0.00/0.03	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.03/0.23	% Computer   : n129.star.cs.uiowa.edu
0.03/0.23	% Model      : x86_64 x86_64
0.03/0.23	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.23	% Memory     : 32218.625MB
0.03/0.23	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.23	% CPULimit   : 300
0.03/0.23	% DateTime   : Sat Jul 14 06:02:39 CDT 2018
0.03/0.23	% CPUTime    : 
63.21/63.41	% SZS status Theorem
63.21/63.41	
66.02/66.29	% SZS output start Proof
66.02/66.29	Take the following subset of the input axioms:
66.02/66.29	  fof(additive_associativity, axiom,
66.02/66.29	      ![A, B, C]:
66.02/66.29	        addition(A, addition(B, C))=addition(addition(A, B), C)).
66.02/66.29	  fof(additive_commutativity, axiom,
66.02/66.29	      ![A, B]: addition(B, A)=addition(A, B)).
66.02/66.29	  fof(additive_idempotence, axiom, ![A]: A=addition(A, A)).
66.02/66.29	  fof(additive_identity, axiom, ![A]: addition(A, zero)=A).
66.02/66.29	  fof(codomain1, axiom,
66.02/66.29	      ![X0]: multiplication(X0, coantidomain(X0))=zero).
66.02/66.29	  fof(codomain2, axiom,
66.02/66.29	      ![X0, X1]:
66.02/66.29	        addition(coantidomain(multiplication(X0, X1)),
66.02/66.29	                 coantidomain(multiplication(coantidomain(coantidomain(X0)),
66.02/66.29	                                             X1)))=coantidomain(multiplication(coantidomain(coantidomain(X0)),
66.02/66.29	                                                                               X1))).
66.02/66.29	  fof(codomain3, axiom,
66.02/66.29	      ![X0]:
66.02/66.29	        one=addition(coantidomain(coantidomain(X0)), coantidomain(X0))).
66.02/66.29	  fof(codomain4, axiom,
66.02/66.29	      ![X0]: codomain(X0)=coantidomain(coantidomain(X0))).
66.02/66.29	  fof(domain1, axiom,
66.02/66.29	      ![X0]: zero=multiplication(antidomain(X0), X0)).
66.02/66.29	  fof(domain2, axiom,
66.02/66.29	      ![X0, X1]:
66.02/66.29	        antidomain(multiplication(X0,
66.02/66.29	                                  antidomain(antidomain(X1))))=addition(antidomain(multiplication(X0,
66.02/66.29	                                                                                                  X1)),
66.02/66.29	                                                                        antidomain(multiplication(X0,
66.02/66.29	                                                                                                  antidomain(antidomain(X1)))))).
66.02/66.29	  fof(domain3, axiom,
66.02/66.29	      ![X0]: one=addition(antidomain(antidomain(X0)), antidomain(X0))).
66.02/66.29	  fof(domain4, axiom, ![X0]: domain(X0)=antidomain(antidomain(X0))).
66.02/66.29	  fof(goals, conjecture,
66.02/66.29	      ![X0, X1, X2]:
66.02/66.29	        addition(domain(multiplication(X0, domain(X1))),
66.02/66.29	                 domain(multiplication(X0, domain(X2))))=domain(multiplication(X0,
66.02/66.29	                                                                               addition(domain(X1),
66.02/66.29	                                                                                        domain(X2))))).
66.02/66.29	  fof(left_annihilation, axiom, ![A]: zero=multiplication(zero, A)).
66.02/66.29	  fof(left_distributivity, axiom,
66.02/66.29	      ![A, B, C]:
66.02/66.29	        addition(multiplication(A, C),
66.02/66.29	                 multiplication(B, C))=multiplication(addition(A, B), C)).
66.02/66.29	  fof(multiplicative_associativity, axiom,
66.02/66.29	      ![A, B, C]:
66.02/66.29	        multiplication(multiplication(A, B), C)=multiplication(A,
66.02/66.29	                                                               multiplication(B, C))).
66.02/66.29	  fof(multiplicative_left_identity, axiom,
66.02/66.29	      ![A]: A=multiplication(one, A)).
66.02/66.29	  fof(multiplicative_right_identity, axiom,
66.02/66.29	      ![A]: A=multiplication(A, one)).
66.02/66.29	  fof(order, axiom, ![A, B]: (leq(A, B) <=> B=addition(A, B))).
66.02/66.29	  fof(right_annihilation, axiom, ![A]: multiplication(A, zero)=zero).
66.02/66.29	  fof(right_distributivity, axiom,
66.02/66.29	      ![A, B, C]:
66.02/66.29	        addition(multiplication(A, B),
66.02/66.29	                 multiplication(A, C))=multiplication(A, addition(B, C))).
66.02/66.29	
66.02/66.29	Now clausify the problem and encode Horn clauses using encoding 3 of
66.02/66.29	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
66.02/66.29	We repeatedly replace C & s=t => u=v by the two clauses:
66.02/66.29	  $$fresh(y, y, x1...xn) = u
66.02/66.29	  C => $$fresh(s, t, x1...xn) = v
66.02/66.29	where $$fresh is a fresh function symbol and x1..xn are the free
66.02/66.29	variables of u and v.
66.02/66.29	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
66.02/66.29	input problem has no model of domain size 1).
66.02/66.29	
66.02/66.29	The encoding turns the above axioms into the following unit equations and goals:
66.02/66.29	
66.02/66.29	Axiom 1 (order): $$fresh2(X, X, Y, Z) = $$true.
66.02/66.29	Axiom 2 (order_1): $$fresh(X, X, Y, Z) = Z.
66.02/66.29	Axiom 3 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
66.02/66.29	Axiom 4 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
66.02/66.29	Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
66.02/66.29	Axiom 6 (multiplicative_left_identity): X = multiplication(one, X).
66.02/66.29	Axiom 7 (multiplicative_right_identity): X = multiplication(X, one).
66.02/66.29	Axiom 8 (left_annihilation): zero = multiplication(zero, X).
66.02/66.29	Axiom 9 (additive_identity): addition(X, zero) = X.
66.02/66.29	Axiom 10 (right_annihilation): multiplication(X, zero) = zero.
66.02/66.29	Axiom 11 (multiplicative_associativity): multiplication(multiplication(X, Y), Z) = multiplication(X, multiplication(Y, Z)).
66.02/66.29	Axiom 12 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
66.02/66.29	Axiom 13 (order_1): $$fresh(leq(X, Y), $$true, X, Y) = addition(X, Y).
66.02/66.29	Axiom 14 (order): $$fresh2(X, addition(Y, X), Y, X) = leq(Y, X).
66.02/66.29	Axiom 15 (additive_idempotence): X = addition(X, X).
66.02/66.29	Axiom 16 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
66.02/66.29	Axiom 17 (codomain3): one = addition(coantidomain(coantidomain(X)), coantidomain(X)).
66.02/66.29	Axiom 18 (domain2): antidomain(multiplication(X, antidomain(antidomain(Y)))) = addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))).
66.02/66.29	Axiom 19 (codomain1): multiplication(X, coantidomain(X)) = zero.
66.02/66.29	Axiom 20 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
66.02/66.29	Axiom 21 (domain1): zero = multiplication(antidomain(X), X).
