by lemma 21 }
% 29.51/4.17 X
% 29.51/4.17
% 29.51/4.17 Lemma 23: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 29.51/4.17 Proof:
% 29.51/4.17 join(zero, join(X, complement(complement(Y))))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(zero, join(complement(complement(Y)), X))
% 29.51/4.17 = { by lemma 20 R->L }
% 29.51/4.17 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 29.51/4.17 join(zero, join(meet(Y, Y), X))
% 29.51/4.17 = { by axiom 7 (maddux2_join_associativity) }
% 29.51/4.17 join(join(zero, meet(Y, Y)), X)
% 29.51/4.17 = { by lemma 22 }
% 29.51/4.17 join(Y, X)
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.17 join(X, Y)
% 29.51/4.17
% 29.51/4.17 Lemma 24: join(X, join(Y, complement(X))) = join(Y, top).
% 29.51/4.17 Proof:
% 29.51/4.17 join(X, join(Y, complement(X)))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(X, join(complement(X), Y))
% 29.51/4.17 = { by axiom 7 (maddux2_join_associativity) }
% 29.51/4.17 join(join(X, complement(X)), Y)
% 29.51/4.17 = { by axiom 5 (def_top) R->L }
% 29.51/4.17 join(top, Y)
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.17 join(Y, top)
% 29.51/4.17
% 29.51/4.17 Lemma 25: join(top, complement(X)) = top.
% 29.51/4.17 Proof:
% 29.51/4.17 join(top, complement(X))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(complement(X), top)
% 29.51/4.17 = { by lemma 24 R->L }
% 29.51/4.17 join(X, join(complement(X), complement(X)))
% 29.51/4.17 = { by lemma 20 }
% 29.51/4.17 join(X, complement(X))
% 29.51/4.17 = { by axiom 5 (def_top) R->L }
% 29.51/4.17 top
% 29.51/4.17
% 29.51/4.17 Lemma 26: join(Y, top) = join(X, top).
% 29.51/4.17 Proof:
% 29.51/4.17 join(Y, top)
% 29.51/4.17 = { by lemma 25 R->L }
% 29.51/4.17 join(Y, join(top, complement(Y)))
% 29.51/4.17 = { by lemma 24 }
% 29.51/4.17 join(top, top)
% 29.51/4.17 = { by lemma 24 R->L }
% 29.51/4.17 join(X, join(top, complement(X)))
% 29.51/4.17 = { by lemma 25 }
% 29.51/4.17 join(X, top)
% 29.51/4.17
% 29.51/4.17 Lemma 27: join(zero, complement(complement(X))) = X.
% 29.51/4.17 Proof:
% 29.51/4.17 join(zero, complement(complement(X)))
% 29.51/4.17 = { by axiom 4 (def_zero) }
% 29.51/4.17 join(meet(X, complement(X)), complement(complement(X)))
% 29.51/4.17 = { by lemma 20 R->L }
% 29.51/4.17 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 29.51/4.17 = { by lemma 21 }
% 29.51/4.17 X
% 29.51/4.17
% 29.51/4.17 Lemma 28: join(X, zero) = join(X, X).
% 29.51/4.17 Proof:
% 29.51/4.17 join(X, zero)
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(zero, X)
% 29.51/4.17 = { by lemma 27 R->L }
% 29.51/4.17 join(zero, join(zero, complement(complement(X))))
% 29.51/4.17 = { by lemma 20 R->L }
% 29.51/4.17 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 29.51/4.17 = { by lemma 23 }
% 29.51/4.17 join(zero, join(complement(complement(X)), X))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.17 join(zero, join(X, complement(complement(X))))
% 29.51/4.17 = { by lemma 23 }
% 29.51/4.17 join(X, X)
% 29.51/4.17
% 29.51/4.17 Lemma 29: join(zero, complement(X)) = complement(X).
% 29.51/4.17 Proof:
% 29.51/4.17 join(zero, complement(X))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(complement(X), zero)
% 29.51/4.17 = { by lemma 28 }
% 29.51/4.17 join(complement(X), complement(X))
% 29.51/4.17 = { by lemma 20 }
% 29.51/4.17 complement(X)
% 29.51/4.17
% 29.51/4.17 Lemma 30: join(X, zero) = X.
% 29.51/4.17 Proof:
% 29.51/4.17 join(X, zero)
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(zero, X)
% 29.51/4.17 = { by lemma 23 R->L }
% 29.51/4.17 join(zero, join(zero, complement(complement(X))))
% 29.51/4.17 = { by lemma 29 }
% 29.51/4.17 join(zero, complement(complement(X)))
% 29.51/4.17 = { by lemma 27 }
% 29.51/4.17 X
% 29.51/4.17
% 29.51/4.17 Lemma 31: join(X, top) = top.
% 29.51/4.17 Proof:
% 29.51/4.17 join(X, top)
% 29.51/4.17 = { by lemma 26 }
% 29.51/4.17 join(join(zero, zero), top)
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(top, join(zero, zero))
% 29.51/4.17 = { by lemma 30 }
% 29.51/4.17 join(top, zero)
% 29.51/4.17 = { by lemma 30 }
% 29.51/4.17 top
% 29.51/4.17
% 29.51/4.17 Lemma 32: join(X, join(complement(X), Y)) = top.
% 29.51/4.17 Proof:
% 29.51/4.17 join(X, join(complement(X), Y))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(X, join(Y, complement(X)))
% 29.51/4.17 = { by lemma 24 }
% 29.51/4.17 join(Y, top)
% 29.51/4.17 = { by lemma 26 R->L }
% 29.51/4.17 join(Z, top)
% 29.51/4.17 = { by lemma 31 }
% 29.51/4.17 top
% 29.51/4.17
% 29.51/4.17 Lemma 33: join(X, complement(zero)) = top.
% 29.51/4.17 Proof:
% 29.51/4.17 join(X, complement(zero))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(complement(zero), X)
% 29.51/4.17 = { by lemma 23 R->L }
% 29.51/4.17 join(zero, join(complement(zero), complement(complement(X))))
% 29.51/4.17 = { by lemma 32 }
% 29.51/4.17 top
% 29.51/4.17
% 29.51/4.17 Lemma 34: complement(zero) = top.
% 29.51/4.17 Proof:
% 29.51/4.17 complement(zero)
% 29.51/4.17 = { by lemma 20 R->L }
% 29.51/4.17 join(complement(zero), complement(zero))
% 29.51/4.17 = { by lemma 33 }
% 29.51/4.17 top
% 29.51/4.17
% 29.51/4.17 Lemma 35: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 29.51/4.17 Proof:
% 29.51/4.17 converse(join(X, converse(Y)))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 converse(join(converse(Y), X))
% 29.51/4.17 = { by axiom 6 (converse_additivity) }
% 29.51/4.17 join(converse(converse(Y)), converse(X))
% 29.51/4.17 = { by axiom 2 (converse_idempotence) }
% 29.51/4.17 join(Y, converse(X))
% 29.51/4.17
% 29.51/4.17 Lemma 36: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 29.51/4.17 Proof:
% 29.51/4.17 converse(join(converse(X), Y))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 converse(join(Y, converse(X)))
% 29.51/4.17 = { by lemma 35 }
% 29.51/4.17 join(X, converse(Y))
% 29.51/4.17
% 29.51/4.17 Lemma 37: join(X, converse(complement(converse(X)))) = converse(top).
% 29.51/4.17 Proof:
% 29.51/4.17 join(X, converse(complement(converse(X))))
% 29.51/4.17 = { by lemma 36 R->L }
% 29.51/4.17 converse(join(converse(X), complement(converse(X))))
% 29.51/4.17 = { by axiom 5 (def_top) R->L }
% 29.51/4.17 converse(top)
% 29.51/4.17
% 29.51/4.17 Lemma 38: join(X, converse(top)) = top.
