0.04/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n023.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 13:33:37 EDT 2019
0.12/0.33	% CPUTime    : 
8.47/8.66	% SZS status Unsatisfiable
8.47/8.66	
8.47/8.66	% SZS output start Proof
8.47/8.66	Take the following subset of the input axioms:
8.47/8.66	  fof(composition_associativity_5, axiom, ![A, C, B]: composition(composition(A, B), C)=composition(A, composition(B, C))).
8.47/8.66	  fof(composition_distributivity_7, axiom, ![A, C, B]: join(composition(A, C), composition(B, C))=composition(join(A, B), C)).
8.47/8.66	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
8.47/8.66	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
8.47/8.66	  fof(converse_cancellativity_11, axiom, ![A, B]: complement(B)=join(composition(converse(A), complement(composition(A, B))), complement(B))).
8.47/8.66	  fof(converse_idempotence_8, axiom, ![A]: A=converse(converse(A))).
8.47/8.66	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
8.47/8.66	  fof(def_top_12, axiom, ![A]: top=join(A, complement(A))).
8.47/8.66	  fof(def_zero_13, axiom, ![A]: meet(A, complement(A))=zero).
8.47/8.66	  fof(goals_17, negated_conjecture, join(sk1, one)=one).
8.47/8.66	  fof(goals_18, negated_conjecture, join(sk2, one)=one).
8.47/8.66	  fof(goals_19, negated_conjecture, tuple(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))!=tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))))).
8.47/8.66	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(A, B)=join(B, A)).
8.47/8.66	  fof(maddux2_join_associativity_2, axiom, ![A, C, B]: join(A, join(B, C))=join(join(A, B), C)).
8.47/8.66	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: A=join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))).
8.47/8.66	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
8.47/8.66	
8.47/8.66	Now clausify the problem and encode Horn clauses using encoding 3 of
8.47/8.66	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
8.47/8.66	We repeatedly replace C & s=t => u=v by the two clauses:
8.47/8.66	  fresh(y, y, x1...xn) = u
8.47/8.66	  C => fresh(s, t, x1...xn) = v
8.47/8.66	where fresh is a fresh function symbol and x1..xn are the free
8.47/8.66	variables of u and v.
8.47/8.66	A predicate p(X) is encoded as p(X)=true (this is sound, because the
8.47/8.66	input problem has no model of domain size 1).
8.47/8.66	
8.47/8.66	The encoding turns the above axioms into the following unit equations and goals:
8.47/8.66	
8.47/8.66	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
8.47/8.66	Axiom 2 (composition_identity_6): X = composition(X, one).
8.47/8.66	Axiom 3 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
8.47/8.66	Axiom 4 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
8.47/8.66	Axiom 5 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
8.47/8.66	Axiom 6 (goals_17): join(sk1, one) = one.
8.47/8.66	Axiom 7 (goals_18): join(sk2, one) = one.
8.47/8.66	Axiom 8 (converse_idempotence_8): X = converse(converse(X)).
8.47/8.66	Axiom 9 (converse_cancellativity_11): complement(X) = join(composition(converse(Y), complement(composition(Y, X))), complement(X)).
8.47/8.66	Axiom 10 (def_top_12): top = join(X, complement(X)).
8.47/8.66	Axiom 11 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
8.47/8.66	Axiom 12 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
8.47/8.66	Axiom 13 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
8.47/8.66	Axiom 14 (def_zero_13): meet(X, complement(X)) = zero.
8.47/8.70	Axiom 15 (composition_distributivity_7): join(composition(X, Y), composition(Z, Y)) = composition(join(X, Z), Y).
8.47/8.70	
8.47/8.70	Lemma 16: meet(X, Y) = meet(Y, X).
8.47/8.70	Proof:
8.47/8.70	  meet(X, Y)
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  complement(join(complement(X), complement(Y)))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  complement(join(complement(Y), complement(X)))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  meet(Y, X)
8.47/8.70	
8.47/8.70	Lemma 17: complement(top) = zero.
8.47/8.70	Proof:
8.47/8.70	  complement(top)
8.47/8.70	= { by axiom 10 (def_top_12) }
8.47/8.70	  complement(join(complement(?), complement(complement(?))))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  meet(?, complement(?))
8.47/8.70	= { by axiom 14 (def_zero_13) }
8.47/8.70	  zero
8.47/8.70	
8.47/8.70	Lemma 18: complement(join(zero, complement(X))) = meet(X, top).
8.47/8.70	Proof:
8.47/8.70	  complement(join(zero, complement(X)))
8.47/8.70	= { by lemma 17 }
8.47/8.70	  complement(join(complement(top), complement(X)))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  meet(top, X)
8.47/8.70	= { by lemma 16 }
8.47/8.70	  meet(X, top)
8.47/8.70	
8.47/8.70	Lemma 19: composition(converse(one), X) = X.
8.47/8.70	Proof:
8.47/8.70	  composition(converse(one), X)
8.47/8.70	= { by axiom 8 (converse_idempotence_8) }
8.47/8.70	  composition(converse(one), converse(converse(X)))
8.47/8.70	= { by axiom 12 (converse_multiplicativity_10) }
8.47/8.70	  converse(composition(converse(X), one))
8.47/8.70	= { by axiom 2 (composition_identity_6) }
8.47/8.70	  converse(converse(X))
8.47/8.70	= { by axiom 8 (converse_idempotence_8) }
8.47/8.70	  X
8.47/8.70	
8.47/8.70	Lemma 20: composition(one, X) = X.
8.47/8.70	Proof:
8.47/8.70	  composition(one, X)
8.47/8.70	= { by lemma 19 }
8.47/8.70	  composition(converse(one), composition(one, X))
8.47/8.70	= { by axiom 4 (composition_associativity_5) }
8.47/8.70	  composition(composition(converse(one), one), X)
8.47/8.70	= { by axiom 2 (composition_identity_6) }
8.47/8.70	  composition(converse(one), X)
8.47/8.70	= { by lemma 19 }
8.47/8.70	  X
8.47/8.70	
8.47/8.70	Lemma 21: join(complement(X), complement(X)) = complement(X).
