TSTP Solution File: REL038+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL038+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 : n002.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 13:44:24 EDT 2023
% Result : Theorem 97.91s 12.87s
% Output : Proof 101.00s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : REL038+1 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n002.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 19:30:47 EDT 2023
% 0.14/0.34 % CPUTime :
% 97.91/12.87 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 97.91/12.87
% 97.91/12.87 % SZS status Theorem
% 97.91/12.87
% 100.17/13.12 % SZS output start Proof
% 100.17/13.12 Axiom 1 (composition_identity): composition(X, one) = X.
% 100.17/13.12 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 100.17/13.12 Axiom 3 (converse_idempotence): converse(converse(X)) = X.
% 100.17/13.12 Axiom 4 (def_top): top = join(X, complement(X)).
% 100.17/13.12 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 100.17/13.12 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 100.17/13.12 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 100.17/13.12 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 100.17/13.12 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 100.17/13.12 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 100.17/13.12 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 100.17/13.12 Axiom 12 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 100.17/13.12 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 100.17/13.12
% 100.17/13.12 Lemma 14: complement(top) = zero.
% 100.17/13.12 Proof:
% 100.17/13.12 complement(top)
% 100.17/13.12 = { by axiom 4 (def_top) }
% 100.17/13.12 complement(join(complement(X), complement(complement(X))))
% 100.17/13.12 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 100.17/13.12 meet(X, complement(X))
% 100.17/13.12 = { by axiom 5 (def_zero) R->L }
% 100.17/13.12 zero
% 100.17/13.12
% 100.17/13.12 Lemma 15: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 100.17/13.12 Proof:
% 100.17/13.12 converse(join(X, converse(Y)))
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.12 converse(join(converse(Y), X))
% 100.17/13.12 = { by axiom 8 (converse_additivity) }
% 100.17/13.12 join(converse(converse(Y)), converse(X))
% 100.17/13.12 = { by axiom 3 (converse_idempotence) }
% 100.17/13.12 join(Y, converse(X))
% 100.17/13.12
% 100.17/13.12 Lemma 16: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 100.17/13.12 Proof:
% 100.17/13.12 converse(join(converse(X), Y))
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.12 converse(join(Y, converse(X)))
% 100.17/13.12 = { by lemma 15 }
% 100.17/13.12 join(X, converse(Y))
% 100.17/13.12
% 100.17/13.12 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 100.17/13.12 Proof:
% 100.17/13.12 join(X, join(Y, complement(X)))
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.12 join(X, join(complement(X), Y))
% 100.17/13.12 = { by axiom 9 (maddux2_join_associativity) }
% 100.17/13.12 join(join(X, complement(X)), Y)
% 100.17/13.12 = { by axiom 4 (def_top) R->L }
% 100.17/13.12 join(top, Y)
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.12 join(Y, top)
% 100.17/13.12
% 100.17/13.12 Lemma 18: composition(converse(one), X) = X.
% 100.17/13.12 Proof:
% 100.17/13.12 composition(converse(one), X)
% 100.17/13.12 = { by axiom 3 (converse_idempotence) R->L }
% 100.17/13.12 composition(converse(one), converse(converse(X)))
% 100.17/13.12 = { by axiom 6 (converse_multiplicativity) R->L }
% 100.17/13.12 converse(composition(converse(X), one))
% 100.17/13.12 = { by axiom 1 (composition_identity) }
% 100.17/13.12 converse(converse(X))
% 100.17/13.12 = { by axiom 3 (converse_idempotence) }
% 100.17/13.12 X
% 100.17/13.12
% 100.17/13.12 Lemma 19: join(complement(X), composition(Y, complement(composition(converse(Y), X)))) = complement(X).
% 100.17/13.12 Proof:
% 100.17/13.12 join(complement(X), composition(Y, complement(composition(converse(Y), X))))
% 100.17/13.12 = { by axiom 3 (converse_idempotence) R->L }
% 100.17/13.12 join(complement(X), composition(converse(converse(Y)), complement(composition(converse(Y), X))))
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.12 join(composition(converse(converse(Y)), complement(composition(converse(Y), X))), complement(X))
% 100.17/13.12 = { by axiom 12 (converse_cancellativity) }
% 100.17/13.12 complement(X)
% 100.17/13.12
% 100.17/13.12 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 100.17/13.12 Proof:
% 100.17/13.12 join(complement(X), complement(X))
% 100.17/13.12 = { by lemma 18 R->L }
% 100.17/13.12 join(complement(X), complement(composition(converse(one), X)))
% 100.17/13.12 = { by axiom 1 (composition_identity) R->L }
% 100.17/13.12 join(complement(X), complement(composition(composition(converse(one), one), X)))
% 100.17/13.12 = { by axiom 7 (composition_associativity) R->L }
% 100.17/13.12 join(complement(X), complement(composition(converse(one), composition(one, X))))
% 100.17/13.12 = { by lemma 18 }
% 100.17/13.12 join(complement(X), complement(composition(one, X)))
% 100.17/13.12 = { by axiom 3 (converse_idempotence) R->L }
% 100.17/13.12 join(complement(X), complement(composition(converse(converse(one)), X)))
% 100.17/13.12 = { by lemma 18 R->L }
% 100.17/13.12 join(complement(X), composition(converse(one), complement(composition(converse(converse(one)), X))))
% 100.17/13.12 = { by lemma 19 }
% 100.17/13.12 complement(X)
% 100.17/13.12
% 100.17/13.12 Lemma 21: join(X, join(Y, complement(join(X, Y)))) = top.
% 100.17/13.12 Proof:
% 100.17/13.12 join(X, join(Y, complement(join(X, Y))))
% 100.17/13.12 = { by axiom 9 (maddux2_join_associativity) }
% 100.17/13.12 join(join(X, Y), complement(join(X, Y)))
% 100.17/13.12 = { by axiom 4 (def_top) R->L }
% 100.17/13.12 top
% 100.17/13.12
% 100.17/13.12 Lemma 22: join(X, top) = top.
% 100.17/13.12 Proof:
% 100.17/13.12 join(X, top)
% 100.17/13.12 = { by axiom 4 (def_top) }
% 100.17/13.12 join(X, join(complement(X), complement(complement(X))))
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.12 join(X, join(complement(complement(X)), complement(X)))
% 100.17/13.12 = { by lemma 17 }
% 100.17/13.12 join(complement(complement(X)), top)
% 100.17/13.12 = { by lemma 17 R->L }
% 100.17/13.12 join(complement(complement(X)), join(complement(complement(X)), complement(complement(complement(X)))))
% 100.17/13.12 = { by lemma 20 R->L }
% 100.17/13.12 join(complement(complement(X)), join(complement(complement(X)), complement(join(complement(complement(X)), complement(complement(X))))))
% 100.17/13.12 = { by lemma 21 }
% 100.17/13.12 top
% 100.17/13.12
% 100.17/13.12 Lemma 23: join(top, X) = top.
% 100.17/13.12 Proof:
% 100.17/13.12 join(top, X)
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.12 join(X, top)
% 100.17/13.12 = { by lemma 22 }
% 100.17/13.12 top
% 100.17/13.12
% 100.17/13.12 Lemma 24: converse(top) = top.