% 29.51/4.17 Proof:
% 29.51/4.17 join(X, converse(top))
% 29.51/4.17 = { by lemma 37 R->L }
% 29.51/4.17 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 29.51/4.17 = { by lemma 32 }
% 29.51/4.17 top
% 29.51/4.17
% 29.51/4.17 Lemma 39: converse(top) = top.
% 29.51/4.17 Proof:
% 29.51/4.17 converse(top)
% 29.51/4.17 = { by lemma 31 R->L }
% 29.51/4.17 converse(join(X, top))
% 29.51/4.17 = { by axiom 6 (converse_additivity) }
% 29.51/4.17 join(converse(X), converse(top))
% 29.51/4.17 = { by lemma 38 }
% 29.51/4.17 top
% 29.51/4.17
% 29.51/4.17 Lemma 40: join(zero, X) = X.
% 29.51/4.17 Proof:
% 29.51/4.17 join(zero, X)
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(X, zero)
% 29.51/4.17 = { by lemma 30 }
% 29.51/4.17 X
% 29.51/4.17
% 29.51/4.17 Lemma 41: meet(Y, X) = meet(X, Y).
% 29.51/4.17 Proof:
% 29.51/4.17 meet(Y, X)
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.17 complement(join(complement(Y), complement(X)))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 complement(join(complement(X), complement(Y)))
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 29.51/4.17 meet(X, Y)
% 29.51/4.17
% 29.51/4.17 Lemma 42: complement(join(zero, complement(X))) = meet(X, top).
% 29.51/4.17 Proof:
% 29.51/4.17 complement(join(zero, complement(X)))
% 29.51/4.17 = { by lemma 15 R->L }
% 29.51/4.17 complement(join(complement(top), complement(X)))
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 29.51/4.17 meet(top, X)
% 29.51/4.17 = { by lemma 41 R->L }
% 29.51/4.17 meet(X, top)
% 29.51/4.17
% 29.51/4.17 Lemma 43: meet(X, zero) = zero.
% 29.51/4.17 Proof:
% 29.51/4.17 meet(X, zero)
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.17 complement(join(complement(X), complement(zero)))
% 29.51/4.17 = { by lemma 33 }
% 29.51/4.17 complement(top)
% 29.51/4.17 = { by lemma 15 }
% 29.51/4.17 zero
% 29.51/4.17
% 29.51/4.17 Lemma 44: join(meet(X, Y), meet(X, complement(Y))) = X.
% 29.51/4.17 Proof:
% 29.51/4.17 join(meet(X, Y), meet(X, complement(Y)))
% 29.51/4.17 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.17 join(meet(X, complement(Y)), meet(X, Y))
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.17 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 29.51/4.17 = { by lemma 21 }
% 29.51/4.17 X
% 29.51/4.17
% 29.51/4.17 Lemma 45: meet(X, top) = X.
% 29.51/4.17 Proof:
% 29.51/4.17 meet(X, top)
% 29.51/4.17 = { by lemma 42 R->L }
% 29.51/4.17 complement(join(zero, complement(X)))
% 29.51/4.17 = { by lemma 29 R->L }
% 29.51/4.17 join(zero, complement(join(zero, complement(X))))
% 29.51/4.17 = { by lemma 42 }
% 29.51/4.17 join(zero, meet(X, top))
% 29.51/4.17 = { by lemma 34 R->L }
% 29.51/4.17 join(zero, meet(X, complement(zero)))
% 29.51/4.17 = { by lemma 43 R->L }
% 29.51/4.17 join(meet(X, zero), meet(X, complement(zero)))
% 29.51/4.17 = { by lemma 44 }
% 29.51/4.17 X
% 29.51/4.17
% 29.51/4.17 Lemma 46: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 29.51/4.17 Proof:
% 29.51/4.17 join(meet(X, Y), meet(X, Y))
% 29.51/4.17 = { by lemma 41 }
% 29.51/4.17 join(meet(Y, X), meet(X, Y))
% 29.51/4.17 = { by lemma 41 }
% 29.51/4.17 join(meet(Y, X), meet(Y, X))
% 29.51/4.17 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.17 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 29.51/4.18 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.18 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 29.51/4.18 = { by lemma 20 }
% 29.51/4.18 complement(join(complement(Y), complement(X)))
% 29.51/4.18 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 29.51/4.18 meet(Y, X)
% 29.51/4.18 = { by lemma 41 R->L }
% 29.51/4.18 meet(X, Y)
% 29.51/4.18
% 29.51/4.18 Lemma 47: converse(zero) = zero.
% 29.51/4.18 Proof:
% 29.51/4.18 converse(zero)
% 29.51/4.18 = { by lemma 40 R->L }
% 29.51/4.18 join(zero, converse(zero))
% 29.51/4.18 = { by lemma 36 R->L }
% 29.51/4.18 converse(join(converse(zero), zero))
% 29.51/4.18 = { by lemma 28 }
% 29.51/4.18 converse(join(converse(zero), converse(zero)))
% 29.51/4.18 = { by lemma 35 }
% 29.51/4.18 join(zero, converse(converse(zero)))
% 29.51/4.18 = { by axiom 2 (converse_idempotence) }
% 29.51/4.18 join(zero, zero)
% 29.51/4.18 = { by lemma 45 R->L }
% 29.51/4.18 join(zero, meet(zero, top))
% 29.51/4.18 = { by lemma 45 R->L }
% 29.51/4.18 join(meet(zero, top), meet(zero, top))
% 29.51/4.18 = { by lemma 46 }
% 29.51/4.18 meet(zero, top)
% 29.51/4.18 = { by lemma 45 }
% 29.51/4.18 zero
% 29.51/4.18
% 29.51/4.18 Lemma 48: join(top, X) = top.
% 29.51/4.18 Proof:
% 29.51/4.18 join(top, X)
% 29.51/4.18 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.18 join(X, top)
% 29.51/4.18 = { by lemma 26 R->L }
% 29.51/4.18 join(Y, top)
% 29.51/4.18 = { by lemma 31 }
% 29.51/4.18 top
% 29.51/4.18
% 29.51/4.18 Lemma 49: complement(complement(X)) = X.
% 29.51/4.18 Proof:
% 29.51/4.18 complement(complement(X))
% 29.51/4.18 = { by lemma 29 R->L }
% 29.51/4.18 join(zero, complement(complement(X)))
% 29.51/4.18 = { by lemma 27 }
% 29.51/4.18 X
% 29.51/4.18
% 29.51/4.18 Lemma 50: meet(zero, X) = zero.
% 29.51/4.18 Proof:
% 29.51/4.18 meet(zero, X)
% 29.51/4.18 = { by lemma 41 }
% 29.51/4.18 meet(X, zero)
% 29.51/4.18 = { by lemma 43 }
% 29.51/4.18 zero
% 29.51/4.18
% 29.51/4.18 Lemma 51: composition(top, zero) = zero.
% 29.51/4.18 Proof:
% 29.51/4.18 composition(top, zero)
% 29.51/4.18 = { by lemma 39 R->L }
% 29.51/4.18 composition(converse(top), zero)
% 29.51/4.18 = { by lemma 40 R->L }
% 29.51/4.18 join(zero, composition(converse(top), zero))
% 29.51/4.18 = { by lemma 15 R->L }
% 29.51/4.18 join(complement(top), composition(converse(top), zero))
% 29.51/4.18 = { by lemma 15 R->L }
% 29.51/4.18 join(complement(top), composition(converse(top), complement(top)))
% 29.51/4.18 = { by lemma 48 R->L }
% 29.51/4.18 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 29.51/4.18 = { by lemma 39 R->L }
% 29.51/4.18 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 29.51/4.18 = { by lemma 18 R->L }
% 29.51/4.18 join(complement(top), composition(converse(top), complement(join(composition(one, top), composition(converse(top), top)))))
% 29.51/4.18 = { by axiom 11 (composition_distributivity) R->L }
% 29.51/4.18 join(complement(top), composition(converse(top), complement(composition(join(one, converse(top)), top))))
% 29.51/4.18 = { by lemma 38 }
% 29.51/4.18 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 29.51/4.18 = { by lemma 19 }
% 29.51/4.18 complement(top)
% 29.51/4.18 = { by lemma 15 }
% 29.51/4.18 zero
% 29.51/4.18
% 29.51/4.18 Lemma 52: composition(X, zero) = zero.