8.47/8.70	Proof:
8.47/8.70	  join(complement(X), complement(X))
8.47/8.70	= { by lemma 19 }
8.47/8.70	  join(complement(X), composition(converse(one), complement(X)))
8.47/8.70	= { by lemma 20 }
8.47/8.70	  join(complement(X), composition(converse(one), complement(composition(one, X))))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(composition(converse(one), complement(composition(one, X))), complement(X))
8.47/8.70	= { by axiom 9 (converse_cancellativity_11) }
8.47/8.70	  complement(X)
8.47/8.70	
8.47/8.70	Lemma 22: join(X, join(Y, Z)) = join(Z, join(X, Y)).
8.47/8.70	Proof:
8.47/8.70	  join(X, join(Y, Z))
8.47/8.70	= { by axiom 5 (maddux2_join_associativity_2) }
8.47/8.70	  join(join(X, Y), Z)
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(Z, join(X, Y))
8.47/8.70	
8.47/8.70	Lemma 23: join(X, join(Y, complement(X))) = join(Y, top).
8.47/8.70	Proof:
8.47/8.70	  join(X, join(Y, complement(X)))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(X, join(complement(X), Y))
8.47/8.70	= { by axiom 5 (maddux2_join_associativity_2) }
8.47/8.70	  join(join(X, complement(X)), Y)
8.47/8.70	= { by axiom 10 (def_top_12) }
8.47/8.70	  join(top, Y)
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(Y, top)
8.47/8.70	
8.47/8.70	Lemma 24: join(top, complement(X)) = top.
8.47/8.70	Proof:
8.47/8.70	  join(top, complement(X))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(complement(X), top)
8.47/8.70	= { by lemma 23 }
8.47/8.70	  join(X, join(complement(X), complement(X)))
8.47/8.70	= { by lemma 21 }
8.47/8.70	  join(X, complement(X))
8.47/8.70	= { by axiom 10 (def_top_12) }
8.47/8.70	  top
8.47/8.70	
8.47/8.70	Lemma 25: join(X, top) = top.
8.47/8.70	Proof:
8.47/8.70	  join(X, top)
8.47/8.70	= { by axiom 10 (def_top_12) }
8.47/8.70	  join(X, join(complement(X), complement(complement(X))))
8.47/8.70	= { by lemma 22 }
8.47/8.70	  join(complement(X), join(complement(complement(X)), X))
8.47/8.70	= { by lemma 22 }
8.47/8.70	  join(complement(complement(X)), join(X, complement(X)))
8.47/8.70	= { by axiom 10 (def_top_12) }
8.47/8.70	  join(complement(complement(X)), top)
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(top, complement(complement(X)))
8.47/8.70	= { by lemma 24 }
8.47/8.70	  top
8.47/8.70	
8.47/8.70	Lemma 26: join(meet(X, Y), complement(join(complement(X), Y))) = X.
8.47/8.70	Proof:
8.47/8.70	  join(meet(X, Y), complement(join(complement(X), Y)))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
8.47/8.70	= { by axiom 13 (maddux3_a_kind_of_de_Morgan_3) }
8.47/8.70	  X
8.47/8.70	
8.47/8.70	Lemma 27: join(zero, meet(X, top)) = X.
8.47/8.70	Proof:
8.47/8.70	  join(zero, meet(X, top))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(meet(X, top), zero)
8.47/8.70	= { by lemma 17 }
8.47/8.70	  join(meet(X, top), complement(top))
8.47/8.70	= { by lemma 25 }
8.47/8.70	  join(meet(X, top), complement(join(complement(X), top)))
8.47/8.70	= { by lemma 26 }
8.47/8.70	  X
8.47/8.70	
8.47/8.70	Lemma 28: join(zero, meet(X, X)) = X.
8.47/8.70	Proof:
8.47/8.70	  join(zero, meet(X, X))
8.47/8.70	= { by axiom 14 (def_zero_13) }
8.47/8.70	  join(meet(X, complement(X)), meet(X, X))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
8.47/8.70	= { by lemma 26 }
8.47/8.70	  X
8.47/8.70	
8.47/8.70	Lemma 29: join(X, meet(Y, Y)) = join(Y, meet(X, X)).
8.47/8.70	Proof:
8.47/8.70	  join(X, meet(Y, Y))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(meet(Y, Y), X)
8.47/8.70	= { by lemma 28 }
8.47/8.70	  join(meet(Y, Y), join(zero, meet(X, X)))
8.47/8.70	= { by axiom 5 (maddux2_join_associativity_2) }
8.47/8.70	  join(join(meet(Y, Y), zero), meet(X, X))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(join(zero, meet(Y, Y)), meet(X, X))
8.47/8.70	= { by lemma 28 }
8.47/8.70	  join(Y, meet(X, X))
8.47/8.70	
8.47/8.70	Lemma 30: join(X, zero) = X.
