0.08/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.08/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.33	% Computer   : n016.cluster.edu
0.13/0.33	% Model      : x86_64 x86_64
0.13/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.33	% Memory     : 8042.1875MB
0.13/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.33	% CPULimit   : 180
0.13/0.33	% DateTime   : Thu Aug 29 14:03:08 EDT 2019
0.13/0.33	% CPUTime    : 
59.51/59.71	% SZS status Unsatisfiable
59.51/59.71	
59.51/59.71	% SZS output start Proof
59.51/59.71	Take the following subset of the input axioms:
59.67/59.83	  fof(composition_associativity_5, axiom, ![A, B, C]: composition(composition(A, B), C)=composition(A, composition(B, C))).
59.67/59.83	  fof(composition_distributivity_7, axiom, ![A, B, C]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))).
59.67/59.83	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
59.67/59.83	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
59.67/59.83	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
59.67/59.83	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
59.67/59.83	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
59.67/59.83	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
59.67/59.83	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
59.67/59.83	  fof(goals_14, negated_conjecture, sk1=composition(sk1, top)).
59.67/59.83	  fof(goals_15, negated_conjecture, composition(meet(sk1, sk2), sk3)!=meet(sk1, composition(sk2, sk3))).
59.67/59.83	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
59.67/59.83	  fof(maddux2_join_associativity_2, axiom, ![A, B, C]: join(join(A, B), C)=join(A, join(B, C))).
59.67/59.83	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
59.67/59.83	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
59.67/59.83	
59.67/59.83	Now clausify the problem and encode Horn clauses using encoding 3 of
59.67/59.83	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
59.67/59.83	We repeatedly replace C & s=t => u=v by the two clauses:
59.67/59.83	  fresh(y, y, x1...xn) = u
59.67/59.83	  C => fresh(s, t, x1...xn) = v
59.67/59.83	where fresh is a fresh function symbol and x1..xn are the free
59.67/59.83	variables of u and v.
59.67/59.83	A predicate p(X) is encoded as p(X)=true (this is sound, because the
59.67/59.83	input problem has no model of domain size 1).
59.67/59.83	
59.67/59.83	The encoding turns the above axioms into the following unit equations and goals:
59.67/59.83	
59.67/59.83	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
59.67/59.83	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
59.67/59.83	Axiom 3 (composition_identity_6): X = composition(X, one).
59.67/59.83	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
59.67/59.83	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
59.67/59.83	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
59.67/59.83	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
59.67/59.83	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
59.67/59.83	Axiom 9 (def_top_12): join(X, complement(X)) = top.
59.67/59.83	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
59.67/59.83	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
59.67/59.83	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
59.67/59.83	Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
59.67/59.86	Axiom 14 (goals_14): sk1 = composition(sk1, top).
59.67/59.86	
59.67/59.86	Lemma 15: meet(X, Y) = meet(Y, X).
59.67/59.86	Proof:
59.67/59.86	  meet(X, Y)
59.67/59.86	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.86	  complement(join(complement(X), complement(Y)))
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  complement(join(complement(Y), complement(X)))
59.67/59.86	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.86	  meet(Y, X)
59.67/59.86	
59.67/59.86	Lemma 16: complement(top) = zero.
59.67/59.86	Proof:
59.67/59.86	  complement(top)
59.67/59.86	= { by axiom 9 (def_top_12) }
59.67/59.86	  complement(join(complement(?), complement(complement(?))))
59.67/59.86	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.86	  meet(?, complement(?))
59.67/59.86	= { by axiom 8 (def_zero_13) }
59.67/59.86	  zero
59.67/59.86	
59.67/59.86	Lemma 17: complement(join(zero, complement(X))) = meet(X, top).
59.67/59.86	Proof:
59.67/59.86	  complement(join(zero, complement(X)))
59.67/59.86	= { by lemma 16 }
59.67/59.86	  complement(join(complement(top), complement(X)))
59.67/59.86	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.86	  meet(top, X)
59.67/59.86	= { by lemma 15 }
59.67/59.86	  meet(X, top)
59.67/59.86	
59.67/59.86	Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
59.67/59.86	Proof:
59.67/59.86	  converse(composition(converse(X), Y))
59.67/59.86	= { by axiom 10 (converse_multiplicativity_10) }
59.67/59.86	  composition(converse(Y), converse(converse(X)))
59.67/59.86	= { by axiom 5 (converse_idempotence_8) }
59.67/59.86	  composition(converse(Y), X)
59.67/59.86	
59.67/59.86	Lemma 19: composition(converse(one), X) = X.
59.67/59.86	Proof:
59.67/59.86	  composition(converse(one), X)
59.67/59.86	= { by lemma 18 }
59.67/59.86	  converse(composition(converse(X), one))
59.67/59.86	= { by axiom 3 (composition_identity_6) }
59.67/59.86	  converse(converse(X))
59.67/59.86	= { by axiom 5 (converse_idempotence_8) }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 20: composition(one, X) = X.
59.67/59.86	Proof:
59.67/59.86	  composition(one, X)
59.67/59.86	= { by lemma 19 }
59.67/59.86	  composition(converse(one), composition(one, X))
59.67/59.86	= { by axiom 12 (composition_associativity_5) }
59.67/59.86	  composition(composition(converse(one), one), X)
59.67/59.86	= { by axiom 3 (composition_identity_6) }
59.67/59.86	  composition(converse(one), X)
59.67/59.86	= { by lemma 19 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 21: join(complement(Y), composition(converse(X), complement(composition(X, Y)))) = complement(Y).