% 100.17/13.12 Proof:
% 100.17/13.12 converse(top)
% 100.17/13.12 = { by axiom 4 (def_top) }
% 100.17/13.12 converse(join(converse(top), complement(converse(top))))
% 100.17/13.12 = { by lemma 16 }
% 100.17/13.12 join(top, converse(complement(converse(top))))
% 100.17/13.12 = { by lemma 23 }
% 100.17/13.12 top
% 100.17/13.12
% 100.17/13.12 Lemma 25: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 100.17/13.12 Proof:
% 100.17/13.12 join(meet(X, Y), complement(join(complement(X), Y)))
% 100.17/13.12 = { by axiom 10 (maddux4_definiton_of_meet) }
% 100.17/13.12 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 100.17/13.12 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 100.17/13.12 X
% 100.17/13.12
% 100.17/13.12 Lemma 26: join(meet(X, Y), meet(X, complement(Y))) = X.
% 100.17/13.12 Proof:
% 100.17/13.12 join(meet(X, Y), meet(X, complement(Y)))
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.12 join(meet(X, complement(Y)), meet(X, Y))
% 100.17/13.12 = { by axiom 10 (maddux4_definiton_of_meet) }
% 100.17/13.12 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 100.17/13.12 = { by lemma 25 }
% 100.17/13.12 X
% 100.17/13.12
% 100.17/13.12 Lemma 27: meet(Y, X) = meet(X, Y).
% 100.17/13.12 Proof:
% 100.17/13.12 meet(Y, X)
% 100.17/13.12 = { by axiom 10 (maddux4_definiton_of_meet) }
% 100.17/13.12 complement(join(complement(Y), complement(X)))
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.12 complement(join(complement(X), complement(Y)))
% 100.17/13.12 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 100.17/13.12 meet(X, Y)
% 100.17/13.12
% 100.17/13.12 Lemma 28: complement(complement(X)) = X.
% 100.17/13.12 Proof:
% 100.17/13.12 complement(complement(X))
% 100.17/13.12 = { by lemma 26 R->L }
% 100.17/13.12 join(meet(complement(complement(X)), X), meet(complement(complement(X)), complement(X)))
% 100.17/13.12 = { by lemma 27 R->L }
% 100.17/13.12 join(meet(complement(complement(X)), X), meet(complement(X), complement(complement(X))))
% 100.17/13.12 = { by axiom 5 (def_zero) R->L }
% 100.17/13.12 join(meet(complement(complement(X)), X), zero)
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.12 join(zero, meet(complement(complement(X)), X))
% 100.17/13.12 = { by lemma 27 R->L }
% 100.17/13.12 join(zero, meet(X, complement(complement(X))))
% 100.17/13.12 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.12 join(meet(X, complement(complement(X))), zero)
% 100.17/13.12 = { by lemma 14 R->L }
% 100.17/13.12 join(meet(X, complement(complement(X))), complement(top))
% 100.17/13.12 = { by axiom 4 (def_top) }
% 100.17/13.12 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 100.17/13.12 = { by lemma 25 }
% 100.17/13.12 X
% 100.17/13.12
% 100.17/13.12 Lemma 29: join(X, X) = X.
% 100.17/13.12 Proof:
% 100.17/13.13 join(X, X)
% 100.17/13.13 = { by lemma 28 R->L }
% 100.17/13.13 join(X, complement(complement(X)))
% 100.17/13.13 = { by lemma 28 R->L }
% 100.17/13.13 join(complement(complement(X)), complement(complement(X)))
% 100.17/13.13 = { by lemma 20 }
% 100.17/13.13 complement(complement(X))
% 100.17/13.13 = { by lemma 28 }
% 100.17/13.13 X
% 100.17/13.13
% 100.17/13.13 Lemma 30: join(X, zero) = X.
% 100.17/13.13 Proof:
% 100.17/13.13 join(X, zero)
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.13 join(zero, X)
% 100.17/13.13 = { by lemma 28 R->L }
% 100.17/13.13 join(zero, complement(complement(X)))
% 100.17/13.13 = { by axiom 5 (def_zero) }
% 100.17/13.13 join(meet(X, complement(X)), complement(complement(X)))
% 100.17/13.13 = { by lemma 29 R->L }
% 100.17/13.13 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 100.17/13.13 = { by lemma 25 }
% 100.17/13.13 X
% 100.17/13.13
% 100.17/13.13 Lemma 31: complement(join(zero, complement(X))) = meet(X, top).
% 100.17/13.13 Proof:
% 100.17/13.13 complement(join(zero, complement(X)))
% 100.17/13.13 = { by lemma 14 R->L }
% 100.17/13.13 complement(join(complement(top), complement(X)))
% 100.17/13.13 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 100.17/13.13 meet(top, X)
% 100.17/13.13 = { by lemma 27 R->L }
% 100.17/13.13 meet(X, top)
% 100.17/13.13
% 100.17/13.13 Lemma 32: join(zero, complement(X)) = complement(X).
% 100.17/13.13 Proof:
% 100.17/13.13 join(zero, complement(X))
% 100.17/13.13 = { by lemma 29 R->L }
% 100.17/13.13 join(zero, complement(join(X, X)))
% 100.17/13.13 = { by lemma 28 R->L }
% 100.17/13.13 join(zero, complement(join(complement(complement(X)), X)))
% 100.17/13.13 = { by axiom 5 (def_zero) }
% 100.17/13.13 join(meet(X, complement(X)), complement(join(complement(complement(X)), X)))
% 100.17/13.13 = { by lemma 27 }
% 100.17/13.13 join(meet(complement(X), X), complement(join(complement(complement(X)), X)))
% 100.17/13.13 = { by lemma 25 }
% 100.17/13.13 complement(X)
% 100.17/13.13
% 100.17/13.13 Lemma 33: meet(X, top) = X.
% 100.17/13.13 Proof:
% 100.17/13.13 meet(X, top)
% 100.17/13.13 = { by lemma 31 R->L }
% 100.17/13.13 complement(join(zero, complement(X)))
% 100.17/13.13 = { by lemma 32 }
% 100.17/13.13 complement(complement(X))
% 100.17/13.13 = { by lemma 28 }
% 100.17/13.13 X
% 100.17/13.13
% 100.17/13.13 Lemma 34: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 100.17/13.13 Proof:
% 100.17/13.13 join(meet(X, Y), meet(Y, complement(X)))
% 100.17/13.13 = { by lemma 27 }
% 100.17/13.13 join(meet(Y, X), meet(Y, complement(X)))
% 100.17/13.13 = { by lemma 26 }
% 100.17/13.13 Y
% 100.17/13.13
% 100.17/13.13 Lemma 35: join(X, converse(complement(converse(X)))) = top.
% 100.17/13.13 Proof:
% 100.17/13.13 join(X, converse(complement(converse(X))))
% 100.17/13.13 = { by lemma 16 R->L }
% 100.17/13.13 converse(join(converse(X), complement(converse(X))))
% 100.17/13.13 = { by axiom 4 (def_top) R->L }
% 100.17/13.13 converse(top)
% 100.17/13.13 = { by lemma 24 }
% 100.17/13.13 top
% 100.17/13.13
% 100.17/13.13 Lemma 36: meet(X, converse(complement(converse(complement(X))))) = X.