8.47/8.70	Proof:
8.47/8.70	  join(X, zero)
8.47/8.70	= { by lemma 17 }
8.47/8.70	  join(X, complement(top))
8.47/8.70	= { by lemma 24 }
8.47/8.70	  join(X, complement(join(top, complement(zero))))
8.47/8.70	= { by lemma 26 }
8.47/8.70	  join(X, complement(join(join(meet(top, complement(complement(top))), complement(join(complement(top), complement(complement(top))))), complement(zero))))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  join(X, complement(join(join(meet(top, complement(complement(top))), meet(top, complement(top))), complement(zero))))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(X, complement(join(join(meet(top, complement(top)), meet(top, complement(complement(top)))), complement(zero))))
8.47/8.70	= { by axiom 14 (def_zero_13) }
8.47/8.70	  join(X, complement(join(join(zero, meet(top, complement(complement(top)))), complement(zero))))
8.47/8.70	= { by lemma 21 }
8.47/8.70	  join(X, complement(join(join(zero, meet(top, complement(join(complement(top), complement(top))))), complement(zero))))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  join(X, complement(join(join(zero, meet(top, meet(top, top))), complement(zero))))
8.47/8.70	= { by lemma 16 }
8.47/8.70	  join(X, complement(join(join(zero, meet(meet(top, top), top)), complement(zero))))
8.47/8.70	= { by lemma 27 }
8.47/8.70	  join(X, complement(join(meet(top, top), complement(zero))))
8.47/8.70	= { by lemma 18 }
8.47/8.70	  join(X, complement(join(complement(join(zero, complement(top))), complement(zero))))
8.47/8.70	= { by lemma 17 }
8.47/8.70	  join(X, complement(join(complement(join(complement(top), complement(top))), complement(zero))))
8.47/8.70	= { by lemma 21 }
8.47/8.70	  join(X, complement(join(complement(complement(top)), complement(zero))))
8.47/8.70	= { by lemma 17 }
8.47/8.70	  join(X, complement(join(complement(zero), complement(zero))))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  join(X, meet(zero, zero))
8.47/8.70	= { by lemma 29 }
8.47/8.70	  join(zero, meet(X, X))
8.47/8.70	= { by lemma 28 }
8.47/8.70	  X
8.47/8.70	
8.47/8.70	Lemma 31: join(zero, X) = X.
8.47/8.70	Proof:
8.47/8.70	  join(zero, X)
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  join(X, zero)
8.47/8.70	= { by lemma 30 }
8.47/8.70	  X
8.47/8.70	
8.47/8.70	Lemma 32: meet(X, top) = X.
8.47/8.70	Proof:
8.47/8.70	  meet(X, top)
8.47/8.70	= { by lemma 31 }
8.47/8.70	  join(zero, meet(X, top))
8.47/8.70	= { by lemma 27 }
8.47/8.70	  X
8.47/8.70	
8.47/8.70	Lemma 33: complement(complement(X)) = X.
8.47/8.70	Proof:
8.47/8.70	  complement(complement(X))
8.47/8.70	= { by lemma 31 }
8.47/8.70	  complement(join(zero, complement(X)))
8.47/8.70	= { by lemma 18 }
8.47/8.70	  meet(X, top)
8.47/8.70	= { by lemma 32 }
8.47/8.70	  X
8.47/8.70	
8.47/8.70	Lemma 34: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
8.47/8.70	Proof:
8.47/8.70	  complement(join(complement(X), meet(Y, Z)))
8.47/8.70	= { by lemma 16 }
8.47/8.70	  complement(join(complement(X), meet(Z, Y)))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  complement(join(meet(Z, Y), complement(X)))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
8.47/8.70	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.70	  meet(join(complement(Z), complement(Y)), X)
8.47/8.70	= { by lemma 16 }
8.47/8.70	  meet(X, join(complement(Z), complement(Y)))
8.47/8.70	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.70	  meet(X, join(complement(Y), complement(Z)))
8.47/8.70	
8.47/8.70	Lemma 35: join(complement(X), complement(Y)) = complement(meet(X, Y)).
8.47/8.70	Proof:
8.47/8.70	  join(complement(X), complement(Y))
8.47/8.70	= { by lemma 32 }
8.47/8.70	  meet(join(complement(X), complement(Y)), top)
8.47/8.70	= { by lemma 16 }
8.47/8.70	  meet(top, join(complement(X), complement(Y)))
8.47/8.70	= { by lemma 34 }
8.47/8.70	  complement(join(complement(top), meet(X, Y)))
8.47/8.70	= { by lemma 17 }
8.47/8.70	  complement(join(zero, meet(X, Y)))
8.47/8.70	= { by lemma 31 }
8.47/8.71	  complement(meet(X, Y))
8.47/8.71	
8.47/8.71	Lemma 36: complement(meet(X, complement(Y))) = join(Y, complement(X)).
8.47/8.71	Proof:
8.47/8.71	  complement(meet(X, complement(Y)))
8.47/8.71	= { by lemma 16 }
8.47/8.71	  complement(meet(complement(Y), X))
8.47/8.71	= { by lemma 31 }
8.47/8.71	  complement(meet(join(zero, complement(Y)), X))
8.47/8.71	= { by lemma 35 }
8.47/8.71	  join(complement(join(zero, complement(Y))), complement(X))
8.47/8.71	= { by lemma 18 }
8.47/8.71	  join(meet(Y, top), complement(X))
8.47/8.71	= { by lemma 32 }
8.47/8.71	  join(Y, complement(X))
8.47/8.71	
8.47/8.71	Lemma 37: meet(X, meet(Y, Z)) = meet(Z, meet(X, Y)).
8.47/8.71	Proof:
8.47/8.71	  meet(X, meet(Y, Z))
8.47/8.71	= { by lemma 32 }
8.47/8.71	  meet(meet(X, meet(Y, Z)), top)
8.47/8.71	= { by lemma 18 }
8.47/8.71	  complement(join(zero, complement(meet(X, meet(Y, Z)))))
8.47/8.71	= { by lemma 35 }
8.47/8.71	  complement(join(zero, join(complement(X), complement(meet(Y, Z)))))
8.47/8.71	= { by lemma 35 }
8.47/8.71	  complement(join(zero, join(complement(X), join(complement(Y), complement(Z)))))
8.47/8.71	= { by axiom 5 (maddux2_join_associativity_2) }
8.47/8.71	  complement(join(zero, join(join(complement(X), complement(Y)), complement(Z))))
8.47/8.71	= { by lemma 36 }
8.47/8.71	  complement(join(zero, complement(meet(Z, complement(join(complement(X), complement(Y)))))))
8.47/8.71	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.47/8.71	  complement(join(zero, complement(meet(Z, meet(X, Y)))))
8.47/8.71	= { by lemma 18 }
8.47/8.71	  meet(meet(Z, meet(X, Y)), top)
8.47/8.71	= { by lemma 32 }
8.47/8.71	  meet(Z, meet(X, Y))
8.47/8.71	
8.47/8.71	Lemma 38: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
8.47/8.71	Proof:
8.47/8.71	  join(meet(X, Y), complement(join(Y, complement(X))))
8.47/8.71	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.71	  join(meet(X, Y), complement(join(complement(X), Y)))
8.47/8.71	= { by lemma 26 }
8.47/8.71	  X
8.47/8.71	
8.47/8.71	Lemma 39: meet(sk2, meet(one, X)) = meet(X, sk2).