59.67/59.86	Proof:
59.67/59.86	  join(complement(Y), composition(converse(X), complement(composition(X, Y))))
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  join(composition(converse(X), complement(composition(X, Y))), complement(Y))
59.67/59.86	= { by axiom 7 (converse_cancellativity_11) }
59.67/59.86	  complement(Y)
59.67/59.86	
59.67/59.86	Lemma 22: join(complement(X), complement(X)) = complement(X).
59.67/59.86	Proof:
59.67/59.86	  join(complement(X), complement(X))
59.67/59.86	= { by lemma 19 }
59.67/59.86	  join(complement(X), composition(converse(one), complement(X)))
59.67/59.86	= { by lemma 20 }
59.67/59.86	  join(complement(X), composition(converse(one), complement(composition(one, X))))
59.67/59.86	= { by lemma 21 }
59.67/59.86	  complement(X)
59.67/59.86	
59.67/59.86	Lemma 23: meet(X, X) = complement(complement(X)).
59.67/59.86	Proof:
59.67/59.86	  meet(X, X)
59.67/59.86	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.86	  complement(join(complement(X), complement(X)))
59.67/59.86	= { by lemma 22 }
59.67/59.86	  complement(complement(X))
59.67/59.86	
59.67/59.86	Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
59.67/59.86	Proof:
59.67/59.86	  join(meet(X, Y), complement(join(complement(X), Y)))
59.67/59.86	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.86	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
59.67/59.86	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 25: join(meet(X, Y), meet(X, complement(Y))) = X.
59.67/59.86	Proof:
59.67/59.86	  join(meet(X, Y), meet(X, complement(Y)))
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  join(meet(X, complement(Y)), meet(X, Y))
59.67/59.86	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.86	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
59.67/59.86	= { by lemma 24 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 26: join(zero, complement(complement(X))) = X.
59.67/59.86	Proof:
59.67/59.86	  join(zero, complement(complement(X)))
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  join(complement(complement(X)), zero)
59.67/59.86	= { by lemma 23 }
59.67/59.86	  join(meet(X, X), zero)
59.67/59.86	= { by axiom 8 (def_zero_13) }
59.67/59.86	  join(meet(X, X), meet(X, complement(X)))
59.67/59.86	= { by lemma 25 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 27: join(X, join(Y, Z)) = join(Z, join(X, Y)).
59.67/59.86	Proof:
59.67/59.86	  join(X, join(Y, Z))
59.67/59.86	= { by axiom 6 (maddux2_join_associativity_2) }
59.67/59.86	  join(join(X, Y), Z)
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  join(Z, join(X, Y))
59.67/59.86	
59.67/59.86	Lemma 28: join(X, join(complement(X), Y)) = join(Y, top).
59.67/59.86	Proof:
59.67/59.86	  join(X, join(complement(X), Y))
59.67/59.86	= { by lemma 27 }
59.67/59.86	  join(complement(X), join(Y, X))
59.67/59.86	= { by lemma 27 }
59.67/59.86	  join(Y, join(X, complement(X)))
59.67/59.86	= { by axiom 9 (def_top_12) }
59.67/59.86	  join(Y, top)
59.67/59.86	
59.67/59.86	Lemma 29: join(X, top) = top.
59.67/59.86	Proof:
59.67/59.86	  join(X, top)
59.67/59.86	= { by axiom 9 (def_top_12) }
59.67/59.86	  join(X, join(complement(X), complement(complement(X))))
59.67/59.86	= { by lemma 28 }
59.67/59.86	  join(complement(complement(X)), top)
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  join(top, complement(complement(X)))
59.67/59.86	= { by axiom 9 (def_top_12) }
59.67/59.86	  join(join(complement(X), complement(complement(X))), complement(complement(X)))
59.67/59.86	= { by axiom 6 (maddux2_join_associativity_2) }
59.67/59.86	  join(complement(X), join(complement(complement(X)), complement(complement(X))))
59.67/59.86	= { by lemma 22 }
59.67/59.86	  join(complement(X), complement(complement(X)))
59.67/59.86	= { by axiom 9 (def_top_12) }
59.67/59.86	  top
59.67/59.86	
59.67/59.86	Lemma 30: join(zero, meet(X, top)) = X.
59.67/59.86	Proof:
59.67/59.86	  join(zero, meet(X, top))
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  join(meet(X, top), zero)
59.67/59.86	= { by lemma 16 }
59.67/59.86	  join(meet(X, top), complement(top))
59.67/59.86	= { by lemma 29 }
59.67/59.86	  join(meet(X, top), complement(join(complement(X), top)))
59.67/59.86	= { by lemma 24 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 31: join(zero, meet(top, X)) = X.
59.67/59.86	Proof:
59.67/59.86	  join(zero, meet(top, X))
59.67/59.86	= { by lemma 15 }
59.67/59.86	  join(zero, meet(X, top))
59.67/59.86	= { by lemma 30 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 32: join(zero, complement(X)) = complement(X).
59.67/59.86	Proof:
59.67/59.86	  join(zero, complement(X))
59.67/59.86	= { by lemma 26 }
59.67/59.86	  join(zero, complement(join(zero, complement(complement(X)))))
59.67/59.86	= { by lemma 17 }
59.67/59.86	  join(zero, meet(complement(X), top))
59.67/59.86	= { by lemma 15 }
59.67/59.86	  join(zero, meet(top, complement(X)))
59.67/59.86	= { by lemma 31 }
59.67/59.86	  complement(X)
59.67/59.86	
59.67/59.86	Lemma 33: complement(complement(X)) = X.