66.02/66.29	Axiom 22 (domain3): one = addition(antidomain(antidomain(X)), antidomain(X)).
66.02/66.29	Axiom 23 (domain4): domain(X) = antidomain(antidomain(X)).
66.02/66.29	
66.02/66.29	Lemma 24: addition(coantidomain(X), coantidomain(coantidomain(X))) = one.
66.02/66.29	Proof:
66.02/66.29	  addition(coantidomain(X), coantidomain(coantidomain(X)))
66.02/66.29	= { by axiom 5 (additive_commutativity) }
66.02/66.29	  addition(coantidomain(coantidomain(X)), coantidomain(X))
66.02/66.29	= { by axiom 17 (codomain3) }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 25: addition(antidomain(X), antidomain(antidomain(X))) = one.
66.02/66.29	Proof:
66.02/66.29	  addition(antidomain(X), antidomain(antidomain(X)))
66.02/66.29	= { by axiom 5 (additive_commutativity) }
66.02/66.29	  addition(antidomain(antidomain(X)), antidomain(X))
66.02/66.29	= { by axiom 22 (domain3) }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 26: coantidomain(one) = zero.
66.02/66.29	Proof:
66.02/66.29	  coantidomain(one)
66.02/66.29	= { by axiom 6 (multiplicative_left_identity) }
66.02/66.29	  multiplication(one, coantidomain(one))
66.02/66.29	= { by axiom 19 (codomain1) }
66.02/66.29	  zero
66.02/66.29	
66.02/66.29	Lemma 27: antidomain(one) = zero.
66.02/66.29	Proof:
66.02/66.29	  antidomain(one)
66.02/66.29	= { by axiom 7 (multiplicative_right_identity) }
66.02/66.29	  multiplication(antidomain(one), one)
66.02/66.29	= { by axiom 21 (domain1) }
66.02/66.29	  zero
66.02/66.29	
66.02/66.29	Lemma 28: addition(zero, X) = X.
66.02/66.29	Proof:
66.02/66.29	  addition(zero, X)
66.02/66.29	= { by axiom 5 (additive_commutativity) }
66.02/66.29	  addition(X, zero)
66.02/66.29	= { by axiom 9 (additive_identity) }
66.02/66.29	  X
66.02/66.29	
66.02/66.29	Lemma 29: codomain(coantidomain(X)) = coantidomain(codomain(X)).
66.02/66.29	Proof:
66.02/66.29	  codomain(coantidomain(X))
66.02/66.29	= { by axiom 20 (codomain4) }
66.02/66.29	  coantidomain(coantidomain(coantidomain(X)))
66.02/66.29	= { by axiom 20 (codomain4) }
66.02/66.29	  coantidomain(codomain(X))
66.02/66.29	
66.02/66.29	Lemma 30: domain(antidomain(X)) = antidomain(domain(X)).
66.02/66.29	Proof:
66.02/66.29	  domain(antidomain(X))
66.02/66.29	= { by axiom 23 (domain4) }
66.02/66.29	  antidomain(antidomain(antidomain(X)))
66.02/66.29	= { by axiom 23 (domain4) }
66.02/66.29	  antidomain(domain(X))
66.02/66.29	
66.02/66.29	Lemma 31: addition(coantidomain(X), codomain(X)) = one.
66.02/66.29	Proof:
66.02/66.29	  addition(coantidomain(X), codomain(X))
66.02/66.29	= { by axiom 20 (codomain4) }
66.02/66.29	  addition(coantidomain(X), coantidomain(coantidomain(X)))
66.02/66.29	= { by lemma 24 }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 32: coantidomain(zero) = one.
66.02/66.29	Proof:
66.02/66.29	  coantidomain(zero)
66.02/66.29	= { by lemma 26 }
66.02/66.29	  coantidomain(coantidomain(one))
66.02/66.29	= { by axiom 20 (codomain4) }
66.02/66.29	  codomain(one)
66.02/66.29	= { by lemma 28 }
66.02/66.29	  addition(zero, codomain(one))
66.02/66.29	= { by lemma 26 }
66.02/66.29	  addition(coantidomain(one), codomain(one))
66.02/66.29	= { by lemma 31 }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 33: multiplication(domain(X), antidomain(X)) = zero.
66.02/66.29	Proof:
66.02/66.29	  multiplication(domain(X), antidomain(X))
66.02/66.29	= { by axiom 23 (domain4) }
66.02/66.29	  multiplication(antidomain(antidomain(X)), antidomain(X))
66.02/66.29	= { by axiom 21 (domain1) }
66.02/66.29	  zero
66.02/66.29	
66.02/66.29	Lemma 34: addition(antidomain(X), domain(X)) = one.
66.02/66.29	Proof:
66.02/66.29	  addition(antidomain(X), domain(X))
66.02/66.29	= { by axiom 23 (domain4) }
66.02/66.29	  addition(antidomain(X), antidomain(antidomain(X)))
66.02/66.29	= { by lemma 25 }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 35: antidomain(zero) = one.
66.02/66.29	Proof:
66.02/66.29	  antidomain(zero)
66.02/66.29	= { by lemma 27 }
66.02/66.29	  antidomain(antidomain(one))
66.02/66.29	= { by axiom 23 (domain4) }
66.02/66.29	  domain(one)
66.02/66.29	= { by lemma 28 }
66.02/66.29	  addition(zero, domain(one))
66.02/66.29	= { by lemma 27 }
66.02/66.29	  addition(antidomain(one), domain(one))
66.02/66.29	= { by lemma 34 }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 36: addition(X, addition(X, Y)) = addition(X, Y).
66.02/66.29	Proof:
66.02/66.29	  addition(X, addition(X, Y))
66.02/66.29	= { by axiom 12 (additive_associativity) }
66.02/66.29	  addition(addition(X, X), Y)
66.02/66.29	= { by axiom 15 (additive_idempotence) }
66.02/66.29	  addition(X, Y)
66.02/66.29	
66.02/66.29	Lemma 37: addition(one, coantidomain(X)) = one.
66.02/66.29	Proof:
66.02/66.29	  addition(one, coantidomain(X))
66.02/66.29	= { by axiom 5 (additive_commutativity) }
66.02/66.29	  addition(coantidomain(X), one)
66.02/66.29	= { by lemma 24 }
66.02/66.29	  addition(coantidomain(X), addition(coantidomain(X), coantidomain(coantidomain(X))))
66.02/66.29	= { by lemma 36 }
66.02/66.29	  addition(coantidomain(X), coantidomain(coantidomain(X)))
66.02/66.29	= { by lemma 24 }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 38: addition(one, antidomain(X)) = one.
66.02/66.29	Proof:
66.02/66.29	  addition(one, antidomain(X))
66.02/66.29	= { by axiom 5 (additive_commutativity) }
66.02/66.29	  addition(antidomain(X), one)
66.02/66.29	= { by lemma 25 }
66.02/66.29	  addition(antidomain(X), addition(antidomain(X), antidomain(antidomain(X))))
66.02/66.29	= { by lemma 36 }
66.02/66.29	  addition(antidomain(X), antidomain(antidomain(X)))
66.02/66.29	= { by lemma 25 }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 39: leq(X, addition(X, Y)) = $$true.