% 100.17/13.13 Proof:
% 100.17/13.13 meet(X, converse(complement(converse(complement(X)))))
% 100.17/13.13 = { by lemma 30 R->L }
% 100.17/13.13 join(meet(X, converse(complement(converse(complement(X))))), zero)
% 100.17/13.13 = { by lemma 14 R->L }
% 100.17/13.13 join(meet(X, converse(complement(converse(complement(X))))), complement(top))
% 100.17/13.13 = { by lemma 35 R->L }
% 100.17/13.13 join(meet(X, converse(complement(converse(complement(X))))), complement(join(complement(X), converse(complement(converse(complement(X)))))))
% 100.17/13.13 = { by lemma 25 }
% 100.17/13.13 X
% 100.17/13.13
% 100.17/13.13 Lemma 37: join(zero, meet(X, Y)) = meet(X, Y).
% 100.17/13.13 Proof:
% 100.17/13.13 join(zero, meet(X, Y))
% 100.17/13.13 = { by lemma 27 }
% 100.17/13.13 join(zero, meet(Y, X))
% 100.17/13.13 = { by axiom 10 (maddux4_definiton_of_meet) }
% 100.17/13.13 join(zero, complement(join(complement(Y), complement(X))))
% 100.17/13.13 = { by lemma 32 }
% 100.17/13.13 complement(join(complement(Y), complement(X)))
% 100.17/13.13 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 100.17/13.13 meet(Y, X)
% 100.17/13.13 = { by lemma 27 R->L }
% 100.17/13.13 meet(X, Y)
% 100.17/13.13
% 100.17/13.13 Lemma 38: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 100.17/13.13 Proof:
% 100.17/13.13 join(complement(X), complement(Y))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.13 join(complement(Y), complement(X))
% 100.17/13.13 = { by lemma 33 R->L }
% 100.17/13.13 meet(join(complement(Y), complement(X)), top)
% 100.17/13.13 = { by lemma 27 R->L }
% 100.17/13.13 meet(top, join(complement(Y), complement(X)))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.13 meet(top, join(complement(X), complement(Y)))
% 100.17/13.13 = { by lemma 27 }
% 100.17/13.13 meet(join(complement(X), complement(Y)), top)
% 100.17/13.13 = { by axiom 10 (maddux4_definiton_of_meet) }
% 100.17/13.13 complement(join(complement(join(complement(X), complement(Y))), complement(top)))
% 100.17/13.13 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 100.17/13.13 complement(join(meet(X, Y), complement(top)))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.13 complement(join(complement(top), meet(X, Y)))
% 100.17/13.13 = { by lemma 27 R->L }
% 100.17/13.13 complement(join(complement(top), meet(Y, X)))
% 100.17/13.13 = { by lemma 14 }
% 100.17/13.13 complement(join(zero, meet(Y, X)))
% 100.17/13.13 = { by lemma 37 }
% 100.17/13.13 complement(meet(Y, X))
% 100.17/13.13 = { by lemma 27 R->L }
% 100.17/13.13 complement(meet(X, Y))
% 100.17/13.13
% 100.17/13.13 Lemma 39: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 100.17/13.13 Proof:
% 100.17/13.13 meet(complement(X), complement(Y))
% 100.17/13.13 = { by axiom 10 (maddux4_definiton_of_meet) }
% 100.17/13.13 complement(join(complement(complement(X)), complement(complement(Y))))
% 100.17/13.13 = { by lemma 28 }
% 100.17/13.13 complement(join(X, complement(complement(Y))))
% 100.17/13.13 = { by lemma 28 }
% 100.17/13.13 complement(join(X, Y))
% 100.17/13.13
% 100.17/13.13 Lemma 40: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 100.17/13.13 Proof:
% 100.17/13.13 complement(meet(X, complement(Y)))
% 100.17/13.13 = { by lemma 27 }
% 100.17/13.13 complement(meet(complement(Y), X))
% 100.17/13.13 = { by lemma 32 R->L }
% 100.17/13.13 complement(meet(join(zero, complement(Y)), X))
% 100.17/13.13 = { by lemma 38 R->L }
% 100.17/13.13 join(complement(join(zero, complement(Y))), complement(X))
% 100.17/13.13 = { by lemma 31 }
% 100.17/13.13 join(meet(Y, top), complement(X))
% 100.17/13.13 = { by lemma 33 }
% 100.17/13.13 join(Y, complement(X))
% 100.17/13.13
% 100.17/13.13 Lemma 41: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
% 100.17/13.13 Proof:
% 100.17/13.13 join(meet(X, Y), complement(join(Y, complement(X))))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.13 join(meet(X, Y), complement(join(complement(X), Y)))
% 100.17/13.13 = { by lemma 25 }
% 100.17/13.13 X
% 100.17/13.13
% 100.17/13.13 Lemma 42: join(meet(X, Y), join(Z, complement(join(complement(X), Y)))) = join(X, Z).
% 100.17/13.13 Proof:
% 100.17/13.13 join(meet(X, Y), join(Z, complement(join(complement(X), Y))))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.13 join(meet(X, Y), join(complement(join(complement(X), Y)), Z))
% 100.17/13.13 = { by axiom 9 (maddux2_join_associativity) }
% 100.17/13.13 join(join(meet(X, Y), complement(join(complement(X), Y))), Z)
% 100.17/13.13 = { by lemma 25 }
% 100.17/13.13 join(X, Z)
% 100.17/13.13
% 100.17/13.13 Lemma 43: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 100.17/13.13 Proof:
% 100.17/13.13 meet(Y, meet(X, Z))
% 100.17/13.13 = { by lemma 28 R->L }
% 100.17/13.13 complement(complement(meet(Y, meet(X, Z))))
% 100.17/13.13 = { by lemma 38 R->L }
% 100.17/13.13 complement(join(complement(Y), complement(meet(X, Z))))
% 100.17/13.13 = { by lemma 38 R->L }
% 100.17/13.13 complement(join(complement(Y), join(complement(X), complement(Z))))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.13 complement(join(complement(Y), join(complement(Z), complement(X))))
% 100.17/13.13 = { by axiom 9 (maddux2_join_associativity) }
% 100.17/13.13 complement(join(join(complement(Y), complement(Z)), complement(X)))
% 100.17/13.13 = { by lemma 38 }
% 100.17/13.13 complement(join(complement(meet(Y, Z)), complement(X)))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.13 complement(join(complement(X), complement(meet(Y, Z))))
% 100.17/13.13 = { by lemma 38 }
% 100.17/13.13 complement(complement(meet(X, meet(Y, Z))))
% 100.17/13.13 = { by lemma 28 }
% 100.17/13.13 meet(X, meet(Y, Z))
% 100.17/13.13
% 100.17/13.13 Lemma 44: converse(complement(converse(X))) = complement(X).