8.47/8.71	Proof:
8.47/8.71	  meet(sk2, meet(one, X))
8.47/8.71	= { by lemma 37 }
8.47/8.71	  meet(one, meet(X, sk2))
8.47/8.71	= { by lemma 37 }
8.47/8.71	  meet(X, meet(sk2, one))
8.47/8.71	= { by lemma 30 }
8.47/8.71	  meet(X, join(meet(sk2, one), zero))
8.47/8.71	= { by lemma 17 }
8.47/8.71	  meet(X, join(meet(sk2, one), complement(top)))
8.47/8.71	= { by lemma 25 }
8.47/8.71	  meet(X, join(meet(sk2, one), complement(join(one, top))))
8.47/8.71	= { by lemma 23 }
8.47/8.71	  meet(X, join(meet(sk2, one), complement(join(sk2, join(one, complement(sk2))))))
8.47/8.71	= { by axiom 5 (maddux2_join_associativity_2) }
8.47/8.71	  meet(X, join(meet(sk2, one), complement(join(join(sk2, one), complement(sk2)))))
8.47/8.71	= { by axiom 7 (goals_18) }
8.47/8.71	  meet(X, join(meet(sk2, one), complement(join(one, complement(sk2)))))
8.47/8.71	= { by lemma 38 }
8.47/8.71	  meet(X, sk2)
8.47/8.71	
8.47/8.71	Lemma 40: meet(X, X) = X.
8.47/8.71	Proof:
8.47/8.71	  meet(X, X)
8.47/8.71	= { by lemma 31 }
8.47/8.71	  join(zero, meet(X, X))
8.47/8.71	= { by lemma 28 }
8.47/8.71	  X
8.47/8.71	
8.47/8.71	Lemma 41: complement(join(Y, complement(X))) = meet(X, complement(Y)).
8.47/8.71	Proof:
8.47/8.71	  complement(join(Y, complement(X)))
8.47/8.71	= { by lemma 40 }
8.47/8.71	  complement(join(Y, meet(complement(X), complement(X))))
8.47/8.71	= { by lemma 29 }
8.47/8.71	  complement(join(complement(X), meet(Y, Y)))
8.47/8.71	= { by lemma 34 }
8.47/8.71	  meet(X, join(complement(Y), complement(Y)))
8.47/8.71	= { by lemma 35 }
8.47/8.71	  meet(X, complement(meet(Y, Y)))
8.47/8.71	= { by lemma 40 }
8.47/8.71	  meet(X, complement(Y))
8.47/8.71	
8.47/8.71	Lemma 42: join(meet(Y, complement(X)), complement(join(X, Y))) = complement(X).
8.47/8.71	Proof:
8.47/8.71	  join(meet(Y, complement(X)), complement(join(X, Y)))
8.47/8.71	= { by lemma 16 }
8.47/8.71	  join(meet(complement(X), Y), complement(join(X, Y)))
8.47/8.71	= { by lemma 31 }
8.47/8.71	  join(meet(join(zero, complement(X)), Y), complement(join(X, Y)))
8.47/8.71	= { by axiom 1 (maddux1_join_commutativity_1) }
8.47/8.71	  join(meet(join(zero, complement(X)), Y), complement(join(Y, X)))
8.47/8.71	= { by lemma 32 }
8.47/8.71	  join(meet(join(zero, complement(X)), Y), complement(join(Y, meet(X, top))))
8.47/8.71	= { by lemma 18 }
8.47/8.71	  join(meet(join(zero, complement(X)), Y), complement(join(Y, complement(join(zero, complement(X))))))
8.47/8.71	= { by lemma 38 }
8.47/8.71	  join(zero, complement(X))
8.47/8.71	= { by lemma 31 }
8.56/8.74	  complement(X)
8.56/8.74	
8.56/8.74	Lemma 43: meet(X, join(Y, complement(X))) = meet(X, Y).