% 29.51/4.18 Proof:
% 29.51/4.18 composition(X, zero)
% 29.51/4.18 = { by lemma 40 R->L }
% 29.51/4.18 join(zero, composition(X, zero))
% 29.51/4.18 = { by lemma 51 R->L }
% 29.51/4.18 join(composition(top, zero), composition(X, zero))
% 29.51/4.18 = { by axiom 11 (composition_distributivity) R->L }
% 29.51/4.18 composition(join(top, X), zero)
% 29.51/4.18 = { by lemma 48 }
% 29.51/4.18 composition(top, zero)
% 29.51/4.18 = { by lemma 51 }
% 29.51/4.18 zero
% 29.51/4.18
% 29.51/4.18 Lemma 53: composition(zero, X) = zero.
% 29.51/4.18 Proof:
% 29.51/4.18 composition(zero, X)
% 29.51/4.18 = { by lemma 47 R->L }
% 29.51/4.18 composition(converse(zero), X)
% 29.51/4.18 = { by lemma 16 R->L }
% 29.51/4.18 converse(composition(converse(X), zero))
% 29.51/4.18 = { by lemma 52 }
% 29.51/4.18 converse(zero)
% 29.51/4.18 = { by lemma 47 }
% 29.51/4.18 zero
% 29.51/4.18
% 29.51/4.18 Lemma 54: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 29.51/4.18 Proof:
% 29.51/4.18 join(meet(X, Y), meet(Y, complement(X)))
% 29.51/4.18 = { by lemma 41 }
% 29.51/4.18 join(meet(Y, X), meet(Y, complement(X)))
% 29.51/4.18 = { by lemma 44 }
% 29.51/4.18 Y
% 29.51/4.18
% 29.51/4.18 Lemma 55: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 29.51/4.18 Proof:
% 29.51/4.18 join(meet(X, Y), meet(complement(X), Y))
% 29.51/4.18 = { by lemma 41 }
% 29.51/4.18 join(meet(X, Y), meet(Y, complement(X)))
% 29.51/4.18 = { by lemma 54 }
% 29.51/4.18 Y
% 29.51/4.18
% 29.51/4.18 Lemma 56: join(complement(one), composition(converse(complement(X)), X)) = complement(one).
% 29.51/4.18 Proof:
% 29.51/4.18 join(complement(one), composition(converse(complement(X)), X))
% 29.51/4.18 = { by lemma 45 R->L }
% 29.51/4.18 join(complement(one), composition(converse(complement(X)), meet(X, top)))
% 29.51/4.18 = { by lemma 29 R->L }
% 29.51/4.18 join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))
% 29.51/4.18 = { by lemma 42 R->L }
% 29.51/4.18 join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))
% 29.51/4.18 = { by axiom 3 (composition_identity) R->L }
% 29.51/4.18 join(complement(one), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), one))))
% 29.51/4.18 = { by lemma 19 }
% 29.51/4.18 complement(one)
% 29.51/4.18
% 29.51/4.18 Lemma 57: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 29.51/4.18 Proof:
% 29.51/4.18 meet(X, join(complement(Y), complement(Z)))
% 29.51/4.18 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.18 meet(X, join(complement(Z), complement(Y)))
% 29.51/4.18 = { by lemma 41 }
% 29.51/4.18 meet(join(complement(Z), complement(Y)), X)
% 29.51/4.18 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.18 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 29.51/4.18 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 29.51/4.18 complement(join(meet(Z, Y), complement(X)))
% 29.51/4.18 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.18 complement(join(complement(X), meet(Z, Y)))
% 29.51/4.18 = { by lemma 41 R->L }
% 29.51/4.18 complement(join(complement(X), meet(Y, Z)))
% 29.51/4.18
% 29.51/4.18 Lemma 58: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 29.51/4.18 Proof:
% 29.51/4.18 complement(join(X, complement(Y)))
% 29.51/4.18 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.18 complement(join(complement(Y), X))
% 29.51/4.18 = { by lemma 45 R->L }
% 29.51/4.18 complement(join(complement(Y), meet(X, top)))
% 29.51/4.18 = { by lemma 41 R->L }
% 29.51/4.18 complement(join(complement(Y), meet(top, X)))
% 29.51/4.18 = { by lemma 57 R->L }
% 29.51/4.18 meet(Y, join(complement(top), complement(X)))
% 29.51/4.18 = { by lemma 15 }
% 29.51/4.18 meet(Y, join(zero, complement(X)))
% 29.51/4.18 = { by lemma 29 }
% 29.51/4.18 meet(Y, complement(X))
% 29.51/4.18
% 29.51/4.18 Lemma 59: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 29.51/4.18 Proof:
% 29.51/4.18 complement(join(complement(X), Y))
% 29.51/4.18 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.18 complement(join(Y, complement(X)))
% 29.51/4.18 = { by lemma 58 }
% 29.51/4.18 meet(X, complement(Y))
% 29.51/4.18
% 29.51/4.18 Lemma 60: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 29.51/4.18 Proof:
% 29.51/4.18 join(complement(X), complement(Y))
% 29.51/4.18 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.18 join(complement(Y), complement(X))
% 29.51/4.18 = { by lemma 22 R->L }
% 29.51/4.18 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 29.51/4.18 = { by lemma 57 }
% 29.51/4.18 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 29.51/4.18 = { by lemma 29 }
% 29.51/4.18 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 29.51/4.18 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 29.51/4.18 complement(join(meet(Y, X), meet(Y, X)))
% 29.51/4.18 = { by lemma 46 }
% 29.51/4.18 complement(meet(Y, X))
% 29.51/4.18 = { by lemma 41 R->L }
% 29.51/4.18 complement(meet(X, Y))
% 29.51/4.18
% 29.51/4.18 Lemma 61: join(X, complement(meet(X, Y))) = top.
% 29.51/4.18 Proof:
% 29.51/4.18 join(X, complement(meet(X, Y)))
% 29.51/4.18 = { by lemma 41 }
% 29.51/4.18 join(X, complement(meet(Y, X)))
% 29.51/4.18 = { by lemma 60 R->L }
% 29.51/4.18 join(X, join(complement(Y), complement(X)))
% 29.51/4.18 = { by lemma 24 }
% 29.51/4.18 join(complement(Y), top)
% 29.51/4.18 = { by lemma 31 }
% 29.51/4.18 top
% 29.51/4.18
% 29.51/4.18 Lemma 62: meet(X, meet(Y, complement(X))) = zero.
% 29.51/4.18 Proof:
% 29.51/4.18 meet(X, meet(Y, complement(X)))
% 29.51/4.18 = { by lemma 41 }
% 29.51/4.18 meet(X, meet(complement(X), Y))
% 29.51/4.18 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.18 complement(join(complement(X), complement(meet(complement(X), Y))))
% 29.51/4.18 = { by lemma 61 }
% 29.51/4.18 complement(top)
% 29.51/4.18 = { by lemma 15 }
% 29.51/4.18 zero
% 29.51/4.18
% 29.51/4.18 Lemma 63: meet(one, composition(converse(complement(X)), X)) = zero.