66.02/66.29	Proof:
66.02/66.29	  leq(X, addition(X, Y))
66.02/66.29	= { by axiom 14 (order) }
66.02/66.29	  $$fresh2(addition(X, Y), addition(X, addition(X, Y)), X, addition(X, Y))
66.02/66.29	= { by lemma 36 }
66.02/66.29	  $$fresh2(addition(X, Y), addition(X, Y), X, addition(X, Y))
66.02/66.29	= { by axiom 1 (order) }
66.02/66.29	  $$true
66.02/66.29	
66.02/66.29	Lemma 40: addition(one, codomain(X)) = one.
66.02/66.29	Proof:
66.02/66.29	  addition(one, codomain(X))
66.02/66.29	= { by axiom 20 (codomain4) }
66.02/66.29	  addition(one, coantidomain(coantidomain(X)))
66.02/66.29	= { by lemma 37 }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 41: addition(one, domain(X)) = one.
66.02/66.29	Proof:
66.02/66.29	  addition(one, domain(X))
66.02/66.29	= { by axiom 23 (domain4) }
66.02/66.29	  addition(one, antidomain(antidomain(X)))
66.02/66.29	= { by lemma 38 }
66.02/66.29	  one
66.02/66.29	
66.02/66.29	Lemma 42: multiplication(X, multiplication(coantidomain(X), Y)) = zero.
66.02/66.29	Proof:
66.02/66.29	  multiplication(X, multiplication(coantidomain(X), Y))
66.02/66.29	= { by axiom 11 (multiplicative_associativity) }
66.02/66.29	  multiplication(multiplication(X, coantidomain(X)), Y)
66.02/66.29	= { by axiom 19 (codomain1) }
66.02/66.29	  multiplication(zero, Y)
66.02/66.29	= { by axiom 8 (left_annihilation) }
66.02/66.29	  zero
66.02/66.29	
66.02/66.29	Lemma 43: multiplication(antidomain(X), multiplication(X, Y)) = zero.
66.02/66.29	Proof:
66.02/66.29	  multiplication(antidomain(X), multiplication(X, Y))
66.02/66.29	= { by axiom 11 (multiplicative_associativity) }
66.02/66.29	  multiplication(multiplication(antidomain(X), X), Y)
66.02/66.29	= { by axiom 21 (domain1) }
66.02/66.29	  multiplication(zero, Y)
66.02/66.29	= { by axiom 8 (left_annihilation) }
66.02/66.30	  zero
66.02/66.30	
66.02/66.30	Lemma 44: addition(codomain(X), coantidomain(codomain(X))) = one.
66.02/66.30	Proof:
66.02/66.30	  addition(codomain(X), coantidomain(codomain(X)))
66.02/66.30	= { by axiom 20 (codomain4) }
66.02/66.30	  addition(coantidomain(coantidomain(X)), coantidomain(codomain(X)))
66.02/66.30	= { by lemma 29 }
66.02/66.30	  addition(coantidomain(coantidomain(X)), codomain(coantidomain(X)))
66.02/66.30	= { by lemma 31 }
66.02/66.30	  one
66.02/66.30	
66.02/66.30	Lemma 45: addition(X, addition(Y, Z)) = addition(Z, addition(X, Y)).
66.02/66.30	Proof:
66.02/66.30	  addition(X, addition(Y, Z))
66.02/66.30	= { by axiom 12 (additive_associativity) }
66.02/66.30	  addition(addition(X, Y), Z)
66.02/66.30	= { by axiom 5 (additive_commutativity) }
66.02/66.30	  addition(Z, addition(X, Y))
66.02/66.30	
66.02/66.30	Lemma 46: addition(Y, multiplication(X, Y)) = multiplication(addition(X, one), Y).
66.02/66.30	Proof:
66.02/66.30	  addition(Y, multiplication(X, Y))
66.02/66.30	= { by axiom 6 (multiplicative_left_identity) }
66.02/66.30	  addition(multiplication(one, Y), multiplication(X, Y))
66.02/66.30	= { by axiom 4 (left_distributivity) }
66.02/66.30	  multiplication(addition(one, X), Y)
66.02/66.30	= { by axiom 5 (additive_commutativity) }
66.02/66.30	  multiplication(addition(X, one), Y)
66.02/66.30	
66.12/66.30	Lemma 47: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
66.12/66.30	Proof:
66.12/66.30	  addition(X, multiplication(X, Y))
66.12/66.30	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.30	  addition(multiplication(X, one), multiplication(X, Y))
66.12/66.30	= { by axiom 3 (right_distributivity) }
66.12/66.30	  multiplication(X, addition(one, Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(X, addition(Y, one))
66.12/66.30	
66.12/66.30	Lemma 48: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(X, addition(Y, coantidomain(X)))
66.12/66.30	= { by axiom 3 (right_distributivity) }
66.12/66.30	  addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
66.12/66.30	= { by axiom 19 (codomain1) }
66.12/66.30	  addition(multiplication(X, Y), zero)
66.12/66.30	= { by axiom 9 (additive_identity) }
66.12/66.30	  multiplication(X, Y)
66.12/66.30	
66.12/66.30	Lemma 49: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(addition(X, antidomain(Y)), Y)
66.12/66.30	= { by axiom 4 (left_distributivity) }
66.12/66.30	  addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
66.12/66.30	= { by axiom 21 (domain1) }
66.12/66.30	  addition(multiplication(X, Y), zero)
66.12/66.30	= { by axiom 9 (additive_identity) }
66.12/66.30	  multiplication(X, Y)
66.12/66.30	
66.12/66.30	Lemma 50: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(antidomain(X), addition(X, Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(antidomain(X), addition(Y, X))
66.12/66.30	= { by axiom 3 (right_distributivity) }
66.12/66.30	  addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
66.12/66.30	= { by axiom 21 (domain1) }
66.12/66.30	  addition(multiplication(antidomain(X), Y), zero)
66.12/66.30	= { by axiom 9 (additive_identity) }
66.12/66.30	  multiplication(antidomain(X), Y)
66.12/66.30	
66.12/66.30	Lemma 51: multiplication(antidomain(X), domain(X)) = zero.
66.12/66.30	Proof:
66.12/66.30	  multiplication(antidomain(X), domain(X))
66.12/66.30	= { by lemma 49 }
66.12/66.30	  multiplication(addition(antidomain(X), antidomain(domain(X))), domain(X))
66.12/66.30	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.30	  multiplication(addition(antidomain(multiplication(one, X)), antidomain(domain(X))), domain(X))
66.12/66.30	= { by lemma 30 }
66.12/66.30	  multiplication(addition(antidomain(multiplication(one, X)), domain(antidomain(X))), domain(X))
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  multiplication(addition(antidomain(multiplication(one, X)), antidomain(antidomain(antidomain(X)))), domain(X))
66.12/66.30	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.30	  multiplication(addition(antidomain(multiplication(one, X)), antidomain(multiplication(one, antidomain(antidomain(X))))), domain(X))
66.12/66.30	= { by axiom 18 (domain2) }
66.12/66.30	  multiplication(antidomain(multiplication(one, antidomain(antidomain(X)))), domain(X))
66.12/66.30	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.30	  multiplication(antidomain(antidomain(antidomain(X))), domain(X))
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  multiplication(domain(antidomain(X)), domain(X))
66.12/66.30	= { by lemma 30 }
66.12/66.30	  multiplication(antidomain(domain(X)), domain(X))
66.12/66.30	= { by axiom 21 (domain1) }
66.12/66.30	  zero
66.12/66.30	
66.12/66.30	Lemma 52: multiplication(X, addition(coantidomain(X), Y)) = multiplication(X, Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(X, addition(coantidomain(X), Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(X, addition(Y, coantidomain(X)))
66.12/66.30	= { by lemma 48 }
66.12/66.30	  multiplication(X, Y)
66.12/66.30	
66.12/66.30	Lemma 53: multiplication(X, codomain(X)) = X.