8.56/8.74	Proof:
8.56/8.74	  meet(X, join(Y, complement(X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  meet(X, join(complement(X), Y))
8.56/8.74	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.56/8.74	  complement(join(complement(X), complement(join(complement(X), Y))))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  complement(join(complement(join(complement(X), Y)), complement(X)))
8.56/8.74	= { by lemma 33 }
8.56/8.74	  complement(join(complement(join(complement(X), complement(complement(Y)))), complement(X)))
8.56/8.74	= { by axiom 8 (converse_idempotence_8) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(converse(converse(complement(Y)))))), complement(X)))
8.56/8.74	= { by lemma 26 }
8.56/8.74	  complement(join(complement(join(complement(X), complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(complement(X))))), complement(join(complement(converse(converse(complement(Y)))), converse(join(converse(complement(Y)), converse(complement(X)))))))))), complement(X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(complement(X))))), complement(join(complement(converse(converse(complement(Y)))), converse(join(converse(complement(X)), converse(complement(Y)))))))))), complement(X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(complement(X))))), complement(join(converse(join(converse(complement(X)), converse(complement(Y)))), complement(converse(converse(complement(Y)))))))))), complement(X)))
8.56/8.74	= { by axiom 3 (converse_additivity_9) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(complement(X))))), complement(join(join(converse(converse(complement(X))), converse(converse(complement(Y)))), complement(converse(converse(complement(Y)))))))))), complement(X)))
8.56/8.74	= { by axiom 5 (maddux2_join_associativity_2) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(complement(X))))), complement(join(converse(converse(complement(X))), join(converse(converse(complement(Y))), complement(converse(converse(complement(Y))))))))))), complement(X)))
8.56/8.74	= { by axiom 10 (def_top_12) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(complement(X))))), complement(join(converse(converse(complement(X))), top)))))), complement(X)))
8.56/8.74	= { by lemma 25 }
8.56/8.74	  complement(join(complement(join(complement(X), complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(complement(X))))), complement(top))))), complement(X)))
8.56/8.74	= { by lemma 17 }
8.56/8.74	  complement(join(complement(join(complement(X), complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(complement(X))))), zero)))), complement(X)))
8.56/8.74	= { by lemma 30 }
8.56/8.74	  complement(join(complement(join(complement(X), complement(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(complement(X)))))))), complement(X)))
8.56/8.74	= { by axiom 8 (converse_idempotence_8) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(meet(complement(Y), converse(join(converse(complement(Y)), converse(complement(X)))))))), complement(X)))
8.56/8.74	= { by axiom 3 (converse_additivity_9) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(meet(complement(Y), converse(converse(join(complement(Y), complement(X)))))))), complement(X)))
8.56/8.74	= { by axiom 8 (converse_idempotence_8) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(meet(complement(Y), join(complement(Y), complement(X)))))), complement(X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  complement(join(complement(join(complement(X), complement(meet(complement(Y), join(complement(X), complement(Y)))))), complement(X)))
8.56/8.74	= { by lemma 16 }
8.56/8.74	  complement(join(complement(join(complement(X), complement(meet(join(complement(X), complement(Y)), complement(Y))))), complement(X)))
8.56/8.74	= { by lemma 36 }
8.56/8.74	  complement(join(complement(join(complement(X), join(Y, complement(join(complement(X), complement(Y)))))), complement(X)))
8.56/8.74	= { by lemma 41 }
8.56/8.74	  complement(join(complement(join(complement(X), join(Y, meet(Y, complement(complement(X)))))), complement(X)))
8.56/8.74	= { by axiom 5 (maddux2_join_associativity_2) }
8.56/8.74	  complement(join(complement(join(join(complement(X), Y), meet(Y, complement(complement(X))))), complement(X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  complement(join(complement(join(meet(Y, complement(complement(X))), join(complement(X), Y))), complement(X)))
8.56/8.74	= { by lemma 32 }
8.56/8.74	  complement(join(complement(join(meet(Y, complement(complement(X))), meet(join(complement(X), Y), top))), complement(X)))
8.56/8.74	= { by lemma 18 }
8.56/8.74	  complement(join(complement(join(meet(Y, complement(complement(X))), complement(join(zero, complement(join(complement(X), Y)))))), complement(X)))
8.56/8.74	= { by lemma 41 }
8.56/8.74	  complement(join(meet(join(zero, complement(join(complement(X), Y))), complement(meet(Y, complement(complement(X))))), complement(X)))
8.56/8.74	= { by lemma 31 }
8.56/8.74	  complement(join(meet(complement(join(complement(X), Y)), complement(meet(Y, complement(complement(X))))), complement(X)))
8.56/8.74	= { by lemma 33 }
8.56/8.74	  complement(join(meet(complement(join(complement(X), Y)), complement(meet(Y, complement(complement(X))))), complement(complement(complement(X)))))
8.56/8.74	= { by lemma 42 }
8.56/8.74	  complement(join(meet(complement(join(complement(X), Y)), complement(meet(Y, complement(complement(X))))), complement(join(meet(Y, complement(complement(X))), complement(join(complement(X), Y))))))
8.56/8.74	= { by lemma 42 }
8.56/8.74	  complement(complement(meet(Y, complement(complement(X)))))
8.56/8.74	= { by lemma 36 }
8.56/8.74	  complement(join(complement(X), complement(Y)))
8.56/8.74	= { by axiom 11 (maddux4_definiton_of_meet_4) }
8.56/8.74	  meet(X, Y)
8.56/8.74	
8.56/8.74	Lemma 44: meet(sk2, join(complement(one), X)) = meet(X, sk2).
8.56/8.74	Proof:
8.56/8.74	  meet(sk2, join(complement(one), X))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  meet(sk2, join(X, complement(one)))
8.56/8.74	= { by lemma 16 }
8.56/8.74	  meet(join(X, complement(one)), sk2)
8.56/8.74	= { by lemma 39 }
8.56/8.74	  meet(sk2, meet(one, join(X, complement(one))))
8.56/8.74	= { by lemma 43 }
8.56/8.74	  meet(sk2, meet(one, X))
8.56/8.74	= { by lemma 39 }
8.56/8.74	  meet(X, sk2)
8.56/8.74	
8.56/8.74	Lemma 45: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
8.56/8.74	Proof:
8.56/8.74	  join(composition(X, Y), composition(X, Z))
8.56/8.74	= { by axiom 8 (converse_idempotence_8) }
8.56/8.74	  converse(converse(join(composition(X, Y), composition(X, Z))))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  converse(converse(join(composition(X, Z), composition(X, Y))))
8.56/8.74	= { by axiom 3 (converse_additivity_9) }
8.56/8.74	  converse(join(converse(composition(X, Z)), converse(composition(X, Y))))
8.56/8.74	= { by axiom 12 (converse_multiplicativity_10) }
8.56/8.74	  converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
8.56/8.74	= { by axiom 12 (converse_multiplicativity_10) }
8.56/8.74	  converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
8.56/8.74	= { by axiom 15 (composition_distributivity_7) }
8.56/8.74	  converse(composition(join(converse(Y), converse(Z)), converse(X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  converse(composition(join(converse(Z), converse(Y)), converse(X)))
8.56/8.74	= { by axiom 3 (converse_additivity_9) }
8.56/8.74	  converse(composition(converse(join(Z, Y)), converse(X)))
8.56/8.74	= { by axiom 12 (converse_multiplicativity_10) }
8.56/8.74	  converse(converse(composition(X, join(Z, Y))))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  converse(converse(composition(X, join(Y, Z))))
8.56/8.74	= { by axiom 8 (converse_idempotence_8) }
8.56/8.74	  composition(X, join(Y, Z))
8.56/8.74	
8.56/8.74	Lemma 46: join(composition(join(Y, one), X), Z) = join(X, join(Z, composition(Y, X))).