59.67/59.86	Proof:
59.67/59.86	  complement(complement(X))
59.67/59.86	= { by lemma 32 }
59.67/59.86	  join(zero, complement(complement(X)))
59.67/59.86	= { by lemma 26 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 34: meet(X, top) = X.
59.67/59.86	Proof:
59.67/59.86	  meet(X, top)
59.67/59.86	= { by lemma 17 }
59.67/59.86	  complement(join(zero, complement(X)))
59.67/59.86	= { by lemma 32 }
59.67/59.86	  join(zero, complement(join(zero, complement(X))))
59.67/59.86	= { by lemma 17 }
59.67/59.86	  join(zero, meet(X, top))
59.67/59.86	= { by lemma 30 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 35: meet(top, X) = X.
59.67/59.86	Proof:
59.67/59.86	  meet(top, X)
59.67/59.86	= { by lemma 15 }
59.67/59.86	  meet(X, top)
59.67/59.86	= { by lemma 34 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 36: complement(join(X, complement(Y))) = meet(Y, complement(X)).
59.67/59.86	Proof:
59.67/59.86	  complement(join(X, complement(Y)))
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  complement(join(complement(Y), X))
59.67/59.86	= { by lemma 35 }
59.67/59.86	  complement(join(complement(Y), meet(top, X)))
59.67/59.86	= { by lemma 15 }
59.67/59.86	  complement(join(complement(Y), meet(X, top)))
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  complement(join(meet(X, top), complement(Y)))
59.67/59.86	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.86	  complement(join(complement(join(complement(X), complement(top))), complement(Y)))
59.67/59.86	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.86	  meet(join(complement(X), complement(top)), Y)
59.67/59.86	= { by lemma 15 }
59.67/59.86	  meet(Y, join(complement(X), complement(top)))
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  meet(Y, join(complement(top), complement(X)))
59.67/59.86	= { by lemma 16 }
59.67/59.86	  meet(Y, join(zero, complement(X)))
59.67/59.86	= { by lemma 32 }
59.67/59.86	  meet(Y, complement(X))
59.67/59.86	
59.67/59.86	Lemma 37: join(meet(Y, X), meet(X, complement(Y))) = X.
59.67/59.86	Proof:
59.67/59.86	  join(meet(Y, X), meet(X, complement(Y)))
59.67/59.86	= { by lemma 15 }
59.67/59.86	  join(meet(X, Y), meet(X, complement(Y)))
59.67/59.86	= { by lemma 25 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 38: join(X, zero) = X.
59.67/59.86	Proof:
59.67/59.86	  join(X, zero)
59.67/59.86	= { by lemma 33 }
59.67/59.86	  join(complement(complement(X)), zero)
59.67/59.86	= { by lemma 23 }
59.67/59.86	  join(meet(X, X), zero)
59.67/59.86	= { by axiom 8 (def_zero_13) }
59.67/59.86	  join(meet(X, X), meet(X, complement(X)))
59.67/59.86	= { by lemma 37 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 39: join(zero, X) = X.
59.67/59.86	Proof:
59.67/59.86	  join(zero, X)
59.67/59.86	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.86	  join(X, zero)
59.67/59.86	= { by lemma 38 }
59.67/59.86	  X
59.67/59.86	
59.67/59.86	Lemma 40: composition(converse(sk1), complement(sk1)) = zero.
59.67/59.86	Proof:
59.67/59.86	  composition(converse(sk1), complement(sk1))
59.67/59.86	= { by lemma 39 }
59.67/59.86	  join(zero, composition(converse(sk1), complement(sk1)))
59.67/59.86	= { by lemma 16 }
59.67/59.86	  join(complement(top), composition(converse(sk1), complement(sk1)))
59.67/59.86	= { by axiom 14 (goals_14) }
59.67/59.86	  join(complement(top), composition(converse(sk1), complement(composition(sk1, top))))
59.67/59.86	= { by lemma 21 }
59.67/59.86	  complement(top)
59.67/59.86	= { by lemma 16 }
59.67/59.87	  zero
59.67/59.87	
59.67/59.87	Lemma 41: converse(join(converse(X), Y)) = join(X, converse(Y)).
59.67/59.87	Proof:
59.67/59.87	  converse(join(converse(X), Y))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  converse(join(Y, converse(X)))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  converse(join(converse(X), Y))
59.67/59.87	= { by axiom 11 (converse_additivity_9) }
59.67/59.87	  join(converse(converse(X)), converse(Y))
59.67/59.87	= { by axiom 5 (converse_idempotence_8) }
59.67/59.87	  join(X, converse(Y))
59.67/59.87	
59.67/59.87	Lemma 42: join(X, converse(top)) = converse(top).
59.67/59.87	Proof:
59.67/59.87	  join(X, converse(top))
59.67/59.87	= { by lemma 41 }
59.67/59.87	  converse(join(converse(X), top))
59.67/59.87	= { by lemma 29 }
59.67/59.87	  converse(top)
59.67/59.87	
59.67/59.87	Lemma 43: join(top, X) = top.
59.67/59.87	Proof:
59.67/59.87	  join(top, X)
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  join(X, top)
59.67/59.87	= { by lemma 29 }
59.67/59.87	  top
59.67/59.87	
59.67/59.87	Lemma 44: composition(top, zero) = zero.