% 29.51/4.18 Proof:
% 29.51/4.18 meet(one, composition(converse(complement(X)), X))
% 29.51/4.18 = { by lemma 41 }
% 29.51/4.18 meet(composition(converse(complement(X)), X), one)
% 29.51/4.18 = { by lemma 49 R->L }
% 29.51/4.18 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 29.51/4.18 = { by lemma 56 R->L }
% 29.51/4.18 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 29.51/4.18 = { by lemma 59 }
% 29.51/4.18 meet(composition(converse(complement(X)), X), meet(one, complement(composition(converse(complement(X)), X))))
% 29.51/4.18 = { by lemma 62 }
% 29.51/4.18 zero
% 29.51/4.18
% 29.51/4.18 Lemma 64: complement(converse(complement(X))) = converse(X).
% 29.51/4.18 Proof:
% 29.51/4.18 complement(converse(complement(X)))
% 29.51/4.18 = { by axiom 2 (converse_idempotence) R->L }
% 29.51/4.18 converse(converse(complement(converse(complement(X)))))
% 29.51/4.18 = { by lemma 55 R->L }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(converse(complement(X)))))))
% 29.51/4.18 = { by lemma 30 R->L }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), join(meet(complement(X), converse(complement(converse(complement(X))))), zero)))
% 29.51/4.18 = { by lemma 50 R->L }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), join(meet(complement(X), converse(complement(converse(complement(X))))), meet(zero, converse(complement(converse(complement(X))))))))
% 29.51/4.18 = { by lemma 53 R->L }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), join(meet(complement(X), converse(complement(converse(complement(X))))), meet(composition(zero, complement(X)), converse(complement(converse(complement(X))))))))
% 29.51/4.18 = { by lemma 18 R->L }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), join(meet(composition(one, complement(X)), converse(complement(converse(complement(X))))), meet(composition(zero, complement(X)), converse(complement(converse(complement(X))))))))
% 29.51/4.18 = { by lemma 63 R->L }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), join(meet(composition(one, complement(X)), converse(complement(converse(complement(X))))), meet(composition(meet(one, composition(converse(complement(converse(complement(X)))), converse(complement(X)))), complement(X)), converse(complement(converse(complement(X))))))))
% 29.51/4.18 = { by axiom 14 (modular_law_2) }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(composition(meet(one, composition(converse(complement(converse(complement(X)))), converse(complement(X)))), complement(X)), converse(complement(converse(complement(X)))))))
% 29.51/4.18 = { by lemma 63 }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(composition(zero, complement(X)), converse(complement(converse(complement(X)))))))
% 29.51/4.18 = { by lemma 53 }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), meet(zero, converse(complement(converse(complement(X)))))))
% 29.51/4.18 = { by lemma 50 }
% 29.51/4.18 converse(join(meet(X, converse(complement(converse(complement(X))))), zero))
% 29.51/4.18 = { by lemma 30 }
% 29.51/4.18 converse(meet(X, converse(complement(converse(complement(X))))))
% 29.51/4.18 = { by axiom 10 (maddux4_definiton_of_meet) }
% 29.51/4.18 converse(complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))
% 29.51/4.18 = { by lemma 29 R->L }
% 29.51/4.18 converse(join(zero, complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))))
% 29.51/4.18 = { by lemma 15 R->L }
% 29.51/4.18 converse(join(complement(top), complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))))
% 29.51/4.18 = { by lemma 39 R->L }
% 29.51/4.18 converse(join(complement(converse(top)), complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))))
% 29.51/4.18 = { by lemma 37 R->L }
% 29.51/4.18 converse(join(complement(join(complement(X), converse(complement(converse(complement(X)))))), complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))))
% 29.51/4.18 = { by lemma 59 }
% 29.51/4.18 converse(join(meet(X, complement(converse(complement(converse(complement(X)))))), complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))))
% 29.51/4.18 = { by lemma 21 }
% 29.51/4.18 converse(X)
% 29.51/4.18
% 29.51/4.18 Lemma 65: converse(complement(X)) = complement(converse(X)).
% 29.51/4.18 Proof:
% 29.51/4.18 converse(complement(X))
% 29.51/4.18 = { by lemma 29 R->L }
% 29.51/4.18 converse(join(zero, complement(X)))
% 29.51/4.18 = { by lemma 64 R->L }
% 29.51/4.18 complement(converse(complement(join(zero, complement(X)))))
% 29.51/4.18 = { by lemma 42 }
% 29.51/4.18 complement(converse(meet(X, top)))
% 29.51/4.18 = { by lemma 45 }
% 29.51/4.18 complement(converse(X))
% 29.51/4.18
% 29.51/4.18 Lemma 66: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 29.51/4.18 Proof:
% 29.51/4.18 complement(meet(X, complement(Y)))
% 29.51/4.18 = { by lemma 40 R->L }
% 29.51/4.18 complement(join(zero, meet(X, complement(Y))))
% 29.51/4.18 = { by lemma 58 R->L }
% 29.51/4.18 complement(join(zero, complement(join(Y, complement(X)))))
% 29.51/4.18 = { by lemma 42 }
% 29.51/4.18 meet(join(Y, complement(X)), top)
% 29.51/4.18 = { by lemma 45 }
% 29.51/4.18 join(Y, complement(X))
% 29.51/4.18
% 29.51/4.18 Lemma 67: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 29.51/4.18 Proof:
% 29.51/4.18 complement(meet(complement(X), Y))
% 29.51/4.18 = { by lemma 41 }
% 29.51/4.18 complement(meet(Y, complement(X)))
% 29.51/4.18 = { by lemma 66 }
% 29.51/4.18 join(X, complement(Y))
% 29.51/4.18
% 29.51/4.18 Lemma 68: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 29.51/4.18 Proof:
% 29.51/4.18 meet(complement(X), complement(Y))
% 29.51/4.18 = { by lemma 41 }
% 29.51/4.18 meet(complement(Y), complement(X))
% 29.51/4.18 = { by lemma 29 R->L }
% 29.51/4.18 meet(join(zero, complement(Y)), complement(X))
% 29.51/4.18 = { by lemma 58 R->L }
% 29.51/4.18 complement(join(X, complement(join(zero, complement(Y)))))
% 29.51/4.18 = { by lemma 42 }
% 29.51/4.18 complement(join(X, meet(Y, top)))
% 29.51/4.18 = { by lemma 45 }
% 29.51/4.18 complement(join(X, Y))
% 29.51/4.18
% 29.51/4.18 Lemma 69: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 29.51/4.19 Proof:
% 29.51/4.19 converse(composition(X, converse(Y)))
% 29.51/4.19 = { by axiom 8 (converse_multiplicativity) }
% 29.51/4.19 composition(converse(converse(Y)), converse(X))
% 29.51/4.19 = { by axiom 2 (converse_idempotence) }
% 29.51/4.19 composition(Y, converse(X))
% 29.51/4.19
% 29.51/4.19 Lemma 70: join(complement(X), meet(Y, complement(Z))) = complement(meet(X, join(Z, complement(Y)))).
% 29.51/4.19 Proof:
% 29.51/4.19 join(complement(X), meet(Y, complement(Z)))
% 29.51/4.19 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.19 join(meet(Y, complement(Z)), complement(X))
% 29.51/4.19 = { by lemma 58 R->L }
% 29.51/4.19 join(complement(join(Z, complement(Y))), complement(X))
% 29.51/4.19 = { by lemma 60 }
% 29.51/4.19 complement(meet(join(Z, complement(Y)), X))
% 29.51/4.19 = { by lemma 41 R->L }
% 29.51/4.19 complement(meet(X, join(Z, complement(Y))))
% 29.51/4.19
% 29.51/4.19 Lemma 71: complement(meet(Y, join(X, complement(Y)))) = complement(meet(X, join(Y, complement(X)))).