% 100.17/13.13 Proof:
% 100.17/13.13 converse(complement(converse(X)))
% 100.17/13.13 = { by lemma 34 R->L }
% 100.17/13.13 join(meet(X, converse(complement(converse(X)))), meet(converse(complement(converse(X))), complement(X)))
% 100.17/13.13 = { by lemma 27 R->L }
% 100.17/13.13 join(meet(X, converse(complement(converse(X)))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 3 (converse_idempotence) R->L }
% 100.17/13.13 join(meet(converse(converse(X)), converse(complement(converse(X)))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by lemma 33 R->L }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), top)), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by lemma 22 R->L }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(converse(complement(join(complement(converse(complement(converse(complement(converse(X)))))), converse(X)))), top))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 4 (def_top) }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(converse(complement(join(complement(converse(complement(converse(complement(converse(X)))))), converse(X)))), join(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))), complement(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X)))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 9 (maddux2_join_associativity) }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(join(converse(complement(join(complement(converse(complement(converse(complement(converse(X)))))), converse(X)))), converse(meet(converse(complement(converse(complement(converse(X))))), converse(X)))), complement(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 8 (converse_additivity) R->L }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(converse(join(complement(join(complement(converse(complement(converse(complement(converse(X)))))), converse(X))), meet(converse(complement(converse(complement(converse(X))))), converse(X)))), complement(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(complement(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X)))), converse(join(complement(join(complement(converse(complement(converse(complement(converse(X)))))), converse(X))), meet(converse(complement(converse(complement(converse(X))))), converse(X))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(complement(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X)))), converse(join(meet(converse(complement(converse(complement(converse(X))))), converse(X)), complement(join(complement(converse(complement(converse(complement(converse(X)))))), converse(X)))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by lemma 25 }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(complement(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X)))), converse(converse(complement(converse(complement(converse(X))))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(converse(converse(complement(converse(complement(converse(X)))))), complement(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 3 (converse_idempotence) }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(complement(converse(complement(converse(X)))), complement(converse(meet(converse(complement(converse(complement(converse(X))))), converse(X))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by lemma 27 R->L }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(complement(converse(complement(converse(X)))), complement(converse(meet(converse(X), converse(complement(converse(complement(converse(X))))))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by lemma 36 }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), join(complement(converse(complement(converse(X)))), complement(converse(converse(X)))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by lemma 38 }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), complement(meet(converse(complement(converse(X))), converse(converse(X)))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by lemma 27 R->L }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), complement(meet(converse(converse(X)), converse(complement(converse(X))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 3 (converse_idempotence) R->L }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(X))), complement(meet(converse(converse(X)), converse(complement(converse(converse(converse(X))))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by axiom 3 (converse_idempotence) R->L }
% 100.17/13.13 join(meet(converse(converse(X)), meet(converse(complement(converse(converse(converse(X))))), complement(meet(converse(converse(X)), converse(complement(converse(converse(converse(X))))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.13 = { by lemma 27 }
% 100.17/13.13 join(meet(converse(converse(X)), meet(complement(meet(converse(converse(X)), converse(complement(converse(converse(converse(X))))))), converse(complement(converse(converse(converse(X))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 30 R->L }
% 100.17/13.14 join(meet(converse(converse(X)), meet(complement(meet(converse(converse(X)), converse(complement(converse(converse(converse(X))))))), join(converse(complement(converse(converse(converse(X))))), zero))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 14 R->L }
% 100.17/13.14 join(meet(converse(converse(X)), meet(complement(meet(converse(converse(X)), converse(complement(converse(converse(converse(X))))))), join(converse(complement(converse(converse(converse(X))))), complement(top)))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 35 R->L }
% 100.17/13.14 join(meet(converse(converse(X)), meet(complement(meet(converse(converse(X)), converse(complement(converse(converse(converse(X))))))), join(converse(complement(converse(converse(converse(X))))), complement(join(converse(converse(X)), converse(complement(converse(converse(converse(X)))))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 39 R->L }
% 100.17/13.14 join(meet(converse(converse(X)), meet(complement(meet(converse(converse(X)), converse(complement(converse(converse(converse(X))))))), join(converse(complement(converse(converse(converse(X))))), meet(complement(converse(converse(X))), complement(converse(complement(converse(converse(converse(X)))))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 38 R->L }
% 100.17/13.14 join(meet(converse(converse(X)), meet(join(complement(converse(converse(X))), complement(converse(complement(converse(converse(converse(X))))))), join(converse(complement(converse(converse(converse(X))))), meet(complement(converse(converse(X))), complement(converse(complement(converse(converse(converse(X)))))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 40 R->L }
% 100.17/13.14 join(meet(converse(converse(X)), meet(complement(meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))), join(converse(complement(converse(converse(converse(X))))), meet(complement(converse(converse(X))), complement(converse(complement(converse(converse(converse(X)))))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 join(meet(converse(converse(X)), meet(complement(meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))), join(meet(complement(converse(converse(X))), complement(converse(complement(converse(converse(converse(X))))))), converse(complement(converse(converse(converse(X)))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 41 R->L }
% 100.17/13.14 join(meet(converse(converse(X)), meet(complement(meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))), join(meet(complement(converse(converse(X))), complement(converse(complement(converse(converse(converse(X))))))), join(meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X))))), complement(join(complement(complement(converse(converse(X)))), complement(converse(complement(converse(converse(converse(X)))))))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 42 }
% 100.17/13.14 join(meet(converse(converse(X)), meet(complement(meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))), join(complement(converse(converse(X))), meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 27 }
% 100.17/13.14 join(meet(converse(converse(X)), meet(join(complement(converse(converse(X))), meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))), complement(meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 43 }
% 100.17/13.14 join(meet(join(complement(converse(converse(X))), meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))), meet(converse(converse(X)), complement(meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 27 }
% 100.17/13.14 join(meet(join(complement(converse(converse(X))), meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))), meet(complement(meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))), converse(converse(X)))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 27 }
% 100.17/13.14 join(meet(join(complement(converse(converse(X))), meet(complement(complement(converse(converse(X)))), converse(complement(converse(converse(converse(X))))))), meet(complement(meet(converse(complement(converse(converse(converse(X))))), complement(complement(converse(converse(X)))))), converse(converse(X)))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 27 }
% 100.17/13.14 join(meet(join(complement(converse(converse(X))), meet(complement(complement(converse(converse(X)))), converse(complement(converse(converse(converse(X))))))), meet(complement(meet(complement(complement(converse(converse(X)))), converse(complement(converse(converse(converse(X))))))), converse(converse(X)))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 join(meet(join(meet(complement(complement(converse(converse(X)))), converse(complement(converse(converse(converse(X)))))), complement(converse(converse(X)))), meet(complement(meet(complement(complement(converse(converse(X)))), converse(complement(converse(converse(converse(X))))))), converse(converse(X)))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 27 }
% 100.17/13.14 join(meet(meet(complement(meet(complement(complement(converse(converse(X)))), converse(complement(converse(converse(converse(X))))))), converse(converse(X))), join(meet(complement(complement(converse(converse(X)))), converse(complement(converse(converse(converse(X)))))), complement(converse(converse(X))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 38 R->L }
% 100.17/13.14 join(meet(meet(join(complement(complement(complement(converse(converse(X))))), complement(converse(complement(converse(converse(converse(X))))))), converse(converse(X))), join(meet(complement(complement(converse(converse(X)))), converse(complement(converse(converse(converse(X)))))), complement(converse(converse(X))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by axiom 10 (maddux4_definiton_of_meet) }
% 100.17/13.14 join(meet(meet(join(complement(complement(complement(converse(converse(X))))), complement(converse(complement(converse(converse(converse(X))))))), converse(converse(X))), join(complement(join(complement(complement(complement(converse(converse(X))))), complement(converse(complement(converse(converse(converse(X)))))))), complement(converse(converse(X))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by lemma 27 }
% 100.17/13.14 join(meet(join(complement(join(complement(complement(complement(converse(converse(X))))), complement(converse(complement(converse(converse(converse(X)))))))), complement(converse(converse(X)))), meet(join(complement(complement(complement(converse(converse(X))))), complement(converse(complement(converse(converse(converse(X))))))), converse(converse(X)))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by axiom 10 (maddux4_definiton_of_meet) }
% 100.17/13.14 join(meet(join(complement(join(complement(complement(complement(converse(converse(X))))), complement(converse(complement(converse(converse(converse(X)))))))), complement(converse(converse(X)))), complement(join(complement(join(complement(complement(complement(converse(converse(X))))), complement(converse(complement(converse(converse(converse(X)))))))), complement(converse(converse(X)))))), meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by axiom 5 (def_zero) R->L }
% 100.17/13.14 join(zero, meet(complement(X), converse(complement(converse(X)))))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 join(meet(complement(X), converse(complement(converse(X)))), zero)
% 100.17/13.14 = { by lemma 30 }
% 100.17/13.14 meet(complement(X), converse(complement(converse(X))))
% 100.17/13.14 = { by lemma 28 R->L }
% 100.17/13.14 meet(complement(X), converse(complement(converse(complement(complement(X))))))
% 100.17/13.14 = { by lemma 36 }
% 100.17/13.14 complement(X)
% 100.17/13.14
% 100.17/13.14 Lemma 45: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 100.17/13.14 Proof:
% 100.17/13.14 join(X, join(Y, Z))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 join(join(Y, Z), X)
% 100.17/13.14 = { by axiom 9 (maddux2_join_associativity) R->L }
% 100.17/13.14 join(Y, join(Z, X))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.14 join(Y, join(X, Z))
% 100.17/13.14
% 100.17/13.14 Lemma 46: join(meet(X, Y), join(Z, complement(join(Y, complement(X))))) = join(Z, X).