66.12/66.30	Proof:
66.12/66.30	  multiplication(X, codomain(X))
66.12/66.30	= { by axiom 20 (codomain4) }
66.12/66.30	  multiplication(X, coantidomain(coantidomain(X)))
66.12/66.30	= { by lemma 52 }
66.12/66.30	  multiplication(X, addition(coantidomain(X), coantidomain(coantidomain(X))))
66.12/66.30	= { by lemma 24 }
66.12/66.30	  multiplication(X, one)
66.12/66.30	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.30	  X
66.12/66.30	
66.12/66.30	Lemma 54: multiplication(domain(X), X) = X.
66.12/66.30	Proof:
66.12/66.30	  multiplication(domain(X), X)
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  multiplication(antidomain(antidomain(X)), X)
66.12/66.30	= { by lemma 49 }
66.12/66.30	  multiplication(addition(antidomain(antidomain(X)), antidomain(X)), X)
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(addition(antidomain(X), antidomain(antidomain(X))), X)
66.12/66.30	= { by lemma 25 }
66.12/66.30	  multiplication(one, X)
66.12/66.30	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.30	  X
66.12/66.30	
66.12/66.30	Lemma 55: multiplication(addition(X, Y), coantidomain(Y)) = multiplication(X, coantidomain(Y)).
66.12/66.30	Proof:
66.12/66.30	  multiplication(addition(X, Y), coantidomain(Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(addition(Y, X), coantidomain(Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(addition(X, Y), coantidomain(Y))
66.12/66.30	= { by axiom 4 (left_distributivity) }
66.12/66.30	  addition(multiplication(X, coantidomain(Y)), multiplication(Y, coantidomain(Y)))
66.12/66.30	= { by axiom 19 (codomain1) }
66.12/66.30	  addition(multiplication(X, coantidomain(Y)), zero)
66.12/66.30	= { by axiom 9 (additive_identity) }
66.12/66.30	  multiplication(X, coantidomain(Y))
66.12/66.30	
66.12/66.30	Lemma 56: coantidomain(codomain(X)) = coantidomain(X).
66.12/66.30	Proof:
66.12/66.30	  coantidomain(codomain(X))
66.12/66.30	= { by lemma 29 }
66.12/66.30	  codomain(coantidomain(X))
66.12/66.30	= { by axiom 20 (codomain4) }
66.12/66.30	  coantidomain(coantidomain(coantidomain(X)))
66.12/66.30	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.30	  multiplication(one, coantidomain(coantidomain(coantidomain(X))))
66.12/66.30	= { by lemma 24 }
66.12/66.30	  multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), coantidomain(coantidomain(coantidomain(X))))
66.12/66.30	= { by lemma 55 }
66.12/66.30	  multiplication(coantidomain(X), coantidomain(coantidomain(coantidomain(X))))
66.12/66.30	= { by axiom 20 (codomain4) }
66.12/66.30	  multiplication(coantidomain(X), codomain(coantidomain(X)))
66.12/66.30	= { by lemma 53 }
66.12/66.30	  coantidomain(X)
66.12/66.30	
66.12/66.30	Lemma 57: codomain(addition(X, one)) = one.
66.12/66.30	Proof:
66.12/66.30	  codomain(addition(X, one))
66.12/66.30	= { by axiom 20 (codomain4) }
66.12/66.30	  coantidomain(coantidomain(addition(X, one)))
66.12/66.30	= { by lemma 28 }
66.12/66.30	  addition(zero, coantidomain(coantidomain(addition(X, one))))
66.12/66.30	= { by axiom 19 (codomain1) }
66.12/66.30	  addition(multiplication(addition(one, X), coantidomain(addition(one, X))), coantidomain(coantidomain(addition(X, one))))
66.12/66.30	= { by lemma 36 }
66.12/66.30	  addition(multiplication(addition(one, addition(one, X)), coantidomain(addition(one, X))), coantidomain(coantidomain(addition(X, one))))
66.12/66.30	= { by lemma 55 }
66.12/66.30	  addition(multiplication(one, coantidomain(addition(one, X))), coantidomain(coantidomain(addition(X, one))))
66.12/66.30	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.30	  addition(coantidomain(addition(one, X)), coantidomain(coantidomain(addition(X, one))))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  addition(coantidomain(addition(X, one)), coantidomain(coantidomain(addition(X, one))))
66.12/66.30	= { by lemma 24 }
66.12/66.30	  one
66.12/66.30	
66.12/66.30	Lemma 58: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(antidomain(X), addition(Y, X))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(antidomain(X), addition(X, Y))
66.12/66.30	= { by lemma 50 }
66.12/66.30	  multiplication(antidomain(X), Y)
66.12/66.30	
66.12/66.30	Lemma 59: antidomain(domain(X)) = antidomain(X).
66.12/66.30	Proof:
66.12/66.30	  antidomain(domain(X))
66.12/66.30	= { by lemma 30 }
66.12/66.30	  domain(antidomain(X))
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  antidomain(antidomain(antidomain(X)))
66.12/66.30	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.30	  multiplication(antidomain(antidomain(antidomain(X))), one)
66.12/66.30	= { by lemma 25 }
66.12/66.30	  multiplication(antidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))
66.12/66.30	= { by lemma 58 }
66.12/66.30	  multiplication(antidomain(antidomain(antidomain(X))), antidomain(X))
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  multiplication(domain(antidomain(X)), antidomain(X))
66.12/66.30	= { by lemma 54 }
66.12/66.30	  antidomain(X)
66.12/66.30	
66.12/66.30	Lemma 60: multiplication(antidomain(addition(X, Y)), X) = zero.
66.12/66.30	Proof:
66.12/66.30	  multiplication(antidomain(addition(X, Y)), X)
66.12/66.30	= { by lemma 58 }
66.12/66.30	  multiplication(antidomain(addition(X, Y)), addition(X, addition(X, Y)))
66.12/66.30	= { by lemma 36 }
66.12/66.30	  multiplication(antidomain(addition(X, Y)), addition(X, Y))
66.12/66.30	= { by axiom 21 (domain1) }
66.12/66.30	  zero
66.12/66.30	
66.12/66.30	Lemma 61: multiplication(domain(X), addition(Y, antidomain(X))) = multiplication(domain(X), Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(domain(X), addition(Y, antidomain(X)))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(domain(X), addition(antidomain(X), Y))
66.12/66.30	= { by axiom 3 (right_distributivity) }
66.12/66.30	  addition(multiplication(domain(X), antidomain(X)), multiplication(domain(X), Y))
66.12/66.30	= { by lemma 33 }
66.12/66.30	  addition(zero, multiplication(domain(X), Y))
66.12/66.30	= { by lemma 28 }
66.12/66.30	  multiplication(domain(X), Y)
66.12/66.30	
66.12/66.30	Lemma 62: multiplication(addition(X, domain(Y)), antidomain(Y)) = multiplication(X, antidomain(Y)).