59.67/59.87	Proof:
59.67/59.87	  composition(top, zero)
59.67/59.87	= { by lemma 39 }
59.67/59.87	  join(zero, composition(top, zero))
59.67/59.87	= { by lemma 16 }
59.67/59.87	  join(complement(top), composition(top, zero))
59.67/59.87	= { by lemma 29 }
59.67/59.87	  join(complement(top), composition(join(converse(top), top), zero))
59.67/59.87	= { by lemma 28 }
59.67/59.87	  join(complement(top), composition(join(?, join(complement(?), converse(top))), zero))
59.67/59.87	= { by lemma 42 }
59.67/59.87	  join(complement(top), composition(join(?, converse(top)), zero))
59.67/59.87	= { by lemma 42 }
59.67/59.87	  join(complement(top), composition(converse(top), zero))
59.67/59.87	= { by lemma 16 }
59.67/59.87	  join(complement(top), composition(converse(top), complement(top)))
59.67/59.87	= { by lemma 43 }
59.67/59.87	  join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
59.67/59.87	= { by lemma 20 }
59.67/59.87	  join(complement(top), composition(converse(top), complement(join(composition(one, top), composition(top, top)))))
59.67/59.87	= { by axiom 13 (composition_distributivity_7) }
59.67/59.87	  join(complement(top), composition(converse(top), complement(composition(join(one, top), top))))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
59.67/59.87	= { by lemma 43 }
59.67/59.87	  join(complement(top), composition(converse(top), complement(composition(top, top))))
59.67/59.87	= { by lemma 21 }
59.67/59.87	  complement(top)
59.67/59.87	= { by lemma 16 }
59.67/59.87	  zero
59.67/59.87	
59.67/59.87	Lemma 45: composition(converse(complement(sk1)), composition(sk1, X)) = zero.
59.67/59.87	Proof:
59.67/59.87	  composition(converse(complement(sk1)), composition(sk1, X))
59.67/59.87	= { by lemma 18 }
59.67/59.87	  converse(composition(converse(composition(sk1, X)), complement(sk1)))
59.67/59.87	= { by axiom 10 (converse_multiplicativity_10) }
59.67/59.87	  converse(composition(composition(converse(X), converse(sk1)), complement(sk1)))
59.67/59.87	= { by axiom 12 (composition_associativity_5) }
59.67/59.87	  converse(composition(converse(X), composition(converse(sk1), complement(sk1))))
59.67/59.87	= { by lemma 40 }
59.67/59.87	  converse(composition(converse(X), zero))
59.67/59.87	= { by lemma 39 }
59.67/59.87	  converse(join(zero, composition(converse(X), zero)))
59.67/59.87	= { by lemma 44 }
59.67/59.87	  converse(join(composition(top, zero), composition(converse(X), zero)))
59.67/59.87	= { by axiom 13 (composition_distributivity_7) }
59.67/59.87	  converse(composition(join(top, converse(X)), zero))
59.67/59.87	= { by lemma 43 }
59.67/59.87	  converse(composition(top, zero))
59.67/59.87	= { by lemma 44 }
59.67/59.87	  converse(zero)
59.67/59.87	= { by lemma 39 }
59.67/59.87	  join(zero, converse(zero))
59.67/59.87	= { by lemma 41 }
59.67/59.87	  converse(join(converse(zero), zero))
59.67/59.87	= { by lemma 38 }
59.67/59.87	  converse(converse(zero))
59.67/59.87	= { by axiom 5 (converse_idempotence_8) }
59.67/59.87	  zero
59.67/59.87	
59.67/59.87	Lemma 46: join(complement(join(X, Y)), meet(Y, complement(X))) = complement(X).
59.67/59.87	Proof:
59.67/59.87	  join(complement(join(X, Y)), meet(Y, complement(X)))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  join(complement(join(Y, X)), meet(Y, complement(X)))
59.67/59.87	= { by lemma 15 }
59.67/59.87	  join(complement(join(Y, X)), meet(complement(X), Y))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  join(meet(complement(X), Y), complement(join(Y, X)))
59.67/59.87	= { by lemma 32 }
59.67/59.87	  join(meet(join(zero, complement(X)), Y), complement(join(Y, X)))
59.67/59.87	= { by lemma 34 }
59.67/59.87	  join(meet(join(zero, complement(X)), Y), complement(join(Y, meet(X, top))))
59.67/59.87	= { by lemma 17 }
59.67/59.87	  join(meet(join(zero, complement(X)), Y), complement(join(Y, complement(join(zero, complement(X))))))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  join(meet(join(zero, complement(X)), Y), complement(join(complement(join(zero, complement(X))), Y)))
59.67/59.87	= { by lemma 24 }
59.67/59.87	  join(zero, complement(X))
59.67/59.87	= { by lemma 32 }
59.67/59.87	  complement(X)
59.67/59.87	
59.67/59.87	Lemma 47: meet(complement(X), complement(Y)) = complement(join(X, Y)).
59.67/59.87	Proof:
59.67/59.87	  meet(complement(X), complement(Y))
59.67/59.87	= { by lemma 32 }
59.67/59.87	  meet(join(zero, complement(X)), complement(Y))
59.67/59.87	= { by lemma 36 }
59.67/59.87	  complement(join(Y, complement(join(zero, complement(X)))))
59.67/59.87	= { by lemma 17 }
59.67/59.87	  complement(join(Y, meet(X, top)))
59.67/59.87	= { by lemma 34 }
59.67/59.87	  complement(join(Y, X))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  complement(join(X, Y))
59.67/59.87	
59.67/59.87	Lemma 48: complement(join(complement(X), Y)) = meet(X, complement(Y)).
59.67/59.87	Proof:
59.67/59.87	  complement(join(complement(X), Y))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  complement(join(Y, complement(X)))
59.67/59.87	= { by lemma 36 }
59.67/59.87	  meet(X, complement(Y))
59.67/59.87	
59.67/59.87	Lemma 49: meet(complement(Z), meet(X, Y)) = meet(X, meet(Y, complement(Z))).