% 100.17/13.14 Proof:
% 100.17/13.14 join(meet(X, Y), join(Z, complement(join(Y, complement(X)))))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 join(meet(X, Y), join(complement(join(Y, complement(X))), Z))
% 100.17/13.14 = { by axiom 9 (maddux2_join_associativity) }
% 100.17/13.14 join(join(meet(X, Y), complement(join(Y, complement(X)))), Z)
% 100.17/13.14 = { by lemma 41 }
% 100.17/13.14 join(X, Z)
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.14 join(Z, X)
% 100.17/13.14
% 100.17/13.14 Lemma 47: join(X, meet(X, Y)) = X.
% 100.17/13.14 Proof:
% 100.17/13.14 join(X, meet(X, Y))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 join(meet(X, Y), X)
% 100.17/13.14 = { by lemma 25 R->L }
% 100.17/13.14 join(meet(X, Y), join(meet(X, Y), complement(join(complement(X), Y))))
% 100.17/13.14 = { by lemma 20 R->L }
% 100.17/13.14 join(meet(X, Y), join(meet(X, Y), join(complement(join(complement(X), Y)), complement(join(complement(X), Y)))))
% 100.17/13.14 = { by lemma 42 }
% 100.17/13.14 join(meet(X, Y), join(X, complement(join(complement(X), Y))))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.14 join(meet(X, Y), join(X, complement(join(Y, complement(X)))))
% 100.17/13.14 = { by lemma 46 }
% 100.17/13.14 join(X, X)
% 100.17/13.14 = { by lemma 29 }
% 100.17/13.14 X
% 100.17/13.14
% 100.17/13.14 Lemma 48: meet(X, join(X, join(Y, Z))) = X.
% 100.17/13.14 Proof:
% 100.17/13.14 meet(X, join(X, join(Y, Z)))
% 100.17/13.14 = { by lemma 45 R->L }
% 100.17/13.14 meet(X, join(Y, join(X, Z)))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 meet(X, join(Y, join(Z, X)))
% 100.17/13.14 = { by lemma 30 R->L }
% 100.17/13.14 join(meet(X, join(Y, join(Z, X))), zero)
% 100.17/13.14 = { by lemma 14 R->L }
% 100.17/13.14 join(meet(X, join(Y, join(Z, X))), complement(top))
% 100.17/13.14 = { by lemma 28 R->L }
% 100.17/13.14 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(top))
% 100.17/13.14 = { by lemma 22 R->L }
% 100.17/13.14 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(Y, top)))
% 100.17/13.14 = { by lemma 22 R->L }
% 100.17/13.14 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(Y, join(Z, top))))
% 100.17/13.14 = { by axiom 9 (maddux2_join_associativity) }
% 100.17/13.14 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(join(Y, Z), top)))
% 100.17/13.14 = { by lemma 17 R->L }
% 100.17/13.14 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(complement(X), join(join(Y, Z), complement(complement(X))))))
% 100.17/13.14 = { by axiom 9 (maddux2_join_associativity) R->L }
% 100.17/13.14 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(complement(X), join(Y, join(Z, complement(complement(X)))))))
% 100.17/13.14 = { by lemma 25 }
% 100.17/13.14 X
% 100.17/13.14
% 100.17/13.14 Lemma 49: meet(X, join(X, Y)) = X.
% 100.17/13.14 Proof:
% 100.17/13.14 meet(X, join(X, Y))
% 100.17/13.14 = { by lemma 25 R->L }
% 100.17/13.14 meet(X, join(X, join(meet(Y, Z), complement(join(complement(Y), Z)))))
% 100.17/13.14 = { by lemma 48 }
% 100.17/13.14 X
% 100.17/13.14
% 100.17/13.14 Lemma 50: meet(meet(X, Y), join(X, Z)) = meet(X, Y).
% 100.17/13.14 Proof:
% 100.17/13.14 meet(meet(X, Y), join(X, Z))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 meet(meet(X, Y), join(Z, X))
% 100.17/13.14 = { by lemma 46 R->L }
% 100.17/13.14 meet(meet(X, Y), join(meet(X, Y), join(Z, complement(join(Y, complement(X))))))
% 100.17/13.14 = { by lemma 48 }
% 100.17/13.14 meet(X, Y)
% 100.17/13.14
% 100.17/13.14 Lemma 51: join(complement(X), meet(meet(X, Y), join(Y, complement(X)))) = join(Y, complement(X)).
% 100.17/13.14 Proof:
% 100.17/13.14 join(complement(X), meet(meet(X, Y), join(Y, complement(X))))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 join(complement(X), meet(meet(X, Y), join(complement(X), Y)))
% 100.17/13.14 = { by lemma 27 }
% 100.17/13.14 join(complement(X), meet(join(complement(X), Y), meet(X, Y)))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 join(meet(join(complement(X), Y), meet(X, Y)), complement(X))
% 100.17/13.14 = { by lemma 25 R->L }
% 100.17/13.14 join(meet(join(complement(X), Y), meet(X, Y)), complement(join(meet(X, Y), complement(join(complement(X), Y)))))
% 100.17/13.14 = { by lemma 41 }
% 100.17/13.14 join(complement(X), Y)
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.14 join(Y, complement(X))
% 100.17/13.14
% 100.17/13.14 Lemma 52: join(meet(X, Y), complement(Y)) = join(X, complement(Y)).