66.12/66.30	Proof:
66.12/66.30	  multiplication(addition(X, domain(Y)), antidomain(Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(addition(domain(Y), X), antidomain(Y))
66.12/66.30	= { by axiom 4 (left_distributivity) }
66.12/66.30	  addition(multiplication(domain(Y), antidomain(Y)), multiplication(X, antidomain(Y)))
66.12/66.30	= { by lemma 33 }
66.12/66.30	  addition(zero, multiplication(X, antidomain(Y)))
66.12/66.30	= { by lemma 28 }
66.12/66.30	  multiplication(X, antidomain(Y))
66.12/66.30	
66.12/66.30	Lemma 63: antidomain(multiplication(X, domain(multiplication(coantidomain(X), Y)))) = one.
66.12/66.30	Proof:
66.12/66.30	  antidomain(multiplication(X, domain(multiplication(coantidomain(X), Y))))
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  antidomain(multiplication(X, antidomain(antidomain(multiplication(coantidomain(X), Y)))))
66.12/66.30	= { by axiom 18 (domain2) }
66.12/66.30	  addition(antidomain(multiplication(X, multiplication(coantidomain(X), Y))), antidomain(multiplication(X, antidomain(antidomain(multiplication(coantidomain(X), Y))))))
66.12/66.30	= { by lemma 42 }
66.12/66.30	  addition(antidomain(zero), antidomain(multiplication(X, antidomain(antidomain(multiplication(coantidomain(X), Y))))))
66.12/66.30	= { by lemma 35 }
66.12/66.30	  addition(one, antidomain(multiplication(X, antidomain(antidomain(multiplication(coantidomain(X), Y))))))
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  addition(one, antidomain(multiplication(X, domain(multiplication(coantidomain(X), Y)))))
66.12/66.30	= { by lemma 38 }
66.12/66.30	  one
66.12/66.30	
66.12/66.30	Lemma 64: multiplication(X, domain(multiplication(coantidomain(X), Y))) = zero.
66.12/66.30	Proof:
66.12/66.30	  multiplication(X, domain(multiplication(coantidomain(X), Y)))
66.12/66.30	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.30	  multiplication(one, multiplication(X, domain(multiplication(coantidomain(X), Y))))
66.12/66.30	= { by lemma 63 }
66.12/66.30	  multiplication(antidomain(multiplication(X, domain(multiplication(coantidomain(X), Y)))), multiplication(X, domain(multiplication(coantidomain(X), Y))))
66.12/66.30	= { by axiom 21 (domain1) }
66.12/66.30	  zero
66.12/66.30	
66.12/66.30	Lemma 65: antidomain(multiplication(antidomain(X), domain(multiplication(X, Y)))) = one.
66.12/66.30	Proof:
66.12/66.30	  antidomain(multiplication(antidomain(X), domain(multiplication(X, Y))))
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  antidomain(multiplication(antidomain(X), antidomain(antidomain(multiplication(X, Y)))))
66.12/66.30	= { by axiom 18 (domain2) }
66.12/66.30	  addition(antidomain(multiplication(antidomain(X), multiplication(X, Y))), antidomain(multiplication(antidomain(X), antidomain(antidomain(multiplication(X, Y))))))
66.12/66.30	= { by lemma 43 }
66.12/66.30	  addition(antidomain(zero), antidomain(multiplication(antidomain(X), antidomain(antidomain(multiplication(X, Y))))))
66.12/66.30	= { by lemma 35 }
66.12/66.30	  addition(one, antidomain(multiplication(antidomain(X), antidomain(antidomain(multiplication(X, Y))))))
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  addition(one, antidomain(multiplication(antidomain(X), domain(multiplication(X, Y)))))
66.12/66.30	= { by lemma 38 }
66.12/66.30	  one
66.12/66.30	
66.12/66.30	Lemma 66: multiplication(codomain(X), addition(Y, coantidomain(X))) = multiplication(codomain(X), Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(codomain(X), addition(Y, coantidomain(X)))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(codomain(X), addition(coantidomain(X), Y))
66.12/66.30	= { by axiom 3 (right_distributivity) }
66.12/66.30	  addition(multiplication(codomain(X), coantidomain(X)), multiplication(codomain(X), Y))
66.12/66.30	= { by lemma 48 }
66.12/66.30	  addition(multiplication(codomain(X), addition(coantidomain(X), coantidomain(codomain(X)))), multiplication(codomain(X), Y))
66.12/66.30	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.30	  addition(multiplication(codomain(X), addition(coantidomain(multiplication(X, one)), coantidomain(codomain(X)))), multiplication(codomain(X), Y))
66.12/66.30	= { by lemma 29 }
66.12/66.30	  addition(multiplication(codomain(X), addition(coantidomain(multiplication(X, one)), codomain(coantidomain(X)))), multiplication(codomain(X), Y))
66.12/66.30	= { by axiom 20 (codomain4) }
66.12/66.30	  addition(multiplication(codomain(X), addition(coantidomain(multiplication(X, one)), coantidomain(coantidomain(coantidomain(X))))), multiplication(codomain(X), Y))
66.12/66.30	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.30	  addition(multiplication(codomain(X), addition(coantidomain(multiplication(X, one)), coantidomain(multiplication(coantidomain(coantidomain(X)), one)))), multiplication(codomain(X), Y))
66.12/66.30	= { by axiom 16 (codomain2) }
66.12/66.30	  addition(multiplication(codomain(X), coantidomain(multiplication(coantidomain(coantidomain(X)), one))), multiplication(codomain(X), Y))
66.12/66.30	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.30	  addition(multiplication(codomain(X), coantidomain(coantidomain(coantidomain(X)))), multiplication(codomain(X), Y))
66.12/66.30	= { by axiom 20 (codomain4) }
66.12/66.30	  addition(multiplication(codomain(X), codomain(coantidomain(X))), multiplication(codomain(X), Y))
66.12/66.30	= { by lemma 29 }
66.12/66.30	  addition(multiplication(codomain(X), coantidomain(codomain(X))), multiplication(codomain(X), Y))
66.12/66.30	= { by axiom 19 (codomain1) }
66.12/66.30	  addition(zero, multiplication(codomain(X), Y))
66.12/66.30	= { by lemma 28 }
66.12/66.30	  multiplication(codomain(X), Y)
66.12/66.30	
66.12/66.30	Lemma 67: multiplication(antidomain(X), addition(Y, domain(X))) = multiplication(antidomain(X), Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(antidomain(X), addition(Y, domain(X)))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(antidomain(X), addition(domain(X), Y))
66.12/66.30	= { by axiom 3 (right_distributivity) }
66.12/66.30	  addition(multiplication(antidomain(X), domain(X)), multiplication(antidomain(X), Y))
66.12/66.30	= { by lemma 51 }
66.12/66.30	  addition(zero, multiplication(antidomain(X), Y))
66.12/66.30	= { by lemma 28 }
66.12/66.30	  multiplication(antidomain(X), Y)
66.12/66.30	
66.12/66.30	Lemma 68: multiplication(addition(X, antidomain(Y)), domain(Y)) = multiplication(X, domain(Y)).