59.67/59.87	Proof:
59.67/59.87	  meet(complement(Z), meet(X, Y))
59.67/59.87	= { by lemma 15 }
59.67/59.87	  meet(meet(X, Y), complement(Z))
59.67/59.87	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.67/59.87	  meet(complement(join(complement(X), complement(Y))), complement(Z))
59.67/59.87	= { by lemma 47 }
59.67/59.87	  complement(join(join(complement(X), complement(Y)), Z))
59.67/59.87	= { by axiom 6 (maddux2_join_associativity_2) }
59.67/59.87	  complement(join(complement(X), join(complement(Y), Z)))
59.67/59.87	= { by lemma 48 }
59.67/59.87	  meet(X, complement(join(complement(Y), Z)))
59.67/59.87	= { by lemma 48 }
59.67/59.87	  meet(X, meet(Y, complement(Z)))
59.67/59.87	
59.67/59.87	Lemma 50: meet(X, join(X, Y)) = X.
59.67/59.87	Proof:
59.67/59.87	  meet(X, join(X, Y))
59.67/59.87	= { by axiom 5 (converse_idempotence_8) }
59.67/59.87	  meet(converse(converse(X)), join(X, Y))
59.67/59.87	= { by axiom 5 (converse_idempotence_8) }
59.67/59.87	  meet(converse(converse(X)), converse(converse(join(X, Y))))
59.67/59.87	= { by axiom 11 (converse_additivity_9) }
59.67/59.87	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
59.67/59.87	= { by lemma 38 }
59.67/59.87	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), zero)
59.67/59.87	= { by lemma 16 }
59.67/59.87	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(top))
59.67/59.87	= { by lemma 29 }
59.67/59.87	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(converse(Y)), top)))
59.67/59.87	= { by axiom 9 (def_top_12) }
59.67/59.87	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(converse(Y)), join(converse(converse(X)), complement(converse(converse(X)))))))
59.67/59.87	= { by axiom 6 (maddux2_join_associativity_2) }
59.67/59.87	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(join(converse(converse(Y)), converse(converse(X))), complement(converse(converse(X))))))
59.67/59.87	= { by axiom 11 (converse_additivity_9) }
59.67/59.87	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(join(converse(Y), converse(X))), complement(converse(converse(X))))))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(complement(converse(converse(X))), converse(join(converse(Y), converse(X))))))
59.67/59.87	= { by axiom 1 (maddux1_join_commutativity_1) }
59.67/59.87	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(complement(converse(converse(X))), converse(join(converse(X), converse(Y))))))
59.67/59.87	= { by lemma 24 }
59.67/59.87	  converse(converse(X))
59.67/59.87	= { by axiom 5 (converse_idempotence_8) }
59.72/59.90	  X
59.72/59.90	
59.72/59.90	Lemma 51: meet(sk1, composition(join(complement(sk1), X), Y)) = meet(sk1, composition(X, Y)).
59.72/59.90	Proof:
59.72/59.90	  meet(sk1, composition(join(complement(sk1), X), Y))
59.72/59.90	= { by lemma 33 }
59.72/59.90	  meet(complement(complement(sk1)), composition(join(complement(sk1), X), Y))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  meet(complement(complement(sk1)), composition(join(X, complement(sk1)), Y))
59.72/59.90	= { by axiom 13 (composition_distributivity_7) }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), composition(complement(sk1), Y)))
59.72/59.90	= { by lemma 33 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(complement(composition(complement(sk1), Y)))))
59.72/59.90	= { by lemma 21 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(converse(converse(sk1)), complement(composition(converse(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 5 (converse_idempotence_8) }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, complement(composition(converse(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 12 (composition_associativity_5) }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, complement(composition(composition(converse(sk1), complement(sk1)), Y)))))))
59.72/59.90	= { by lemma 40 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, complement(composition(zero, Y)))))))
59.72/59.90	= { by lemma 45 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, complement(composition(composition(converse(complement(sk1)), composition(sk1, ?)), Y)))))))
59.72/59.90	= { by axiom 12 (composition_associativity_5) }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, complement(composition(converse(complement(sk1)), composition(composition(sk1, ?), Y))))))))
59.72/59.90	= { by axiom 12 (composition_associativity_5) }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, complement(composition(converse(complement(sk1)), composition(sk1, composition(?, Y)))))))))
59.72/59.90	= { by lemma 45 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, complement(zero))))))
59.72/59.90	= { by lemma 16 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, complement(complement(top)))))))
59.72/59.90	= { by lemma 31 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, join(zero, meet(top, complement(complement(top)))))))))
59.72/59.90	= { by axiom 8 (def_zero_13) }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, join(meet(top, complement(top)), meet(top, complement(complement(top)))))))))
59.72/59.90	= { by lemma 25 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), composition(sk1, top)))))
59.72/59.90	= { by axiom 14 (goals_14) }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(complement(composition(complement(sk1), Y)), sk1))))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), complement(join(sk1, complement(composition(complement(sk1), Y))))))
59.72/59.90	= { by lemma 36 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1))))
59.72/59.90	= { by lemma 15 }
59.72/59.90	  meet(complement(complement(sk1)), join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))
59.72/59.90	= { by lemma 15 }
59.72/59.90	  meet(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))), complement(complement(sk1)))
59.72/59.90	= { by lemma 36 }
59.72/59.90	  complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))
59.72/59.90	= { by lemma 46 }
59.72/59.90	  join(complement(join(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))), composition(X, Y))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  join(complement(join(composition(X, Y), join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by lemma 15 }
59.72/59.90	  join(complement(join(composition(X, Y), join(complement(sk1), complement(join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1))))))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  join(complement(join(composition(X, Y), join(complement(join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1)))), complement(sk1)))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by lemma 37 }
59.72/59.90	  join(complement(join(composition(X, Y), join(complement(join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1)))), join(meet(composition(complement(sk1), Y), complement(sk1)), meet(complement(sk1), complement(composition(complement(sk1), Y))))))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 6 (maddux2_join_associativity_2) }