8.56/8.74	Proof:
8.56/8.74	  join(composition(join(Y, one), X), Z)
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  join(composition(join(one, Y), X), Z)
8.56/8.74	= { by axiom 15 (composition_distributivity_7) }
8.56/8.74	  join(join(composition(one, X), composition(Y, X)), Z)
8.56/8.74	= { by lemma 20 }
8.56/8.74	  join(join(X, composition(Y, X)), Z)
8.56/8.74	= { by axiom 5 (maddux2_join_associativity_2) }
8.56/8.74	  join(X, join(composition(Y, X), Z))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  join(X, join(Z, composition(Y, X)))
8.56/8.74	
8.56/8.74	Lemma 47: join(X, composition(sk1, complement(X))) = join(X, composition(sk1, top)).
8.56/8.74	Proof:
8.56/8.74	  join(X, composition(sk1, complement(X)))
8.56/8.74	= { by lemma 20 }
8.56/8.74	  join(composition(one, X), composition(sk1, complement(X)))
8.56/8.74	= { by axiom 6 (goals_17) }
8.56/8.74	  join(composition(join(sk1, one), X), composition(sk1, complement(X)))
8.56/8.74	= { by lemma 46 }
8.56/8.74	  join(X, join(composition(sk1, complement(X)), composition(sk1, X)))
8.56/8.74	= { by lemma 45 }
8.56/8.74	  join(X, composition(sk1, join(complement(X), X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  join(X, composition(sk1, join(X, complement(X))))
8.56/8.74	= { by axiom 10 (def_top_12) }
8.56/8.74	  join(X, composition(sk1, top))
8.56/8.74	
8.56/8.74	Lemma 48: meet(X, join(complement(X), Y)) = meet(X, Y).
8.56/8.74	Proof:
8.56/8.74	  meet(X, join(complement(X), Y))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  meet(X, join(Y, complement(X)))
8.56/8.74	= { by lemma 43 }
8.56/8.74	  meet(X, Y)
8.56/8.74	
8.56/8.74	Lemma 49: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
8.56/8.74	Proof:
8.56/8.74	  join(complement(composition(X, Y)), composition(join(X, Z), Y))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  join(complement(composition(X, Y)), composition(join(Z, X), Y))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  join(composition(join(Z, X), Y), complement(composition(X, Y)))
8.56/8.74	= { by axiom 15 (composition_distributivity_7) }
8.56/8.74	  join(join(composition(Z, Y), composition(X, Y)), complement(composition(X, Y)))
8.56/8.74	= { by axiom 5 (maddux2_join_associativity_2) }
8.56/8.74	  join(composition(Z, Y), join(composition(X, Y), complement(composition(X, Y))))
8.56/8.74	= { by axiom 10 (def_top_12) }
8.56/8.74	  join(composition(Z, Y), top)
8.56/8.74	= { by lemma 25 }
8.56/8.74	  top
8.56/8.74	
8.56/8.74	Lemma 50: meet(X, composition(sk1, X)) = composition(sk1, X).
8.56/8.74	Proof:
8.56/8.74	  meet(X, composition(sk1, X))
8.56/8.74	= { by lemma 16 }
8.56/8.74	  meet(composition(sk1, X), X)
8.56/8.74	= { by lemma 30 }
8.56/8.74	  join(meet(composition(sk1, X), X), zero)
8.56/8.74	= { by lemma 17 }
8.56/8.74	  join(meet(composition(sk1, X), X), complement(top))
8.56/8.74	= { by lemma 49 }
8.56/8.74	  join(meet(composition(sk1, X), X), complement(join(complement(composition(sk1, X)), composition(join(sk1, one), X))))
8.56/8.74	= { by axiom 6 (goals_17) }
8.56/8.74	  join(meet(composition(sk1, X), X), complement(join(complement(composition(sk1, X)), composition(one, X))))
8.56/8.74	= { by lemma 20 }
8.56/8.74	  join(meet(composition(sk1, X), X), complement(join(complement(composition(sk1, X)), X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  join(meet(composition(sk1, X), X), complement(join(X, complement(composition(sk1, X)))))
8.56/8.74	= { by lemma 38 }
8.56/8.74	  composition(sk1, X)
8.56/8.74	
8.56/8.74	Lemma 51: meet(sk1, sk2) = composition(sk1, sk2).
8.56/8.74	Proof:
8.56/8.74	  meet(sk1, sk2)
8.56/8.74	= { by axiom 2 (composition_identity_6) }
8.56/8.74	  meet(composition(sk1, one), sk2)
8.56/8.74	= { by lemma 33 }
8.56/8.74	  meet(composition(sk1, complement(complement(one))), sk2)
8.56/8.74	= { by lemma 44 }
8.56/8.74	  meet(sk2, join(complement(one), composition(sk1, complement(complement(one)))))
8.56/8.74	= { by lemma 47 }
8.56/8.74	  meet(sk2, join(complement(one), composition(sk1, top)))
8.56/8.74	= { by lemma 44 }
8.56/8.74	  meet(composition(sk1, top), sk2)
8.56/8.74	= { by lemma 16 }
8.56/8.74	  meet(sk2, composition(sk1, top))
8.56/8.74	= { by lemma 48 }
8.56/8.74	  meet(sk2, join(complement(sk2), composition(sk1, top)))
8.56/8.74	= { by lemma 47 }
8.56/8.74	  meet(sk2, join(complement(sk2), composition(sk1, complement(complement(sk2)))))
8.56/8.74	= { by lemma 48 }
8.56/8.74	  meet(sk2, composition(sk1, complement(complement(sk2))))
8.56/8.74	= { by lemma 33 }
8.56/8.74	  meet(sk2, composition(sk1, sk2))
8.56/8.74	= { by lemma 50 }
8.56/8.74	  composition(sk1, sk2)
8.56/8.74	
8.56/8.74	Lemma 52: join(X, X) = X.