% 100.17/13.14 Proof:
% 100.17/13.14 join(meet(X, Y), complement(Y))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.14 join(complement(Y), meet(X, Y))
% 100.17/13.14 = { by lemma 50 R->L }
% 100.17/13.14 join(complement(Y), meet(meet(X, Y), join(X, complement(Y))))
% 100.17/13.14 = { by lemma 27 R->L }
% 100.17/13.14 join(complement(Y), meet(meet(Y, X), join(X, complement(Y))))
% 100.17/13.14 = { by lemma 51 }
% 100.17/13.14 join(X, complement(Y))
% 100.17/13.14
% 100.17/13.14 Lemma 53: join(converse(composition(X, Y)), composition(Z, converse(X))) = composition(join(Z, converse(Y)), converse(X)).
% 100.17/13.14 Proof:
% 100.17/13.14 join(converse(composition(X, Y)), composition(Z, converse(X)))
% 100.17/13.14 = { by axiom 6 (converse_multiplicativity) }
% 100.17/13.14 join(composition(converse(Y), converse(X)), composition(Z, converse(X)))
% 100.17/13.14 = { by axiom 11 (composition_distributivity) R->L }
% 100.17/13.14 composition(join(converse(Y), Z), converse(X))
% 100.17/13.14 = { by axiom 2 (maddux1_join_commutativity) }
% 100.17/13.14 composition(join(Z, converse(Y)), converse(X))
% 100.17/13.14
% 100.17/13.14 Lemma 54: join(composition(X, Y), composition(X, Z)) = composition(X, join(Z, Y)).
% 100.17/13.14 Proof:
% 100.17/13.14 join(composition(X, Y), composition(X, Z))
% 100.17/13.14 = { by axiom 3 (converse_idempotence) R->L }
% 100.17/13.15 join(converse(converse(composition(X, Y))), composition(X, Z))
% 100.17/13.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.15 join(composition(X, Z), converse(converse(composition(X, Y))))
% 100.17/13.15 = { by lemma 15 R->L }
% 100.17/13.15 converse(join(converse(composition(X, Y)), converse(composition(X, Z))))
% 100.17/13.15 = { by axiom 6 (converse_multiplicativity) }
% 100.17/13.15 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 100.17/13.15 = { by lemma 53 }
% 100.17/13.15 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 100.17/13.15 = { by axiom 6 (converse_multiplicativity) }
% 100.17/13.15 composition(converse(converse(X)), converse(join(converse(Z), converse(Y))))
% 100.17/13.15 = { by axiom 3 (converse_idempotence) }
% 100.17/13.15 composition(X, converse(join(converse(Z), converse(Y))))
% 100.17/13.15 = { by lemma 16 }
% 100.17/13.15 composition(X, join(Z, converse(converse(Y))))
% 100.17/13.15 = { by axiom 3 (converse_idempotence) }
% 100.17/13.15 composition(X, join(Z, Y))
% 100.17/13.15
% 100.17/13.15 Lemma 55: join(X, join(Y, converse(complement(converse(join(X, Y)))))) = top.
% 100.17/13.15 Proof:
% 100.17/13.15 join(X, join(Y, converse(complement(converse(join(X, Y))))))
% 100.17/13.15 = { by lemma 16 R->L }
% 100.17/13.15 join(X, converse(join(converse(Y), complement(converse(join(X, Y))))))
% 100.17/13.15 = { by lemma 16 R->L }
% 100.17/13.15 converse(join(converse(X), join(converse(Y), complement(converse(join(X, Y))))))
% 100.17/13.15 = { by axiom 8 (converse_additivity) }
% 100.17/13.15 converse(join(converse(X), join(converse(Y), complement(join(converse(X), converse(Y))))))
% 100.17/13.15 = { by lemma 21 }
% 100.17/13.15 converse(top)
% 100.17/13.15 = { by lemma 24 }
% 100.17/13.15 top
% 100.17/13.15
% 100.17/13.15 Lemma 56: join(complement(X), join(Y, composition(Z, complement(composition(converse(Z), X))))) = join(Y, complement(X)).
% 100.17/13.15 Proof:
% 100.17/13.15 join(complement(X), join(Y, composition(Z, complement(composition(converse(Z), X)))))
% 100.17/13.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.17/13.15 join(complement(X), join(composition(Z, complement(composition(converse(Z), X))), Y))
% 100.17/13.15 = { by axiom 9 (maddux2_join_associativity) }
% 100.17/13.15 join(join(complement(X), composition(Z, complement(composition(converse(Z), X)))), Y)
% 100.17/13.15 = { by lemma 19 }
% 100.17/13.15 join(complement(X), Y)
% 100.17/13.15 = { by axiom 2 (maddux1_join_commutativity) }
% 100.66/13.16 join(Y, complement(X))
% 100.66/13.16
% 100.66/13.16 Goal 1 (goals): join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)) = meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2).
% 100.66/13.16 Proof:
% 100.66/13.16 join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))
% 100.66/13.16 = { by lemma 25 R->L }
% 100.66/13.16 join(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))), complement(join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))))
% 100.66/13.16 = { by lemma 45 }
% 100.66/13.16 join(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))), complement(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))))
% 100.66/13.16 = { by lemma 55 }
% 100.66/13.16 join(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))), complement(top))
% 100.66/13.16 = { by lemma 14 }
% 100.66/13.16 join(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))), zero)
% 100.66/13.16 = { by lemma 30 }
% 100.66/13.16 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))
% 100.66/13.16 = { by lemma 49 R->L }
% 100.66/13.16 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), meet(x2, complement(composition(x0, x1))))))))))))
% 100.66/13.16 = { by axiom 2 (maddux1_join_commutativity) }
% 100.66/13.16 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), meet(composition(x0, x1), x2)), meet(x2, complement(composition(x0, x1))))))))))))
% 100.66/13.17 = { by axiom 9 (maddux2_join_associativity) R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), meet(x2, complement(composition(x0, x1)))))))))))))
% 100.66/13.17 = { by lemma 34 }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), x2)))))))))
% 100.66/13.17 = { by axiom 2 (maddux1_join_commutativity) }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))))
% 100.66/13.17 = { by lemma 27 R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(x2, meet(x2, composition(x0, meet(x1, composition(converse(x0), x2)))))))))))))
% 100.66/13.17 = { by lemma 47 }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), x2))))))))
% 100.66/13.17 = { by lemma 27 R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))))
% 100.66/13.17 = { by lemma 38 R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))))
% 100.66/13.17 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(x2))))))))
% 100.66/13.17 = { by lemma 28 R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(complement(complement(x2))))))))))
% 100.66/13.17 = { by lemma 51 R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(complement(complement(x2))), meet(meet(complement(complement(x2)), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))), join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(complement(complement(x2))))))))))))
% 100.66/13.17 = { by lemma 27 R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(complement(complement(x2))), meet(meet(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(complement(x2))), join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(complement(complement(x2))))))))))))
% 100.66/13.17 = { by lemma 28 }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), meet(meet(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(complement(x2))), join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(complement(complement(x2))))))))))))
% 100.66/13.17 = { by lemma 28 }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), meet(meet(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(complement(x2))), join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(x2))))))))))
% 100.66/13.17 = { by lemma 27 R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), meet(join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(x2)), meet(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(complement(x2)))))))))))
% 100.66/13.17 = { by lemma 43 R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), meet(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), meet(join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(x2)), complement(complement(x2)))))))))))
% 100.66/13.17 = { by lemma 27 R->L }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), meet(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), meet(complement(complement(x2)), join(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(x2)))))))))))
% 100.66/13.17 = { by axiom 2 (maddux1_join_commutativity) }
% 100.66/13.17 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), meet(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), meet(complement(complement(x2)), join(complement(x2), complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))))))
% 100.66/13.18 = { by lemma 38 }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), meet(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), meet(complement(complement(x2)), complement(meet(x2, join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))))))
% 100.66/13.18 = { by lemma 39 }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), meet(complement(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(join(complement(x2), meet(x2, join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))))))
% 100.66/13.18 = { by lemma 39 }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), complement(join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(x2, join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))))))
% 100.66/13.18 = { by lemma 38 }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(x2, join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))))))