% 29.51/4.19 Proof:
% 29.51/4.19 complement(meet(Y, join(X, complement(Y))))
% 29.51/4.19 = { by lemma 70 R->L }
% 29.51/4.19 join(complement(Y), meet(Y, complement(X)))
% 29.51/4.19 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.19 join(meet(Y, complement(X)), complement(Y))
% 29.51/4.19 = { by lemma 55 R->L }
% 29.51/4.19 join(meet(Y, complement(X)), join(meet(X, complement(Y)), meet(complement(X), complement(Y))))
% 29.51/4.19 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.19 join(meet(Y, complement(X)), join(meet(complement(X), complement(Y)), meet(X, complement(Y))))
% 29.51/4.19 = { by axiom 7 (maddux2_join_associativity) }
% 29.51/4.19 join(join(meet(Y, complement(X)), meet(complement(X), complement(Y))), meet(X, complement(Y)))
% 29.51/4.19 = { by lemma 54 }
% 29.51/4.19 join(complement(X), meet(X, complement(Y)))
% 29.51/4.19 = { by lemma 70 }
% 29.51/4.19 complement(meet(X, join(Y, complement(X))))
% 29.51/4.19
% 29.51/4.19 Lemma 72: join(complement(Y), meet(Y, X)) = join(X, complement(join(X, Y))).
% 29.51/4.19 Proof:
% 29.51/4.19 join(complement(Y), meet(Y, X))
% 29.51/4.19 = { by lemma 41 }
% 29.51/4.19 join(complement(Y), meet(X, Y))
% 29.51/4.19 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.19 join(meet(X, Y), complement(Y))
% 29.51/4.19 = { by lemma 66 R->L }
% 29.51/4.19 complement(meet(Y, complement(meet(X, Y))))
% 29.51/4.19 = { by lemma 60 R->L }
% 29.51/4.19 complement(meet(Y, join(complement(X), complement(Y))))
% 29.51/4.19 = { by lemma 71 R->L }
% 29.51/4.19 complement(meet(complement(X), join(Y, complement(complement(X)))))
% 29.51/4.19 = { by lemma 67 }
% 29.51/4.19 join(X, complement(join(Y, complement(complement(X)))))
% 29.51/4.19 = { by lemma 58 }
% 29.51/4.19 join(X, meet(complement(X), complement(Y)))
% 29.51/4.19 = { by lemma 68 }
% 29.51/4.19 join(X, complement(join(X, Y)))
% 29.51/4.19
% 29.51/4.19 Lemma 73: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 29.51/4.19 Proof:
% 29.51/4.19 join(X, complement(join(X, Y)))
% 29.51/4.19 = { by lemma 68 R->L }
% 29.51/4.19 join(X, meet(complement(X), complement(Y)))
% 29.51/4.19 = { by lemma 58 R->L }
% 29.51/4.19 join(X, complement(join(Y, complement(complement(X)))))
% 29.51/4.19 = { by lemma 67 R->L }
% 29.51/4.19 complement(meet(complement(X), join(Y, complement(complement(X)))))
% 29.51/4.19 = { by lemma 71 R->L }
% 29.51/4.19 complement(meet(Y, join(complement(X), complement(Y))))
% 29.51/4.19 = { by lemma 60 }
% 29.51/4.19 complement(meet(Y, complement(meet(X, Y))))
% 29.51/4.19 = { by lemma 66 }
% 29.51/4.19 join(meet(X, Y), complement(Y))
% 29.51/4.19 = { by lemma 55 R->L }
% 29.51/4.19 join(meet(X, Y), complement(join(meet(X, Y), meet(complement(X), Y))))
% 29.51/4.19 = { by lemma 72 R->L }
% 29.51/4.19 join(complement(meet(complement(X), Y)), meet(meet(complement(X), Y), meet(X, Y)))
% 29.51/4.19 = { by lemma 67 }
% 29.51/4.19 join(join(X, complement(Y)), meet(meet(complement(X), Y), meet(X, Y)))
% 29.51/4.19 = { by axiom 7 (maddux2_join_associativity) R->L }
% 29.51/4.19 join(X, join(complement(Y), meet(meet(complement(X), Y), meet(X, Y))))
% 29.51/4.19 = { by lemma 41 }
% 29.51/4.19 join(X, join(complement(Y), meet(meet(X, Y), meet(complement(X), Y))))
% 29.51/4.19 = { by lemma 41 }
% 29.51/4.19 join(X, join(complement(Y), meet(meet(X, Y), meet(Y, complement(X)))))
% 29.51/4.19 = { by lemma 45 R->L }
% 29.51/4.19 join(X, join(complement(Y), meet(meet(X, Y), meet(Y, meet(complement(X), top)))))
% 29.51/4.19 = { by lemma 42 R->L }
% 29.51/4.19 join(X, join(complement(Y), meet(meet(X, Y), meet(Y, complement(join(zero, complement(complement(X))))))))
% 29.51/4.19 = { by lemma 59 R->L }
% 29.51/4.19 join(X, join(complement(Y), meet(meet(X, Y), complement(join(complement(Y), join(zero, complement(complement(X))))))))
% 29.51/4.19 = { by lemma 59 R->L }
% 29.51/4.19 join(X, join(complement(Y), complement(join(complement(meet(X, Y)), join(complement(Y), join(zero, complement(complement(X))))))))
% 29.51/4.19 = { by axiom 7 (maddux2_join_associativity) }
% 29.51/4.19 join(X, join(complement(Y), complement(join(join(complement(meet(X, Y)), complement(Y)), join(zero, complement(complement(X)))))))
% 29.51/4.19 = { by lemma 68 R->L }
% 29.51/4.19 join(X, join(complement(Y), meet(complement(join(complement(meet(X, Y)), complement(Y))), complement(join(zero, complement(complement(X)))))))
% 29.51/4.19 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 29.51/4.19 join(X, join(complement(Y), meet(meet(meet(X, Y), Y), complement(join(zero, complement(complement(X)))))))
% 29.51/4.19 = { by lemma 41 R->L }
% 29.51/4.19 join(X, join(complement(Y), meet(complement(join(zero, complement(complement(X)))), meet(meet(X, Y), Y))))
% 29.51/4.19 = { by lemma 41 R->L }
% 29.51/4.19 join(X, join(complement(Y), meet(complement(join(zero, complement(complement(X)))), meet(Y, meet(X, Y)))))
% 29.51/4.19 = { by lemma 42 }
% 29.51/4.19 join(X, join(complement(Y), meet(meet(complement(X), top), meet(Y, meet(X, Y)))))
% 29.51/4.19 = { by lemma 45 }
% 29.51/4.19 join(X, join(complement(Y), meet(complement(X), meet(Y, meet(X, Y)))))
% 29.51/4.19 = { by lemma 41 }
% 29.51/4.19 join(X, join(complement(Y), meet(meet(Y, meet(X, Y)), complement(X))))
% 29.51/4.19 = { by lemma 70 }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(meet(Y, meet(X, Y)))))))
% 29.51/4.19 = { by lemma 41 }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(meet(Y, meet(Y, X)))))))
% 29.51/4.19 = { by lemma 41 }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(meet(meet(Y, X), Y))))))
% 29.51/4.19 = { by lemma 21 R->L }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(meet(meet(Y, X), join(meet(Y, X), complement(join(complement(Y), X)))))))))
% 29.51/4.19 = { by lemma 66 R->L }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(meet(meet(Y, X), complement(meet(join(complement(Y), X), complement(meet(Y, X))))))))))
% 29.51/4.19 = { by lemma 59 R->L }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(complement(join(complement(meet(Y, X)), meet(join(complement(Y), X), complement(meet(Y, X))))))))))
% 29.51/4.19 = { by lemma 29 R->L }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(join(zero, complement(join(complement(meet(Y, X)), meet(join(complement(Y), X), complement(meet(Y, X)))))))))))
% 29.51/4.19 = { by lemma 62 R->L }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(join(meet(meet(Y, X), meet(join(complement(Y), X), complement(meet(Y, X)))), complement(join(complement(meet(Y, X)), meet(join(complement(Y), X), complement(meet(Y, X)))))))))))
% 29.51/4.19 = { by lemma 21 }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(meet(Y, X))))))
% 29.51/4.19 = { by lemma 41 }
% 29.51/4.19 join(X, complement(meet(Y, join(X, complement(meet(X, Y))))))
% 29.51/4.19 = { by lemma 61 }
% 29.51/4.19 join(X, complement(meet(Y, top)))
% 29.51/4.19 = { by lemma 45 }
% 29.51/4.19 join(X, complement(Y))
% 29.51/4.19
% 29.51/4.19 Lemma 74: meet(X, join(Y, complement(X))) = meet(X, Y).