8.56/8.74	Proof:
8.56/8.74	  join(X, X)
8.56/8.74	= { by lemma 33 }
8.56/8.74	  join(complement(complement(X)), X)
8.56/8.74	= { by lemma 33 }
8.56/8.74	  join(complement(complement(X)), complement(complement(X)))
8.56/8.74	= { by lemma 21 }
8.56/8.74	  complement(complement(X))
8.56/8.74	= { by lemma 33 }
8.56/8.74	  X
8.56/8.74	
8.56/8.74	Lemma 53: join(X, composition(sk2, complement(X))) = join(X, composition(sk2, top)).
8.56/8.74	Proof:
8.56/8.74	  join(X, composition(sk2, complement(X)))
8.56/8.74	= { by lemma 20 }
8.56/8.74	  join(composition(one, X), composition(sk2, complement(X)))
8.56/8.74	= { by axiom 7 (goals_18) }
8.56/8.74	  join(composition(join(sk2, one), X), composition(sk2, complement(X)))
8.56/8.74	= { by lemma 46 }
8.56/8.74	  join(X, join(composition(sk2, complement(X)), composition(sk2, X)))
8.56/8.74	= { by lemma 45 }
8.56/8.74	  join(X, composition(sk2, join(complement(X), X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  join(X, composition(sk2, join(X, complement(X))))
8.56/8.74	= { by axiom 10 (def_top_12) }
8.56/8.74	  join(X, composition(sk2, top))
8.56/8.74	
8.56/8.74	Lemma 54: meet(X, composition(sk2, top)) = composition(sk2, X).
8.56/8.74	Proof:
8.56/8.74	  meet(X, composition(sk2, top))
8.56/8.74	= { by lemma 48 }
8.56/8.74	  meet(X, join(complement(X), composition(sk2, top)))
8.56/8.74	= { by lemma 53 }
8.56/8.74	  meet(X, join(complement(X), composition(sk2, complement(complement(X)))))
8.56/8.74	= { by lemma 48 }
8.56/8.74	  meet(X, composition(sk2, complement(complement(X))))
8.56/8.74	= { by lemma 33 }
8.56/8.74	  meet(X, composition(sk2, X))
8.56/8.74	= { by lemma 16 }
8.56/8.74	  meet(composition(sk2, X), X)
8.56/8.74	= { by lemma 30 }
8.56/8.74	  join(meet(composition(sk2, X), X), zero)
8.56/8.74	= { by lemma 17 }
8.56/8.74	  join(meet(composition(sk2, X), X), complement(top))
8.56/8.74	= { by lemma 49 }
8.56/8.74	  join(meet(composition(sk2, X), X), complement(join(complement(composition(sk2, X)), composition(join(sk2, one), X))))
8.56/8.74	= { by axiom 7 (goals_18) }
8.56/8.74	  join(meet(composition(sk2, X), X), complement(join(complement(composition(sk2, X)), composition(one, X))))
8.56/8.74	= { by lemma 20 }
8.56/8.74	  join(meet(composition(sk2, X), X), complement(join(complement(composition(sk2, X)), X)))
8.56/8.74	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.74	  join(meet(composition(sk2, X), X), complement(join(X, complement(composition(sk2, X)))))
8.56/8.74	= { by lemma 38 }
8.56/8.75	  composition(sk2, X)
8.56/8.75	
8.56/8.75	Lemma 55: meet(sk1, meet(one, X)) = meet(X, sk1).
8.56/8.75	Proof:
8.56/8.75	  meet(sk1, meet(one, X))
8.56/8.75	= { by lemma 37 }
8.56/8.75	  meet(one, meet(X, sk1))
8.56/8.75	= { by lemma 37 }
8.56/8.75	  meet(X, meet(sk1, one))
8.56/8.75	= { by lemma 30 }
8.56/8.75	  meet(X, join(meet(sk1, one), zero))
8.56/8.75	= { by lemma 17 }
8.56/8.75	  meet(X, join(meet(sk1, one), complement(top)))
8.56/8.75	= { by lemma 25 }
8.56/8.75	  meet(X, join(meet(sk1, one), complement(join(one, top))))
8.56/8.75	= { by lemma 23 }
8.56/8.75	  meet(X, join(meet(sk1, one), complement(join(sk1, join(one, complement(sk1))))))
8.56/8.75	= { by axiom 5 (maddux2_join_associativity_2) }
8.56/8.75	  meet(X, join(meet(sk1, one), complement(join(join(sk1, one), complement(sk1)))))
8.56/8.75	= { by axiom 6 (goals_17) }
8.56/8.75	  meet(X, join(meet(sk1, one), complement(join(one, complement(sk1)))))
8.56/8.75	= { by lemma 38 }
8.56/8.75	  meet(X, sk1)
8.56/8.75	
8.56/8.75	Lemma 56: meet(sk1, join(complement(one), X)) = meet(X, sk1).
8.56/8.75	Proof:
8.56/8.75	  meet(sk1, join(complement(one), X))
8.56/8.75	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.75	  meet(sk1, join(X, complement(one)))
8.56/8.75	= { by lemma 16 }
8.56/8.75	  meet(join(X, complement(one)), sk1)
8.56/8.75	= { by lemma 55 }
8.56/8.75	  meet(sk1, meet(one, join(X, complement(one))))
8.56/8.75	= { by lemma 43 }
8.56/8.75	  meet(sk1, meet(one, X))
8.56/8.75	= { by lemma 55 }
8.56/8.75	  meet(X, sk1)
8.56/8.75	
8.56/8.75	Lemma 57: composition(sk2, sk1) = composition(sk1, sk2).