66.12/66.30	Proof:
66.12/66.30	  multiplication(addition(X, antidomain(Y)), domain(Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(addition(antidomain(Y), X), domain(Y))
66.12/66.30	= { by axiom 4 (left_distributivity) }
66.12/66.30	  addition(multiplication(antidomain(Y), domain(Y)), multiplication(X, domain(Y)))
66.12/66.30	= { by lemma 51 }
66.12/66.30	  addition(zero, multiplication(X, domain(Y)))
66.12/66.30	= { by lemma 28 }
66.12/66.30	  multiplication(X, domain(Y))
66.12/66.30	
66.12/66.30	Lemma 69: multiplication(domain(X), addition(antidomain(X), Y)) = multiplication(domain(X), Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(domain(X), addition(antidomain(X), Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(domain(X), addition(Y, antidomain(X)))
66.12/66.30	= { by lemma 61 }
66.12/66.30	  multiplication(domain(X), Y)
66.12/66.30	
66.12/66.30	Lemma 70: addition(Z, multiplication(X, multiplication(Y, Z))) = multiplication(addition(one, multiplication(X, Y)), Z).
66.12/66.30	Proof:
66.12/66.30	  addition(Z, multiplication(X, multiplication(Y, Z)))
66.12/66.30	= { by axiom 11 (multiplicative_associativity) }
66.12/66.30	  addition(Z, multiplication(multiplication(X, Y), Z))
66.12/66.30	= { by lemma 46 }
66.12/66.30	  multiplication(addition(multiplication(X, Y), one), Z)
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(addition(one, multiplication(X, Y)), Z)
66.12/66.30	
66.12/66.30	Lemma 71: addition(multiplication(X, addition(Y, one)), Z) = addition(X, addition(Z, multiplication(X, Y))).
66.12/66.30	Proof:
66.12/66.30	  addition(multiplication(X, addition(Y, one)), Z)
66.12/66.30	= { by lemma 47 }
66.12/66.30	  addition(addition(X, multiplication(X, Y)), Z)
66.12/66.30	= { by axiom 12 (additive_associativity) }
66.12/66.30	  addition(X, addition(multiplication(X, Y), Z))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  addition(X, addition(Z, multiplication(X, Y)))
66.12/66.30	
66.12/66.30	Lemma 72: multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y))) = zero.
66.12/66.30	Proof:
66.12/66.30	  multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y)))
66.12/66.30	= { by axiom 23 (domain4) }
66.12/66.30	  multiplication(antidomain(multiplication(X, Y)), multiplication(X, antidomain(antidomain(Y))))
66.12/66.30	= { by lemma 49 }
66.12/66.30	  multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))), multiplication(X, antidomain(antidomain(Y))))
66.12/66.30	= { by axiom 18 (domain2) }
66.12/66.30	  multiplication(antidomain(multiplication(X, antidomain(antidomain(Y)))), multiplication(X, antidomain(antidomain(Y))))
66.12/66.30	= { by axiom 21 (domain1) }
66.12/66.30	  zero
66.12/66.30	
66.12/66.30	Lemma 73: multiplication(antidomain(X), addition(domain(X), Y)) = multiplication(antidomain(X), Y).
66.12/66.30	Proof:
66.12/66.30	  multiplication(antidomain(X), addition(domain(X), Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(antidomain(X), addition(Y, domain(X)))
66.12/66.30	= { by lemma 67 }
66.12/66.30	  multiplication(antidomain(X), Y)
66.12/66.30	
66.12/66.30	Lemma 74: multiplication(addition(antidomain(Y), X), domain(Y)) = multiplication(X, domain(Y)).
66.12/66.30	Proof:
66.12/66.30	  multiplication(addition(antidomain(Y), X), domain(Y))
66.12/66.30	= { by axiom 5 (additive_commutativity) }
66.12/66.30	  multiplication(addition(X, antidomain(Y)), domain(Y))
66.12/66.30	= { by lemma 68 }
66.12/66.31	  multiplication(X, domain(Y))
66.12/66.31	
66.12/66.31	Lemma 75: leq(multiplication(X, Y), multiplication(X, addition(Y, Z))) = $$true.
66.12/66.31	Proof:
66.12/66.31	  leq(multiplication(X, Y), multiplication(X, addition(Y, Z)))
66.12/66.31	= { by axiom 3 (right_distributivity) }
66.12/66.31	  leq(multiplication(X, Y), addition(multiplication(X, Y), multiplication(X, Z)))
66.12/66.31	= { by lemma 39 }
66.12/66.31	  $$true
66.12/66.31	
66.12/66.31	Lemma 76: leq(multiplication(X, Y), multiplication(addition(X, Z), Y)) = $$true.
66.12/66.31	Proof:
66.12/66.31	  leq(multiplication(X, Y), multiplication(addition(X, Z), Y))
66.12/66.31	= { by axiom 4 (left_distributivity) }
66.12/66.31	  leq(multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y)))
66.12/66.31	= { by lemma 39 }
66.12/66.31	  $$true
66.12/66.31	
66.12/66.31	Lemma 77: leq(multiplication(coantidomain(X), Y), Y) = $$true.
66.12/66.31	Proof:
66.12/66.31	  leq(multiplication(coantidomain(X), Y), Y)
66.12/66.31	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.31	  leq(multiplication(coantidomain(X), Y), multiplication(one, Y))
66.12/66.31	= { by lemma 24 }
66.12/66.31	  leq(multiplication(coantidomain(X), Y), multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), Y))
66.12/66.31	= { by lemma 76 }
66.12/66.31	  $$true
66.12/66.31	
66.12/66.31	Lemma 78: leq(multiplication(antidomain(X), Y), Y) = $$true.
66.12/66.31	Proof:
66.12/66.31	  leq(multiplication(antidomain(X), Y), Y)
66.12/66.31	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.31	  leq(multiplication(antidomain(X), Y), multiplication(one, Y))
66.12/66.31	= { by lemma 25 }
66.12/66.31	  leq(multiplication(antidomain(X), Y), multiplication(addition(antidomain(X), antidomain(antidomain(X))), Y))
66.12/66.31	= { by lemma 76 }
66.12/66.31	  $$true
66.12/66.31	
66.12/66.31	Lemma 79: addition(multiplication(addition(one, Z), X), Y) = addition(X, addition(Y, multiplication(Z, X))).
66.12/66.31	Proof:
66.12/66.31	  addition(multiplication(addition(one, Z), X), Y)
66.12/66.31	= { by axiom 5 (additive_commutativity) }
66.12/66.31	  addition(multiplication(addition(Z, one), X), Y)
66.12/66.31	= { by lemma 46 }
66.12/66.31	  addition(addition(X, multiplication(Z, X)), Y)
66.12/66.31	= { by axiom 12 (additive_associativity) }
66.12/66.31	  addition(X, addition(multiplication(Z, X), Y))
66.12/66.31	= { by axiom 5 (additive_commutativity) }
66.12/66.31	  addition(X, addition(Y, multiplication(Z, X)))
66.12/66.31	
66.12/66.31	Lemma 81: addition(X, addition(Y, multiplication(X, Z))) = addition(multiplication(X, addition(one, Z)), Y).