% 29.51/4.19 Proof:
% 29.51/4.19 meet(X, join(Y, complement(X)))
% 29.51/4.19 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.19 meet(X, join(complement(X), Y))
% 29.51/4.19 = { by lemma 49 R->L }
% 29.51/4.19 meet(X, complement(complement(join(complement(X), Y))))
% 29.51/4.19 = { by lemma 59 R->L }
% 29.51/4.19 complement(join(complement(X), complement(join(complement(X), Y))))
% 29.51/4.19 = { by lemma 73 }
% 29.51/4.19 complement(join(complement(X), complement(Y)))
% 29.51/4.19 = { by lemma 59 }
% 29.51/4.19 meet(X, complement(complement(Y)))
% 29.51/4.19 = { by lemma 49 }
% 29.51/4.19 meet(X, Y)
% 29.51/4.19
% 29.51/4.19 Lemma 75: composition(converse(X), composition(Y, converse(Z))) = converse(composition(composition(Z, converse(Y)), X)).
% 29.51/4.19 Proof:
% 29.51/4.19 composition(converse(X), composition(Y, converse(Z)))
% 29.51/4.19 = { by axiom 9 (composition_associativity) }
% 29.51/4.19 composition(composition(converse(X), Y), converse(Z))
% 29.51/4.19 = { by lemma 16 R->L }
% 29.51/4.19 composition(converse(composition(converse(Y), X)), converse(Z))
% 29.51/4.19 = { by axiom 8 (converse_multiplicativity) R->L }
% 29.51/4.19 converse(composition(Z, composition(converse(Y), X)))
% 29.51/4.19 = { by axiom 9 (composition_associativity) }
% 29.51/4.19 converse(composition(composition(Z, converse(Y)), X))
% 29.51/4.19
% 29.51/4.19 Lemma 76: composition(converse(X), complement(composition(X, top))) = zero.
% 29.51/4.19 Proof:
% 29.51/4.19 composition(converse(X), complement(composition(X, top)))
% 29.51/4.19 = { by lemma 40 R->L }
% 29.51/4.19 join(zero, composition(converse(X), complement(composition(X, top))))
% 29.51/4.19 = { by lemma 15 R->L }
% 29.51/4.19 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 29.51/4.19 = { by lemma 19 }
% 29.51/4.19 complement(top)
% 29.51/4.19 = { by lemma 15 }
% 29.51/4.19 zero
% 29.51/4.19
% 29.51/4.19 Lemma 77: composition(X, complement(composition(converse(X), top))) = zero.
% 29.51/4.19 Proof:
% 29.51/4.19 composition(X, complement(composition(converse(X), top)))
% 29.51/4.19 = { by axiom 2 (converse_idempotence) R->L }
% 29.51/4.19 composition(converse(converse(X)), complement(composition(converse(X), top)))
% 29.51/4.19 = { by lemma 76 }
% 29.51/4.19 zero
% 29.51/4.19
% 29.51/4.19 Lemma 78: composition(top, join(X, composition(Y, X))) = converse(composition(converse(X), top)).
% 29.51/4.19 Proof:
% 29.51/4.19 composition(top, join(X, composition(Y, X)))
% 29.51/4.19 = { by axiom 2 (converse_idempotence) R->L }
% 29.51/4.19 converse(converse(composition(top, join(X, composition(Y, X)))))
% 29.51/4.19 = { by axiom 8 (converse_multiplicativity) }
% 29.51/4.19 converse(composition(converse(join(X, composition(Y, X))), converse(top)))
% 29.51/4.19 = { by lemma 39 }
% 29.51/4.19 converse(composition(converse(join(X, composition(Y, X))), top))
% 29.51/4.19 = { by axiom 6 (converse_additivity) }
% 29.51/4.19 converse(composition(join(converse(X), converse(composition(Y, X))), top))
% 29.51/4.19 = { by axiom 11 (composition_distributivity) }
% 29.51/4.19 converse(join(composition(converse(X), top), composition(converse(composition(Y, X)), top)))
% 29.51/4.19 = { by lemma 34 R->L }
% 29.51/4.19 converse(join(composition(converse(X), top), composition(converse(composition(Y, X)), complement(zero))))
% 29.51/4.19 = { by lemma 49 R->L }
% 29.51/4.19 converse(join(complement(complement(composition(converse(X), top))), composition(converse(composition(Y, X)), complement(zero))))
% 29.51/4.19 = { by lemma 52 R->L }
% 29.51/4.19 converse(join(complement(complement(composition(converse(X), top))), composition(converse(composition(Y, X)), complement(composition(Y, zero)))))
% 29.51/4.19 = { by lemma 77 R->L }
% 29.51/4.19 converse(join(complement(complement(composition(converse(X), top))), composition(converse(composition(Y, X)), complement(composition(Y, composition(X, complement(composition(converse(X), top))))))))
% 29.51/4.19 = { by axiom 9 (composition_associativity) }
% 29.51/4.19 converse(join(complement(complement(composition(converse(X), top))), composition(converse(composition(Y, X)), complement(composition(composition(Y, X), complement(composition(converse(X), top)))))))
% 29.51/4.19 = { by lemma 19 }
% 29.51/4.19 converse(complement(complement(composition(converse(X), top))))
% 29.51/4.19 = { by lemma 49 }
% 29.51/4.19 converse(composition(converse(X), top))
% 29.51/4.19
% 29.51/4.19 Lemma 79: composition(top, composition(X, converse(X))) = composition(top, converse(X)).
% 29.51/4.19 Proof:
% 29.51/4.19 composition(top, composition(X, converse(X)))
% 29.51/4.19 = { by lemma 39 R->L }
% 29.51/4.19 composition(converse(top), composition(X, converse(X)))
% 29.51/4.19 = { by lemma 75 }
% 29.51/4.19 converse(composition(composition(X, converse(X)), top))
% 29.51/4.19 = { by axiom 9 (composition_associativity) R->L }
% 29.51/4.19 converse(composition(X, composition(converse(X), top)))
% 29.51/4.19 = { by axiom 8 (converse_multiplicativity) }
% 29.51/4.19 composition(converse(composition(converse(X), top)), converse(X))
% 29.51/4.19 = { by lemma 78 R->L }
% 29.51/4.19 composition(composition(top, join(X, composition(Y, X))), converse(X))
% 29.51/4.19 = { by lemma 40 R->L }
% 29.51/4.19 join(zero, composition(composition(top, join(X, composition(Y, X))), converse(X)))
% 29.51/4.19 = { by lemma 47 R->L }
% 29.51/4.19 join(converse(zero), composition(composition(top, join(X, composition(Y, X))), converse(X)))
% 29.51/4.19 = { by lemma 77 R->L }
% 29.51/4.19 join(converse(composition(X, complement(composition(converse(X), top)))), composition(composition(top, join(X, composition(Y, X))), converse(X)))
% 29.51/4.19 = { by axiom 8 (converse_multiplicativity) }
% 29.51/4.19 join(composition(converse(complement(composition(converse(X), top))), converse(X)), composition(composition(top, join(X, composition(Y, X))), converse(X)))
% 29.51/4.19 = { by axiom 11 (composition_distributivity) R->L }
% 29.51/4.19 composition(join(converse(complement(composition(converse(X), top))), composition(top, join(X, composition(Y, X)))), converse(X))
% 29.51/4.19 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.19 composition(join(composition(top, join(X, composition(Y, X))), converse(complement(composition(converse(X), top)))), converse(X))
% 29.51/4.19 = { by lemma 65 }
% 29.51/4.19 composition(join(composition(top, join(X, composition(Y, X))), complement(converse(composition(converse(X), top)))), converse(X))
% 29.51/4.19 = { by lemma 78 R->L }
% 29.51/4.19 composition(join(composition(top, join(X, composition(Y, X))), complement(composition(top, join(X, composition(Y, X))))), converse(X))
% 29.51/4.19 = { by axiom 5 (def_top) R->L }
% 29.51/4.19 composition(top, converse(X))
% 29.51/4.19
% 29.51/4.19 Lemma 80: composition(join(X, complement(composition(top, composition(Y, converse(Y))))), Y) = composition(X, Y).