% 100.66/13.18 = { by lemma 27 }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), x2)))))))))))
% 100.66/13.18 = { by lemma 28 R->L }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(complement(x2)))))))))))))
% 100.66/13.18 = { by lemma 27 }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(complement(complement(x2)), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))))))
% 100.66/13.18 = { by lemma 49 R->L }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(complement(complement(x2)), meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(x2))))))))))))))
% 100.66/13.18 = { by axiom 2 (maddux1_join_commutativity) }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(complement(complement(x2)), meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))))))))))
% 100.66/13.18 = { by lemma 27 }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(complement(complement(x2)), meet(join(complement(x2), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))))))))
% 100.66/13.18 = { by lemma 43 }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(join(complement(x2), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), meet(complement(complement(x2)), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))))))))
% 100.66/13.18 = { by lemma 27 }
% 100.66/13.18 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(meet(complement(complement(x2)), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), join(complement(x2), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))))))))
% 100.66/13.18 = { by lemma 28 R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(x2), meet(meet(complement(complement(x2)), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), join(complement(complement(complement(x2))), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))))))))
% 100.66/13.19 = { by lemma 28 R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(complement(complement(x2))), meet(meet(complement(complement(x2)), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), join(complement(complement(complement(x2))), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))))))))
% 100.66/13.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(complement(complement(complement(x2))), meet(meet(complement(complement(x2)), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(complement(complement(x2)))))))))))))))
% 100.66/13.19 = { by lemma 51 }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(complement(complement(x2)))))))))))))
% 100.66/13.19 = { by lemma 28 }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(x2)))))))))))
% 100.66/13.19 = { by axiom 9 (maddux2_join_associativity) }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(x2))))))))))
% 100.66/13.19 = { by lemma 29 }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(x2, join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(x2))))))))))
% 100.66/13.19 = { by lemma 38 R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), complement(join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(x2))))))))))
% 100.66/13.19 = { by lemma 44 R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(x2))))))))))))
% 100.66/13.19 = { by lemma 33 R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), top)), complement(x2))))))))))))
% 100.66/13.19 = { by lemma 23 R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(top, complement(composition(x0, meet(x1, composition(converse(x0), x2))))))), complement(x2))))))))))))
% 100.66/13.19 = { by lemma 21 R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), complement(join(meet(composition(x0, x1), x2), join(complement(x2), complement(composition(x0, x1))))))), complement(composition(x0, meet(x1, composition(converse(x0), x2))))))), complement(x2))))))))))))
% 100.66/13.19 = { by axiom 9 (maddux2_join_associativity) R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(join(complement(x2), complement(composition(x0, x1))), complement(join(meet(composition(x0, x1), x2), join(complement(x2), complement(composition(x0, x1)))))), complement(composition(x0, meet(x1, composition(converse(x0), x2)))))))), complement(x2))))))))))))
% 100.66/13.19 = { by axiom 9 (maddux2_join_associativity) R->L }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), join(complement(join(meet(composition(x0, x1), x2), join(complement(x2), complement(composition(x0, x1))))), complement(composition(x0, meet(x1, composition(converse(x0), x2))))))))), complement(x2))))))))))))
% 100.66/13.19 = { by axiom 2 (maddux1_join_commutativity) }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), join(complement(composition(x0, meet(x1, composition(converse(x0), x2)))), complement(join(meet(composition(x0, x1), x2), join(complement(x2), complement(composition(x0, x1)))))))))), complement(x2))))))))))))
% 100.66/13.19 = { by lemma 38 }
% 100.66/13.19 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), join(meet(composition(x0, x1), x2), join(complement(x2), complement(composition(x0, x1)))))))))), complement(x2))))))))))))
% 100.66/13.20 = { by lemma 45 R->L }
% 100.66/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), join(complement(x2), join(meet(composition(x0, x1), x2), complement(composition(x0, x1)))))))))), complement(x2))))))))))))
% 100.66/13.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 100.66/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), join(complement(x2), join(complement(composition(x0, x1)), meet(composition(x0, x1), x2))))))))), complement(x2))))))))))))
% 100.66/13.20 = { by lemma 45 R->L }
% 100.66/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), join(complement(composition(x0, x1)), join(complement(x2), meet(composition(x0, x1), x2))))))))), complement(x2))))))))))))
% 100.66/13.20 = { by axiom 9 (maddux2_join_associativity) }
% 100.66/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), join(join(complement(composition(x0, x1)), complement(x2)), meet(composition(x0, x1), x2)))))))), complement(x2))))))))))))
% 100.66/13.20 = { by axiom 10 (maddux4_definiton_of_meet) }
% 101.00/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), join(join(complement(composition(x0, x1)), complement(x2)), complement(join(complement(composition(x0, x1)), complement(x2)))))))))), complement(x2))))))))))))
% 101.00/13.20 = { by axiom 4 (def_top) R->L }
% 101.00/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(join(complement(x2), complement(composition(x0, x1))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), top)))))), complement(x2))))))))))))
% 101.00/13.20 = { by axiom 9 (maddux2_join_associativity) R->L }
% 101.00/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(complement(x2), join(complement(composition(x0, x1)), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), top))))))), complement(x2))))))))))))
% 101.00/13.20 = { by lemma 38 }
% 101.00/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), join(complement(x2), complement(meet(composition(x0, x1), meet(composition(x0, meet(x1, composition(converse(x0), x2))), top))))))), complement(x2))))))))))))
% 101.00/13.20 = { by lemma 38 }
% 101.00/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, x1), meet(composition(x0, meet(x1, composition(converse(x0), x2))), top))))))), complement(x2))))))))))))
% 101.00/13.20 = { by lemma 33 }
% 101.00/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, x1), composition(x0, meet(x1, composition(converse(x0), x2))))))))), complement(x2))))))))))))
% 101.00/13.20 = { by lemma 27 R->L }
% 101.00/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), composition(x0, x1))))))), complement(x2))))))))))))
% 101.00/13.20 = { by axiom 3 (converse_idempotence) R->L }
% 101.00/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), converse(converse(composition(x0, x1))))))))), complement(x2))))))))))))
% 101.00/13.20 = { by axiom 6 (converse_multiplicativity) }
% 101.00/13.20 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), converse(composition(converse(x1), converse(x0))))))))), complement(x2))))))))))))
% 101.00/13.20 = { by lemma 47 R->L }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), converse(composition(converse(join(x1, meet(x1, composition(converse(x0), x2)))), converse(x0))))))))), complement(x2))))))))))))
% 101.00/13.21 = { by axiom 8 (converse_additivity) }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), converse(composition(join(converse(x1), converse(meet(x1, composition(converse(x0), x2)))), converse(x0))))))))), complement(x2))))))))))))
% 101.00/13.21 = { by lemma 53 R->L }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), converse(join(converse(composition(x0, meet(x1, composition(converse(x0), x2)))), composition(converse(x1), converse(x0)))))))))), complement(x2))))))))))))
% 101.00/13.21 = { by axiom 6 (converse_multiplicativity) R->L }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), converse(join(converse(composition(x0, meet(x1, composition(converse(x0), x2)))), converse(composition(x0, x1)))))))))), complement(x2))))))))))))
% 101.00/13.21 = { by axiom 8 (converse_additivity) R->L }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), converse(converse(join(composition(x0, meet(x1, composition(converse(x0), x2))), composition(x0, x1)))))))))), complement(x2))))))))))))