66.12/66.31	Proof:
66.12/66.31	  addition(X, addition(Y, multiplication(X, Z)))
66.12/66.31	= { by lemma 71 }
66.12/66.31	  addition(multiplication(X, addition(Z, one)), Y)
66.12/66.31	= { by axiom 5 (additive_commutativity) }
66.12/66.31	  addition(multiplication(X, addition(one, Z)), Y)
66.12/66.31	
66.12/66.31	Lemma 81: addition(multiplication(X, addition(one, Z)), Y) = addition(X, addition(Y, multiplication(X, Z))).
66.12/66.31	Proof:
66.12/66.31	  addition(multiplication(X, addition(one, Z)), Y)
66.12/66.31	= { by axiom 5 (additive_commutativity) }
66.12/66.31	  addition(multiplication(X, addition(Z, one)), Y)
66.12/66.31	= { by lemma 71 }
66.12/66.31	  addition(X, addition(Y, multiplication(X, Z)))
66.12/66.31	
66.12/66.31	Lemma 82: multiplication(X, multiplication(domain(multiplication(coantidomain(X), Y)), Z)) = zero.
66.12/66.31	Proof:
66.12/66.31	  multiplication(X, multiplication(domain(multiplication(coantidomain(X), Y)), Z))
66.12/66.31	= { by axiom 11 (multiplicative_associativity) }
66.12/66.31	  multiplication(multiplication(X, domain(multiplication(coantidomain(X), Y))), Z)
66.12/66.31	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.31	  multiplication(one, multiplication(multiplication(X, domain(multiplication(coantidomain(X), Y))), Z))
66.12/66.31	= { by lemma 63 }
66.12/66.31	  multiplication(antidomain(multiplication(X, domain(multiplication(coantidomain(X), Y)))), multiplication(multiplication(X, domain(multiplication(coantidomain(X), Y))), Z))
66.12/66.31	= { by lemma 43 }
66.12/66.31	  zero
66.12/66.31	
66.12/66.31	Lemma 83: addition(multiplication(W, multiplication(Z, X)), multiplication(Y, X)) = multiplication(addition(Y, multiplication(W, Z)), X).
66.12/66.31	Proof:
66.12/66.31	  addition(multiplication(W, multiplication(Z, X)), multiplication(Y, X))
66.12/66.31	= { by axiom 11 (multiplicative_associativity) }
66.12/66.31	  addition(multiplication(multiplication(W, Z), X), multiplication(Y, X))
66.12/66.31	= { by axiom 4 (left_distributivity) }
66.12/66.31	  multiplication(addition(multiplication(W, Z), Y), X)
66.12/66.31	= { by axiom 5 (additive_commutativity) }
66.12/66.31	  multiplication(addition(Y, multiplication(W, Z)), X)
66.12/66.31	
66.12/66.31	Lemma 84: antidomain(codomain(X)) = coantidomain(X).
66.12/66.31	Proof:
66.12/66.31	  antidomain(codomain(X))
66.12/66.31	= { by axiom 2 (order_1) }
66.12/66.31	  $$fresh($$true, $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by lemma 77 }
66.12/66.31	  $$fresh(leq(multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X)))), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by lemma 28 }
66.12/66.31	  $$fresh(leq(addition(zero, multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X))))), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by axiom 21 (domain1) }
66.12/66.31	  $$fresh(leq(addition(multiplication(antidomain(multiplication(coantidomain(X), domain(coantidomain(coantidomain(X))))), multiplication(coantidomain(X), domain(coantidomain(coantidomain(X))))), multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X))))), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.31	  $$fresh(leq(addition(multiplication(antidomain(multiplication(coantidomain(X), domain(multiplication(coantidomain(coantidomain(X)), one)))), multiplication(coantidomain(X), domain(coantidomain(coantidomain(X))))), multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X))))), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by lemma 63 }
66.12/66.31	  $$fresh(leq(addition(multiplication(one, multiplication(coantidomain(X), domain(coantidomain(coantidomain(X))))), multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X))))), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.31	  $$fresh(leq(addition(multiplication(coantidomain(X), domain(coantidomain(coantidomain(X)))), multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X))))), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by axiom 3 (right_distributivity) }
66.12/66.31	  $$fresh(leq(multiplication(coantidomain(X), addition(domain(coantidomain(coantidomain(X))), antidomain(coantidomain(coantidomain(X))))), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by axiom 5 (additive_commutativity) }
66.12/66.31	  $$fresh(leq(multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(X))))), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by lemma 34 }
66.12/66.31	  $$fresh(leq(multiplication(coantidomain(X), one), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.31	  $$fresh(leq(coantidomain(X), antidomain(coantidomain(coantidomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by axiom 20 (codomain4) }
66.12/66.31	  $$fresh(leq(coantidomain(X), antidomain(codomain(X))), $$true, coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by axiom 13 (order_1) }
66.12/66.31	  addition(coantidomain(X), antidomain(codomain(X)))
66.12/66.31	= { by axiom 5 (additive_commutativity) }
66.12/66.31	  addition(antidomain(codomain(X)), coantidomain(X))
66.12/66.31	= { by axiom 13 (order_1) }
66.12/66.31	  $$fresh(leq(antidomain(codomain(X)), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
66.12/66.31	= { by axiom 20 (codomain4) }
66.12/66.31	  $$fresh(leq(antidomain(coantidomain(coantidomain(X))), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
66.12/66.31	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.31	  $$fresh(leq(multiplication(antidomain(coantidomain(coantidomain(X))), one), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
66.12/66.31	= { by lemma 24 }
66.12/66.31	  $$fresh(leq(multiplication(antidomain(coantidomain(coantidomain(X))), addition(coantidomain(X), coantidomain(coantidomain(X)))), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
66.12/66.31	= { by lemma 58 }
66.12/66.31	  $$fresh(leq(multiplication(antidomain(coantidomain(coantidomain(X))), coantidomain(X)), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
66.12/66.31	= { by axiom 20 (codomain4) }
66.12/66.31	  $$fresh(leq(multiplication(antidomain(codomain(X)), coantidomain(X)), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
66.12/66.31	= { by lemma 78 }
66.12/66.31	  $$fresh($$true, $$true, antidomain(codomain(X)), coantidomain(X))
66.12/66.31	= { by axiom 2 (order_1) }
66.12/66.31	  coantidomain(X)
66.12/66.31	
66.12/66.31	Lemma 85: domain(codomain(X)) = codomain(X).
66.12/66.31	Proof:
66.12/66.31	  domain(codomain(X))
66.12/66.31	= { by axiom 23 (domain4) }
66.12/66.31	  antidomain(antidomain(codomain(X)))
66.12/66.31	= { by lemma 84 }
66.12/66.31	  antidomain(coantidomain(X))
66.12/66.31	= { by lemma 56 }
66.12/66.31	  antidomain(coantidomain(codomain(X)))
66.12/66.31	= { by lemma 29 }
66.12/66.31	  antidomain(codomain(coantidomain(X)))
66.12/66.31	= { by lemma 84 }
66.12/66.31	  coantidomain(coantidomain(X))
66.12/66.31	= { by axiom 20 (codomain4) }
66.12/66.34	  codomain(X)
66.12/66.34	
66.12/66.34	Lemma 86: coantidomain(domain(X)) = antidomain(X).