% 29.51/4.19 Proof:
% 29.51/4.19 composition(join(X, complement(composition(top, composition(Y, converse(Y))))), Y)
% 29.51/4.19 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.19 composition(join(complement(composition(top, composition(Y, converse(Y)))), X), Y)
% 29.51/4.19 = { by lemma 79 }
% 29.51/4.19 composition(join(complement(composition(top, converse(Y))), X), Y)
% 29.51/4.19 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.19 composition(join(X, complement(composition(top, converse(Y)))), Y)
% 29.51/4.19 = { by lemma 39 R->L }
% 29.51/4.19 composition(join(X, complement(composition(converse(top), converse(Y)))), Y)
% 29.51/4.19 = { by axiom 8 (converse_multiplicativity) R->L }
% 29.51/4.19 composition(join(X, complement(converse(composition(Y, top)))), Y)
% 29.51/4.19 = { by lemma 65 R->L }
% 29.51/4.19 composition(join(X, converse(complement(composition(Y, top)))), Y)
% 29.51/4.19 = { by lemma 35 R->L }
% 29.51/4.19 composition(converse(join(complement(composition(Y, top)), converse(X))), Y)
% 29.51/4.19 = { by lemma 16 R->L }
% 29.51/4.19 converse(composition(converse(Y), join(complement(composition(Y, top)), converse(X))))
% 29.51/4.19 = { by axiom 8 (converse_multiplicativity) }
% 29.51/4.19 composition(converse(join(complement(composition(Y, top)), converse(X))), converse(converse(Y)))
% 29.51/4.19 = { by lemma 35 }
% 29.51/4.19 composition(join(X, converse(complement(composition(Y, top)))), converse(converse(Y)))
% 29.51/4.19 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.19 composition(join(converse(complement(composition(Y, top))), X), converse(converse(Y)))
% 29.51/4.19 = { by axiom 11 (composition_distributivity) }
% 29.51/4.19 join(composition(converse(complement(composition(Y, top))), converse(converse(Y))), composition(X, converse(converse(Y))))
% 29.51/4.19 = { by axiom 8 (converse_multiplicativity) R->L }
% 29.51/4.19 join(converse(composition(converse(Y), complement(composition(Y, top)))), composition(X, converse(converse(Y))))
% 29.51/4.19 = { by lemma 76 }
% 29.51/4.19 join(converse(zero), composition(X, converse(converse(Y))))
% 29.51/4.19 = { by lemma 47 }
% 29.51/4.20 join(zero, composition(X, converse(converse(Y))))
% 29.51/4.20 = { by lemma 40 }
% 29.51/4.20 composition(X, converse(converse(Y)))
% 29.51/4.20 = { by axiom 2 (converse_idempotence) }
% 29.51/4.20 composition(X, Y)
% 29.51/4.20
% 29.51/4.20 Goal 1 (goals): join(x0, composition(composition(x0, converse(x0)), x0)) = composition(composition(x0, converse(x0)), x0).
% 29.51/4.20 Proof:
% 29.51/4.20 join(x0, composition(composition(x0, converse(x0)), x0))
% 29.51/4.20 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), x0)
% 29.51/4.20 = { by lemma 18 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(one, x0))
% 29.51/4.20 = { by lemma 80 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(join(one, complement(composition(top, composition(x0, converse(x0))))), x0))
% 29.51/4.20 = { by lemma 73 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(join(one, complement(join(one, composition(top, composition(x0, converse(x0)))))), x0))
% 29.51/4.20 = { by lemma 72 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(join(complement(composition(top, composition(x0, converse(x0)))), meet(composition(top, composition(x0, converse(x0))), one)), x0))
% 29.51/4.20 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(join(meet(composition(top, composition(x0, converse(x0))), one), complement(composition(top, composition(x0, converse(x0))))), x0))
% 29.51/4.20 = { by lemma 80 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(composition(top, composition(x0, converse(x0))), one), x0))
% 29.51/4.20 = { by lemma 74 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(composition(top, composition(x0, converse(x0))), join(one, complement(composition(top, composition(x0, converse(x0)))))), x0))
% 29.51/4.20 = { by lemma 66 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(composition(top, composition(x0, converse(x0))), complement(meet(composition(top, composition(x0, converse(x0))), complement(one)))), x0))
% 29.51/4.20 = { by lemma 59 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(complement(join(complement(composition(top, composition(x0, converse(x0)))), meet(composition(top, composition(x0, converse(x0))), complement(one)))), x0))
% 29.51/4.20 = { by lemma 72 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(complement(join(complement(one), complement(join(complement(one), composition(top, composition(x0, converse(x0))))))), x0))
% 29.51/4.20 = { by lemma 58 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(join(complement(one), composition(top, composition(x0, converse(x0)))), complement(complement(one))), x0))
% 29.51/4.20 = { by lemma 41 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(complement(one), composition(top, composition(x0, converse(x0))))), x0))
% 29.51/4.20 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(top, composition(x0, converse(x0))), complement(one))), x0))
% 29.51/4.20 = { by lemma 79 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(top, converse(x0)), complement(one))), x0))
% 29.51/4.20 = { by axiom 5 (def_top) }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(join(x0, complement(x0)), converse(x0)), complement(one))), x0))
% 29.51/4.20 = { by axiom 11 (composition_distributivity) }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(join(composition(x0, converse(x0)), composition(complement(x0), converse(x0))), complement(one))), x0))
% 29.51/4.20 = { by axiom 7 (maddux2_join_associativity) R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(x0, converse(x0)), join(composition(complement(x0), converse(x0)), complement(one)))), x0))
% 29.51/4.20 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(x0, converse(x0)), join(complement(one), composition(complement(x0), converse(x0))))), x0))
% 29.51/4.20 = { by axiom 2 (converse_idempotence) R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(x0, converse(x0)), join(complement(one), composition(complement(converse(converse(x0))), converse(x0))))), x0))
% 29.51/4.20 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(x0, converse(x0)), join(composition(complement(converse(converse(x0))), converse(x0)), complement(one)))), x0))
% 29.51/4.20 = { by lemma 65 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(x0, converse(x0)), join(composition(converse(complement(converse(x0))), converse(x0)), complement(one)))), x0))
% 29.51/4.20 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(x0, converse(x0)), join(complement(one), composition(converse(complement(converse(x0))), converse(x0))))), x0))
% 29.51/4.20 = { by lemma 56 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(complement(complement(one)), join(composition(x0, converse(x0)), complement(one))), x0))
% 29.51/4.20 = { by lemma 41 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(join(composition(x0, converse(x0)), complement(one)), complement(complement(one))), x0))
% 29.51/4.20 = { by lemma 58 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(complement(join(complement(one), complement(join(composition(x0, converse(x0)), complement(one))))), x0))
% 29.51/4.20 = { by lemma 58 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(complement(join(complement(one), meet(one, complement(composition(x0, converse(x0)))))), x0))
% 29.51/4.20 = { by lemma 59 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(one, complement(meet(one, complement(composition(x0, converse(x0)))))), x0))
% 29.51/4.20 = { by lemma 66 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(one, join(composition(x0, converse(x0)), complement(one))), x0))
% 29.51/4.20 = { by lemma 74 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(one, composition(x0, converse(x0))), x0))