% 101.00/13.21 = { by axiom 3 (converse_idempotence) }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, meet(composition(x0, meet(x1, composition(converse(x0), x2))), join(composition(x0, meet(x1, composition(converse(x0), x2))), composition(x0, x1)))))))), complement(x2))))))))))))
% 101.00/13.21 = { by lemma 49 }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(x2, composition(x0, meet(x1, composition(converse(x0), x2)))))))), complement(x2))))))))))))
% 101.00/13.21 = { by lemma 27 }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))), complement(x2))))))))))))
% 101.00/13.21 = { by lemma 40 R->L }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(composition(x0, x1), x2)))))), complement(x2))))))))))))
% 101.00/13.21 = { by lemma 38 R->L }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(complement(meet(composition(x0, x1), x2)))))), complement(x2))))))))))))
% 101.00/13.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(complement(meet(composition(x0, x1), x2))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))), complement(x2))))))))))))
% 101.00/13.21 = { by lemma 27 }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), meet(join(complement(complement(meet(composition(x0, x1), x2))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))), complement(x2))))))))))))
% 101.00/13.21 = { by axiom 10 (maddux4_definiton_of_meet) }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), complement(join(complement(join(complement(complement(meet(composition(x0, x1), x2))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))), complement(x2))))))))))))
% 101.00/13.21 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), complement(join(meet(complement(meet(composition(x0, x1), x2)), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))), complement(x2))))))))))))
% 101.00/13.21 = { by axiom 2 (maddux1_join_commutativity) }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), complement(join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), meet(complement(meet(composition(x0, x1), x2)), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))), complement(x2))))))))))))
% 101.00/13.21 = { by lemma 27 R->L }
% 101.00/13.21 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(join(meet(composition(x0, x1), x2), complement(join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(composition(x0, x1), x2)))))), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 40 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(meet(join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(composition(x0, x1), x2)))), complement(meet(composition(x0, x1), x2)))), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 27 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(meet(complement(meet(composition(x0, x1), x2)), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(composition(x0, x1), x2)))))), complement(x2))))))))))))
% 101.00/13.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(meet(complement(meet(composition(x0, x1), x2)), join(meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(composition(x0, x1), x2))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 40 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(meet(complement(meet(composition(x0, x1), x2)), complement(meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(meet(composition(x0, x1), x2)))))))), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 38 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(meet(complement(meet(composition(x0, x1), x2)), complement(meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(complement(meet(composition(x0, x1), x2)))))))), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 37 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(join(zero, meet(complement(meet(composition(x0, x1), x2)), complement(meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(complement(meet(composition(x0, x1), x2))))))))), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 38 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(join(zero, meet(complement(meet(composition(x0, x1), x2)), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(complement(meet(composition(x0, x1), x2))))))))), complement(x2))))))))))))
% 101.00/13.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(join(zero, meet(complement(meet(composition(x0, x1), x2)), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(join(complement(complement(meet(composition(x0, x1), x2))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))))), complement(x2))))))))))))
% 101.00/13.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(join(meet(complement(meet(composition(x0, x1), x2)), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(join(complement(complement(meet(composition(x0, x1), x2))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))), zero)), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 14 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(join(meet(complement(meet(composition(x0, x1), x2)), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(join(complement(complement(meet(composition(x0, x1), x2))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))), complement(top))), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 21 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(join(meet(complement(meet(composition(x0, x1), x2)), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(join(complement(complement(meet(composition(x0, x1), x2))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))))), complement(join(complement(complement(meet(composition(x0, x1), x2))), join(complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), complement(join(complement(complement(meet(composition(x0, x1), x2))), complement(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))))))))), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 25 }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(complement(meet(composition(x0, x1), x2))), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 28 }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(meet(composition(x0, x1), x2), complement(x2))))))))))))
% 101.00/13.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(x2), meet(composition(x0, x1), x2))))))))))))
% 101.00/13.22 = { by lemma 50 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(x2), meet(meet(composition(x0, x1), x2), join(composition(x0, x1), complement(x2))))))))))))))
% 101.00/13.22 = { by lemma 27 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(x2), meet(meet(x2, composition(x0, x1)), join(composition(x0, x1), complement(x2))))))))))))))
% 101.00/13.22 = { by lemma 51 }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(composition(x0, x1), complement(x2))))))))))))
% 101.00/13.22 = { by lemma 56 R->L }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(x2), join(composition(x0, x1), composition(x0, complement(composition(converse(x0), x2)))))))))))))))
% 101.00/13.22 = { by lemma 54 }
% 101.00/13.22 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(x2), composition(x0, join(complement(composition(converse(x0), x2)), x1)))))))))))))
% 101.00/13.23 = { by axiom 2 (maddux1_join_commutativity) }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(x2), composition(x0, join(x1, complement(composition(converse(x0), x2)))))))))))))))
% 101.00/13.23 = { by lemma 52 R->L }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(x2), composition(x0, join(meet(x1, composition(converse(x0), x2)), complement(composition(converse(x0), x2)))))))))))))))
% 101.00/13.23 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(x2), composition(x0, join(complement(composition(converse(x0), x2)), meet(x1, composition(converse(x0), x2)))))))))))))))
% 101.00/13.23 = { by lemma 54 R->L }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(complement(x2), join(composition(x0, meet(x1, composition(converse(x0), x2))), composition(x0, complement(composition(converse(x0), x2)))))))))))))))
% 101.00/13.23 = { by lemma 56 }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(composition(x0, meet(x1, composition(converse(x0), x2))), complement(x2))))))))))))
% 101.00/13.23 = { by lemma 52 R->L }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(complement(x2), converse(complement(converse(join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(x2))))))))))))
% 101.00/13.23 = { by lemma 55 }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), converse(complement(converse(top)))))
% 101.00/13.23 = { by lemma 44 }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), complement(top)))
% 101.00/13.23 = { by lemma 14 }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), zero))
% 101.00/13.23 = { by lemma 30 }
% 101.00/13.23 meet(join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2))
% 101.00/13.23 = { by lemma 27 }
% 101.00/13.23 meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, x1), x2), meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)))
% 101.00/13.23 = { by axiom 2 (maddux1_join_commutativity) }
% 101.00/13.23 meet(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), join(meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2), meet(composition(x0, x1), x2)))
% 101.00/13.23 = { by lemma 49 }
% 101.00/13.23 meet(composition(x0, meet(x1, composition(converse(x0), x2))), x2)
% 101.00/13.23 % SZS output end Proof
% 101.00/13.23
% 101.00/13.23 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------