8.56/8.75	Proof:
8.56/8.75	  composition(sk2, sk1)
8.56/8.75	= { by lemma 54 }
8.56/8.75	  meet(sk1, composition(sk2, top))
8.56/8.75	= { by lemma 16 }
8.56/8.75	  meet(composition(sk2, top), sk1)
8.56/8.75	= { by lemma 56 }
8.56/8.75	  meet(sk1, join(complement(one), composition(sk2, top)))
8.56/8.75	= { by lemma 53 }
8.56/8.75	  meet(sk1, join(complement(one), composition(sk2, complement(complement(one)))))
8.56/8.75	= { by lemma 56 }
8.56/8.75	  meet(composition(sk2, complement(complement(one))), sk1)
8.56/8.75	= { by lemma 33 }
8.56/8.75	  meet(composition(sk2, one), sk1)
8.56/8.75	= { by axiom 2 (composition_identity_6) }
8.56/8.75	  meet(sk2, sk1)
8.56/8.75	= { by lemma 16 }
8.56/8.75	  meet(sk1, sk2)
8.56/8.75	= { by lemma 51 }
8.56/8.75	  composition(sk1, sk2)
8.56/8.75	
8.56/8.75	Lemma 58: join(composition(X, W), composition(Y, composition(Z, W))) = composition(join(X, composition(Y, Z)), W).
8.56/8.75	Proof:
8.56/8.75	  join(composition(X, W), composition(Y, composition(Z, W)))
8.56/8.75	= { by axiom 4 (composition_associativity_5) }
8.56/8.75	  join(composition(X, W), composition(composition(Y, Z), W))
8.56/8.75	= { by axiom 15 (composition_distributivity_7) }
8.56/8.75	  composition(join(X, composition(Y, Z)), W)
8.56/8.75	
8.56/8.75	Lemma 59: meet(X, composition(sk2, Y)) = composition(sk2, meet(X, Y)).
8.56/8.75	Proof:
8.56/8.75	  meet(X, composition(sk2, Y))
8.56/8.75	= { by lemma 54 }
8.56/8.75	  meet(X, meet(Y, composition(sk2, top)))
8.56/8.75	= { by lemma 16 }
8.56/8.75	  meet(X, meet(composition(sk2, top), Y))
8.56/8.75	= { by lemma 37 }
8.56/8.75	  meet(composition(sk2, top), meet(Y, X))
8.56/8.75	= { by lemma 16 }
8.56/8.75	  meet(meet(Y, X), composition(sk2, top))
8.56/8.75	= { by lemma 54 }
8.56/8.75	  composition(sk2, meet(Y, X))
8.56/8.75	= { by lemma 16 }
8.56/8.75	  composition(sk2, meet(X, Y))
8.56/8.75	
8.56/8.75	Goal 1 (goals_19): tuple(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))) = tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))).
8.56/8.75	Proof:
8.56/8.75	  tuple(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by lemma 51 }
8.56/8.75	  tuple(composition(composition(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by lemma 52 }
8.56/8.75	  tuple(composition(join(composition(sk1, sk2), composition(sk1, sk2)), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by lemma 57 }
8.56/8.75	  tuple(composition(join(composition(sk1, sk2), composition(sk2, sk1)), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by lemma 58 }
8.56/8.75	  tuple(join(composition(composition(sk1, sk2), sk3), composition(sk2, composition(sk1, sk3))), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by lemma 51 }
8.56/8.75	  tuple(join(composition(meet(sk1, sk2), sk3), composition(sk2, composition(sk1, sk3))), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by lemma 50 }
8.56/8.75	  tuple(join(composition(meet(sk1, sk2), sk3), composition(sk2, meet(sk3, composition(sk1, sk3)))), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by lemma 16 }
8.56/8.75	  tuple(join(composition(meet(sk1, sk2), sk3), composition(sk2, meet(composition(sk1, sk3), sk3))), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by lemma 59 }
8.56/8.75	  tuple(join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by axiom 1 (maddux1_join_commutativity_1) }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), meet(composition(sk1, sk3), composition(sk2, sk3)))
8.56/8.75	= { by lemma 59 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), composition(sk2, meet(composition(sk1, sk3), sk3)))
8.56/8.75	= { by lemma 16 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), composition(sk2, meet(sk3, composition(sk1, sk3))))
8.56/8.75	= { by lemma 50 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), composition(sk2, composition(sk1, sk3)))
8.56/8.75	= { by axiom 4 (composition_associativity_5) }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), composition(composition(sk2, sk1), sk3))
8.56/8.75	= { by lemma 57 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), composition(composition(sk1, sk2), sk3))
8.56/8.75	= { by lemma 52 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), composition(join(composition(sk1, sk2), composition(sk1, sk2)), sk3))
8.56/8.75	= { by lemma 57 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), composition(join(composition(sk1, sk2), composition(sk2, sk1)), sk3))
8.56/8.75	= { by lemma 58 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(composition(sk1, sk2), sk3), composition(sk2, composition(sk1, sk3))))
8.56/8.75	= { by lemma 51 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), composition(sk2, composition(sk1, sk3))))
8.56/8.75	= { by lemma 50 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), composition(sk2, meet(sk3, composition(sk1, sk3)))))
8.56/8.75	= { by lemma 16 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), composition(sk2, meet(composition(sk1, sk3), sk3))))
8.56/8.75	= { by lemma 59 }
8.56/8.75	  tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))))
8.56/8.75	% SZS output end Proof
8.56/8.75	
8.56/8.75	RESULT: Unsatisfiable (the axioms are contradictory).
8.56/8.76	EOF