59.72/59.90	  join(complement(join(composition(X, Y), join(join(complement(join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1)))), meet(composition(complement(sk1), Y), complement(sk1))), meet(complement(sk1), complement(composition(complement(sk1), Y)))))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  join(complement(join(composition(X, Y), join(join(meet(composition(complement(sk1), Y), complement(sk1)), complement(join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1))))), meet(complement(sk1), complement(composition(complement(sk1), Y)))))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 6 (maddux2_join_associativity_2) }
59.72/59.90	  join(complement(join(join(composition(X, Y), join(meet(composition(complement(sk1), Y), complement(sk1)), complement(join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1)))))), meet(complement(sk1), complement(composition(complement(sk1), Y))))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  join(complement(join(meet(complement(sk1), complement(composition(complement(sk1), Y))), join(composition(X, Y), join(meet(composition(complement(sk1), Y), complement(sk1)), complement(join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1)))))))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 6 (maddux2_join_associativity_2) }
59.72/59.90	  join(complement(join(meet(complement(sk1), complement(composition(complement(sk1), Y))), join(join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1))), complement(join(composition(X, Y), meet(composition(complement(sk1), Y), complement(sk1))))))), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by axiom 9 (def_top_12) }
59.72/59.90	  join(complement(join(meet(complement(sk1), complement(composition(complement(sk1), Y))), top)), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by lemma 29 }
59.72/59.90	  join(complement(top), meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by lemma 16 }
59.72/59.90	  join(zero, meet(composition(X, Y), complement(join(complement(sk1), complement(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))))))))
59.72/59.90	= { by lemma 36 }
59.72/59.90	  join(zero, meet(composition(X, Y), meet(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))), complement(complement(sk1)))))
59.72/59.90	= { by lemma 39 }
59.72/59.90	  meet(composition(X, Y), meet(join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y))), complement(complement(sk1))))
59.72/59.90	= { by lemma 49 }
59.72/59.90	  meet(complement(complement(sk1)), meet(composition(X, Y), join(composition(X, Y), meet(complement(sk1), composition(complement(sk1), Y)))))
59.72/59.90	= { by lemma 50 }
59.72/59.90	  meet(complement(complement(sk1)), composition(X, Y))
59.72/59.90	= { by lemma 15 }
59.72/59.90	  meet(composition(X, Y), complement(complement(sk1)))
59.72/59.90	= { by lemma 33 }
59.72/59.90	  meet(composition(X, Y), sk1)
59.72/59.90	= { by lemma 15 }
59.72/59.90	  meet(sk1, composition(X, Y))
59.72/59.90	
59.72/59.90	Lemma 52: meet(X, join(Y, X)) = X.
59.72/59.90	Proof:
59.72/59.90	  meet(X, join(Y, X))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  meet(X, join(X, Y))
59.72/59.90	= { by lemma 50 }
59.72/59.90	  X
59.72/59.90	
59.72/59.90	Lemma 53: join(sk1, composition(meet(X, sk1), top)) = sk1.
59.72/59.90	Proof:
59.72/59.90	  join(sk1, composition(meet(X, sk1), top))
59.72/59.90	= { by lemma 33 }
59.72/59.90	  join(sk1, complement(complement(composition(meet(X, sk1), top))))
59.72/59.90	= { by lemma 35 }
59.72/59.90	  meet(top, join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by lemma 29 }
59.72/59.90	  meet(join(composition(complement(join(complement(sk1), X)), top), top), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by axiom 9 (def_top_12) }
59.72/59.90	  meet(join(composition(complement(join(complement(sk1), X)), top), join(composition(meet(sk1, X), top), complement(composition(meet(sk1, X), top)))), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by axiom 6 (maddux2_join_associativity_2) }
59.72/59.90	  meet(join(join(composition(complement(join(complement(sk1), X)), top), composition(meet(sk1, X), top)), complement(composition(meet(sk1, X), top))), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by axiom 13 (composition_distributivity_7) }
59.72/59.90	  meet(join(composition(join(complement(join(complement(sk1), X)), meet(sk1, X)), top), complement(composition(meet(sk1, X), top))), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  meet(join(complement(composition(meet(sk1, X), top)), composition(join(complement(join(complement(sk1), X)), meet(sk1, X)), top)), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  meet(join(complement(composition(meet(sk1, X), top)), composition(join(meet(sk1, X), complement(join(complement(sk1), X))), top)), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by lemma 24 }
59.72/59.90	  meet(join(complement(composition(meet(sk1, X), top)), composition(sk1, top)), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by axiom 1 (maddux1_join_commutativity_1) }
59.72/59.90	  meet(join(composition(sk1, top), complement(composition(meet(sk1, X), top))), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by axiom 14 (goals_14) }
59.72/59.90	  meet(join(sk1, complement(composition(meet(sk1, X), top))), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by lemma 15 }
59.72/59.90	  meet(join(sk1, complement(composition(meet(X, sk1), top))), join(sk1, complement(complement(composition(meet(X, sk1), top)))))
59.72/59.90	= { by lemma 34 }
59.72/59.90	  meet(join(sk1, complement(composition(meet(X, sk1), top))), meet(join(sk1, complement(complement(composition(meet(X, sk1), top)))), top))
59.72/59.90	= { by lemma 17 }
59.72/59.90	  meet(join(sk1, complement(composition(meet(X, sk1), top))), complement(join(zero, complement(join(sk1, complement(complement(composition(meet(X, sk1), top))))))))
59.72/59.90	= { by lemma 36 }
59.72/59.90	  meet(join(sk1, complement(composition(meet(X, sk1), top))), complement(join(zero, meet(complement(composition(meet(X, sk1), top)), complement(sk1)))))
59.72/59.90	= { by lemma 39 }
59.72/59.90	  meet(join(sk1, complement(composition(meet(X, sk1), top))), complement(meet(complement(composition(meet(X, sk1), top)), complement(sk1))))
59.72/59.90	= { by lemma 48 }
59.72/59.90	  complement(join(complement(join(sk1, complement(composition(meet(X, sk1), top)))), meet(complement(composition(meet(X, sk1), top)), complement(sk1))))
59.72/59.90	= { by lemma 46 }
59.72/59.90	  complement(complement(sk1))
59.72/59.90	= { by lemma 33 }
59.86/60.02	  sk1
59.86/60.02	
59.86/60.02	Goal 1 (goals_15): composition(meet(sk1, sk2), sk3) = meet(sk1, composition(sk2, sk3)).