% 29.51/4.20 = { by axiom 2 (converse_idempotence) R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(one, composition(x0, converse(x0))), converse(converse(x0))))
% 29.51/4.20 = { by lemma 69 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(meet(one, converse(composition(x0, converse(x0)))), converse(converse(x0))))
% 29.51/4.20 = { by lemma 69 R->L }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), converse(composition(converse(x0), converse(meet(one, converse(composition(x0, converse(x0))))))))
% 29.51/4.20 = { by lemma 16 }
% 29.51/4.20 join(composition(composition(x0, converse(x0)), x0), composition(converse(converse(meet(one, converse(composition(x0, converse(x0)))))), x0))
% 29.51/4.20 = { by axiom 11 (composition_distributivity) R->L }
% 29.51/4.20 composition(join(composition(x0, converse(x0)), converse(converse(meet(one, converse(composition(x0, converse(x0))))))), x0)
% 29.51/4.20 = { by lemma 69 R->L }
% 29.51/4.20 composition(join(converse(composition(x0, converse(x0))), converse(converse(meet(one, converse(composition(x0, converse(x0))))))), x0)
% 29.51/4.20 = { by axiom 6 (converse_additivity) R->L }
% 29.51/4.20 composition(converse(join(composition(x0, converse(x0)), converse(meet(one, converse(composition(x0, converse(x0))))))), x0)
% 29.51/4.20 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.20 composition(converse(join(converse(meet(one, converse(composition(x0, converse(x0))))), composition(x0, converse(x0)))), x0)
% 29.51/4.20 = { by lemma 16 R->L }
% 29.51/4.20 converse(composition(converse(x0), join(converse(meet(one, converse(composition(x0, converse(x0))))), composition(x0, converse(x0)))))
% 29.51/4.20 = { by axiom 1 (maddux1_join_commutativity) }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), converse(meet(one, converse(composition(x0, converse(x0))))))))
% 29.51/4.20 = { by lemma 64 R->L }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))
% 29.51/4.20 = { by lemma 45 R->L }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), meet(complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))), top))))
% 29.51/4.20 = { by lemma 39 R->L }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), meet(complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))), converse(top)))))
% 29.51/4.20 = { by lemma 61 R->L }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), meet(complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))), converse(join(converse(composition(x0, converse(x0))), complement(meet(converse(composition(x0, converse(x0))), one))))))))
% 29.51/4.20 = { by lemma 36 }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), meet(complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))), join(composition(x0, converse(x0)), converse(complement(meet(converse(composition(x0, converse(x0))), one))))))))
% 29.51/4.20 = { by lemma 41 R->L }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), meet(complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))), join(composition(x0, converse(x0)), converse(complement(meet(one, converse(composition(x0, converse(x0)))))))))))
% 29.51/4.20 = { by lemma 41 }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), meet(join(composition(x0, converse(x0)), converse(complement(meet(one, converse(composition(x0, converse(x0))))))), complement(converse(complement(meet(one, converse(composition(x0, converse(x0)))))))))))
% 29.51/4.20 = { by lemma 58 R->L }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), complement(join(converse(complement(meet(one, converse(composition(x0, converse(x0)))))), complement(join(composition(x0, converse(x0)), converse(complement(meet(one, converse(composition(x0, converse(x0)))))))))))))
% 29.51/4.20 = { by lemma 68 R->L }
% 29.51/4.20 converse(composition(converse(x0), join(composition(x0, converse(x0)), complement(join(converse(complement(meet(one, converse(composition(x0, converse(x0)))))), meet(complement(composition(x0, converse(x0))), complement(converse(complement(meet(one, converse(composition(x0, converse(x0)))))))))))))
% 29.51/4.20 = { by lemma 67 R->L }
% 29.51/4.20 converse(composition(converse(x0), complement(meet(complement(composition(x0, converse(x0))), join(converse(complement(meet(one, converse(composition(x0, converse(x0)))))), meet(complement(composition(x0, converse(x0))), complement(converse(complement(meet(one, converse(composition(x0, converse(x0)))))))))))))
% 29.51/4.20 = { by lemma 29 R->L }
% 29.51/4.20 converse(composition(converse(x0), complement(meet(complement(composition(x0, converse(x0))), join(converse(complement(meet(one, converse(composition(x0, converse(x0)))))), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))))))
% 29.51/4.20 = { by lemma 45 R->L }
% 29.51/4.20 converse(composition(converse(x0), complement(meet(complement(composition(x0, converse(x0))), join(meet(converse(complement(meet(one, converse(composition(x0, converse(x0)))))), top), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))))))
% 29.51/4.20 = { by lemma 42 R->L }
% 29.51/4.20 converse(composition(converse(x0), complement(meet(complement(composition(x0, converse(x0))), join(complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))))))
% 29.51/4.21 = { by lemma 30 R->L }
% 29.51/4.21 converse(composition(converse(x0), complement(join(meet(complement(composition(x0, converse(x0))), join(complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))), zero))))
% 29.51/4.21 = { by lemma 15 R->L }
% 29.51/4.21 converse(composition(converse(x0), complement(join(meet(complement(composition(x0, converse(x0))), join(complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))), complement(top)))))
% 29.51/4.21 = { by axiom 5 (def_top) }
% 29.51/4.21 converse(composition(converse(x0), complement(join(meet(complement(composition(x0, converse(x0))), join(complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))), complement(join(join(complement(complement(composition(x0, converse(x0)))), complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0)))))))))), complement(join(complement(complement(composition(x0, converse(x0)))), complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0)))))))))))))))))
% 29.51/4.21 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 29.51/4.21 converse(composition(converse(x0), complement(join(meet(complement(composition(x0, converse(x0))), join(complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))), complement(join(join(complement(complement(composition(x0, converse(x0)))), complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0)))))))))), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0)))))))))))))))
% 29.51/4.21 = { by axiom 7 (maddux2_join_associativity) R->L }
% 29.51/4.21 converse(composition(converse(x0), complement(join(meet(complement(composition(x0, converse(x0))), join(complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))), complement(join(complement(complement(composition(x0, converse(x0)))), join(complement(join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))), meet(complement(composition(x0, converse(x0))), join(zero, complement(converse(complement(meet(one, converse(composition(x0, converse(x0))))))))))))))))
% 29.51/4.21 = { by lemma 21 }
% 29.51/4.21 converse(composition(converse(x0), complement(complement(composition(x0, converse(x0))))))
% 29.51/4.21 = { by lemma 49 }
% 29.51/4.21 converse(composition(converse(x0), composition(x0, converse(x0))))
% 29.51/4.21 = { by lemma 75 }
% 29.51/4.21 converse(converse(composition(composition(x0, converse(x0)), x0)))
% 29.51/4.21 = { by axiom 2 (converse_idempotence) }
% 29.51/4.21 composition(composition(x0, converse(x0)), x0)
% 29.51/4.21 % SZS output end Proof
% 29.51/4.21
% 29.51/4.21 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------