59.86/60.02	Proof:
59.86/60.02	  composition(meet(sk1, sk2), sk3)
59.86/60.02	= { by lemma 15 }
59.86/60.02	  composition(meet(sk2, sk1), sk3)
59.86/60.02	= { by lemma 52 }
59.86/60.02	  meet(composition(meet(sk2, sk1), sk3), join(sk1, composition(meet(sk2, sk1), sk3)))
59.86/60.02	= { by axiom 1 (maddux1_join_commutativity_1) }
59.86/60.02	  meet(composition(meet(sk2, sk1), sk3), join(composition(meet(sk2, sk1), sk3), sk1))
59.86/60.02	= { by lemma 53 }
59.86/60.02	  meet(composition(meet(sk2, sk1), sk3), join(composition(meet(sk2, sk1), sk3), join(sk1, composition(meet(sk2, sk1), top))))
59.86/60.02	= { by axiom 1 (maddux1_join_commutativity_1) }
59.86/60.02	  meet(composition(meet(sk2, sk1), sk3), join(composition(meet(sk2, sk1), sk3), join(composition(meet(sk2, sk1), top), sk1)))
59.86/60.02	= { by axiom 6 (maddux2_join_associativity_2) }
59.86/60.02	  meet(composition(meet(sk2, sk1), sk3), join(join(composition(meet(sk2, sk1), sk3), composition(meet(sk2, sk1), top)), sk1))
59.86/60.02	= { by axiom 5 (converse_idempotence_8) }
59.86/60.02	  meet(composition(meet(sk2, sk1), sk3), join(converse(converse(join(composition(meet(sk2, sk1), sk3), composition(meet(sk2, sk1), top)))), sk1))
59.86/60.02	= { by axiom 1 (maddux1_join_commutativity_1) }
59.86/60.02	  meet(composition(meet(sk2, sk1), sk3), join(converse(converse(join(composition(meet(sk2, sk1), top), composition(meet(sk2, sk1), sk3)))), sk1))
59.86/60.02	= { by axiom 11 (converse_additivity_9) }
59.86/60.02	  meet(composition(meet(sk2, sk1), sk3), join(converse(join(converse(composition(meet(sk2, sk1), top)), converse(composition(meet(sk2, sk1), sk3)))), sk1))
59.86/60.02	= { by axiom 10 (converse_multiplicativity_10) }
59.86/60.02	  meet(composition(meet(sk2, sk1), sk3), join(converse(join(composition(converse(top), converse(meet(sk2, sk1))), converse(composition(meet(sk2, sk1), sk3)))), sk1))
59.86/60.02	= { by axiom 1 (maddux1_join_commutativity_1) }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(converse(join(converse(composition(meet(sk2, sk1), sk3)), composition(converse(top), converse(meet(sk2, sk1))))), sk1))
59.86/60.03	= { by axiom 10 (converse_multiplicativity_10) }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(converse(join(composition(converse(sk3), converse(meet(sk2, sk1))), composition(converse(top), converse(meet(sk2, sk1))))), sk1))
59.86/60.03	= { by axiom 13 (composition_distributivity_7) }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(converse(composition(join(converse(sk3), converse(top)), converse(meet(sk2, sk1)))), sk1))
59.86/60.03	= { by axiom 1 (maddux1_join_commutativity_1) }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(converse(composition(join(converse(top), converse(sk3)), converse(meet(sk2, sk1)))), sk1))
59.86/60.03	= { by axiom 11 (converse_additivity_9) }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(converse(composition(converse(join(top, sk3)), converse(meet(sk2, sk1)))), sk1))
59.86/60.03	= { by axiom 10 (converse_multiplicativity_10) }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(converse(converse(composition(meet(sk2, sk1), join(top, sk3)))), sk1))
59.86/60.03	= { by axiom 1 (maddux1_join_commutativity_1) }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(converse(converse(composition(meet(sk2, sk1), join(sk3, top)))), sk1))
59.86/60.03	= { by axiom 5 (converse_idempotence_8) }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(composition(meet(sk2, sk1), join(sk3, top)), sk1))
59.86/60.03	= { by axiom 1 (maddux1_join_commutativity_1) }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(sk1, composition(meet(sk2, sk1), join(sk3, top))))
59.86/60.03	= { by lemma 29 }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), join(sk1, composition(meet(sk2, sk1), top)))
59.86/60.03	= { by lemma 53 }
59.86/60.03	  meet(composition(meet(sk2, sk1), sk3), sk1)
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(meet(sk2, sk1), sk3))
59.86/60.03	= { by lemma 33 }
59.86/60.03	  meet(sk1, composition(meet(sk2, complement(complement(sk1))), sk3))
59.86/60.03	= { by lemma 51 }
59.86/60.03	  meet(sk1, composition(join(complement(sk1), meet(sk2, complement(complement(sk1)))), sk3))
59.86/60.03	= { by axiom 1 (maddux1_join_commutativity_1) }
59.86/60.03	  meet(sk1, composition(join(meet(sk2, complement(complement(sk1))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), sk2), complement(sk1)), sk3))
59.86/60.03	= { by lemma 52 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), meet(sk2, join(complement(sk1), sk2))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 33 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), meet(sk2, join(complement(sk1), complement(complement(sk2))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), meet(join(complement(sk1), complement(complement(sk2))), sk2)), complement(sk1)), sk3))
59.86/60.03	= { by lemma 34 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), meet(join(complement(sk1), complement(complement(sk2))), meet(sk2, top))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 17 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), meet(join(complement(sk1), complement(complement(sk2))), complement(join(zero, complement(sk2))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 48 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), complement(join(complement(join(complement(sk1), complement(complement(sk2)))), join(zero, complement(sk2))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 48 }
59.86/60.03	  meet(sk1, composition(join(complement(join(complement(complement(complement(sk1))), join(complement(join(complement(sk1), complement(complement(sk2)))), join(zero, complement(sk2))))), complement(sk1)), sk3))
59.86/60.03	= { by axiom 6 (maddux2_join_associativity_2) }
59.86/60.03	  meet(sk1, composition(join(complement(join(join(complement(complement(complement(sk1))), complement(join(complement(sk1), complement(complement(sk2))))), join(zero, complement(sk2)))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 47 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(join(complement(complement(complement(sk1))), complement(join(complement(sk1), complement(complement(sk2)))))), complement(join(zero, complement(sk2)))), complement(sk1)), sk3))
59.86/60.03	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.86/60.03	  meet(sk1, composition(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), complement(join(zero, complement(sk2)))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(join(zero, complement(sk2))), meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 17 }
59.86/60.03	  meet(sk1, composition(join(meet(meet(sk2, top), meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 34 }
59.86/60.03	  meet(sk1, composition(join(meet(sk2, meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), complement(sk1)), sk3))
59.86/60.03	= { by lemma 38 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), zero), complement(sk1)), sk3))
59.86/60.03	= { by axiom 8 (def_zero_13) }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(join(complement(sk1), complement(complement(sk2))), complement(join(complement(sk1), complement(complement(sk2)))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 36 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(join(complement(sk1), complement(complement(sk2))), meet(complement(sk2), complement(complement(sk1))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 48 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(join(complement(sk1), complement(complement(sk2))), complement(join(complement(complement(sk2)), complement(sk1))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 48 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), complement(join(complement(join(complement(sk1), complement(complement(sk2)))), join(complement(complement(sk2)), complement(sk1))))), complement(sk1)), sk3))
59.86/60.03	= { by axiom 6 (maddux2_join_associativity_2) }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), complement(join(join(complement(join(complement(sk1), complement(complement(sk2)))), complement(complement(sk2))), complement(sk1)))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 47 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(complement(join(complement(join(complement(sk1), complement(complement(sk2)))), complement(complement(sk2)))), complement(complement(sk1)))), complement(sk1)), sk3))
59.86/60.03	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(meet(join(complement(sk1), complement(complement(sk2))), complement(sk2)), complement(complement(sk1)))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(complement(complement(sk1)), meet(join(complement(sk1), complement(complement(sk2))), complement(sk2)))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(complement(complement(sk1)), meet(complement(sk2), join(complement(sk1), complement(complement(sk2)))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 49 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(complement(sk2), meet(join(complement(sk1), complement(complement(sk2))), complement(complement(sk1))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(complement(sk2), meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(join(join(meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), sk2), meet(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), complement(sk2))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 25 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), join(complement(sk1), complement(complement(sk2)))), complement(sk1)), sk3))
59.86/60.03	= { by lemma 33 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), join(complement(sk1), sk2)), complement(sk1)), sk3))
59.86/60.03	= { by lemma 50 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), join(complement(sk1), sk2)), meet(complement(sk1), join(complement(sk1), sk2))), sk3))
59.86/60.03	= { by lemma 33 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), join(complement(sk1), sk2)), meet(complement(complement(complement(sk1))), join(complement(sk1), sk2))), sk3))
59.86/60.03	= { by lemma 15 }
59.86/60.03	  meet(sk1, composition(join(meet(complement(complement(sk1)), join(complement(sk1), sk2)), meet(join(complement(sk1), sk2), complement(complement(complement(sk1))))), sk3))
59.86/60.03	= { by lemma 37 }
59.86/60.03	  meet(sk1, composition(join(complement(sk1), sk2), sk3))
59.86/60.03	= { by lemma 51 }
59.86/60.03	  meet(sk1, composition(sk2, sk3))
59.86/60.03	% SZS output end Proof
59.86/60.03	
59.86/60.03	RESULT: Unsatisfiable (the axioms are contradictory).
59.86/60.04	EOF