66.12/66.34	Proof:
66.12/66.34	  coantidomain(domain(X))
66.12/66.34	= { by axiom 2 (order_1) }
66.12/66.34	  $$fresh($$true, $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by lemma 75 }
66.12/66.34	  $$fresh(leq(multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by lemma 28 }
66.12/66.34	  $$fresh(leq(addition(zero, multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 19 (codomain1) }
66.12/66.34	  $$fresh(leq(addition(multiplication(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)), coantidomain(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 20 (codomain4) }
66.12/66.34	  $$fresh(leq(addition(multiplication(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.34	  $$fresh(leq(addition(multiplication(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one)))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 16 (codomain2) }
66.12/66.34	  $$fresh(leq(addition(multiplication(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)), addition(coantidomain(multiplication(antidomain(antidomain(X)), multiplication(antidomain(X), one))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by lemma 43 }
66.12/66.34	  $$fresh(leq(addition(multiplication(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)), addition(coantidomain(zero), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by lemma 32 }
66.12/66.34	  $$fresh(leq(addition(multiplication(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)), addition(one, coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 20 (codomain4) }
66.12/66.34	  $$fresh(leq(addition(multiplication(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)), addition(one, coantidomain(multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by lemma 37 }
66.12/66.34	  $$fresh(leq(addition(multiplication(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)), one), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 11 (multiplicative_associativity) }
66.12/66.34	  $$fresh(leq(addition(multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), one)), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.34	  $$fresh(leq(addition(multiplication(codomain(antidomain(antidomain(X))), antidomain(X)), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 4 (left_distributivity) }
66.12/66.34	  $$fresh(leq(multiplication(addition(codomain(antidomain(antidomain(X))), coantidomain(antidomain(antidomain(X)))), antidomain(X)), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  $$fresh(leq(multiplication(addition(coantidomain(antidomain(antidomain(X))), codomain(antidomain(antidomain(X)))), antidomain(X)), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by lemma 31 }
66.12/66.34	  $$fresh(leq(multiplication(one, antidomain(X)), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.34	  $$fresh(leq(antidomain(X), multiplication(coantidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by lemma 25 }
66.12/66.34	  $$fresh(leq(antidomain(X), multiplication(coantidomain(antidomain(antidomain(X))), one)), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.34	  $$fresh(leq(antidomain(X), coantidomain(antidomain(antidomain(X)))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 23 (domain4) }
66.12/66.34	  $$fresh(leq(antidomain(X), coantidomain(domain(X))), $$true, antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 13 (order_1) }
66.12/66.34	  addition(antidomain(X), coantidomain(domain(X)))
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  addition(coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by axiom 13 (order_1) }
66.12/66.34	  $$fresh(leq(coantidomain(domain(X)), antidomain(X)), $$true, coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by axiom 23 (domain4) }
66.12/66.34	  $$fresh(leq(coantidomain(antidomain(antidomain(X))), antidomain(X)), $$true, coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.34	  $$fresh(leq(multiplication(one, coantidomain(antidomain(antidomain(X)))), antidomain(X)), $$true, coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by lemma 25 }
66.12/66.34	  $$fresh(leq(multiplication(addition(antidomain(X), antidomain(antidomain(X))), coantidomain(antidomain(antidomain(X)))), antidomain(X)), $$true, coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by lemma 55 }
66.12/66.34	  $$fresh(leq(multiplication(antidomain(X), coantidomain(antidomain(antidomain(X)))), antidomain(X)), $$true, coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by axiom 23 (domain4) }
66.12/66.34	  $$fresh(leq(multiplication(antidomain(X), coantidomain(domain(X))), antidomain(X)), $$true, coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.34	  $$fresh(leq(multiplication(antidomain(X), coantidomain(domain(X))), multiplication(antidomain(X), one)), $$true, coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by lemma 24 }
66.12/66.34	  $$fresh(leq(multiplication(antidomain(X), coantidomain(domain(X))), multiplication(antidomain(X), addition(coantidomain(domain(X)), coantidomain(coantidomain(domain(X)))))), $$true, coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by lemma 75 }
66.12/66.34	  $$fresh($$true, $$true, coantidomain(domain(X)), antidomain(X))
66.12/66.34	= { by axiom 2 (order_1) }
66.12/66.34	  antidomain(X)
66.12/66.34	
66.12/66.34	Lemma 87: codomain(antidomain(X)) = antidomain(X).
66.12/66.34	Proof:
66.12/66.34	  codomain(antidomain(X))
66.12/66.34	= { by lemma 86 }
66.12/66.34	  codomain(coantidomain(domain(X)))
66.12/66.34	= { by lemma 29 }
66.12/66.34	  coantidomain(codomain(domain(X)))
66.12/66.34	= { by lemma 56 }
66.12/66.34	  coantidomain(domain(X))
66.12/66.34	= { by lemma 86 }
66.12/66.34	  antidomain(X)
66.12/66.34	
66.12/66.34	Lemma 88: coantidomain(antidomain(X)) = domain(X).
66.12/66.34	Proof:
66.12/66.34	  coantidomain(antidomain(X))
66.12/66.34	= { by lemma 59 }
66.12/66.34	  coantidomain(antidomain(domain(X)))
66.12/66.34	= { by lemma 30 }
66.12/66.34	  coantidomain(domain(antidomain(X)))
66.12/66.34	= { by lemma 86 }
66.12/66.34	  antidomain(antidomain(X))
66.12/66.34	= { by axiom 23 (domain4) }
66.12/66.34	  domain(X)
66.12/66.34	
66.12/66.34	Lemma 89: multiplication(antidomain(X), multiplication(addition(one, Y), X)) = multiplication(antidomain(X), multiplication(Y, X)).
66.12/66.34	Proof:
66.12/66.34	  multiplication(antidomain(X), multiplication(addition(one, Y), X))
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  multiplication(antidomain(X), multiplication(addition(Y, one), X))
66.12/66.34	= { by lemma 46 }
66.12/66.34	  multiplication(antidomain(X), addition(X, multiplication(Y, X)))
66.12/66.34	= { by lemma 50 }
66.12/66.34	  multiplication(antidomain(X), multiplication(Y, X))
66.12/66.34	
66.12/66.34	Lemma 90: multiplication(antidomain(X), multiplication(antidomain(Y), X)) = zero.
66.12/66.34	Proof:
66.12/66.34	  multiplication(antidomain(X), multiplication(antidomain(Y), X))
66.12/66.34	= { by lemma 89 }
66.12/66.34	  multiplication(antidomain(X), multiplication(addition(one, antidomain(Y)), X))
66.12/66.34	= { by lemma 38 }
66.12/66.34	  multiplication(antidomain(X), multiplication(one, X))
66.12/66.34	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.34	  multiplication(antidomain(X), X)
66.12/66.34	= { by axiom 21 (domain1) }
66.12/66.34	  zero
66.12/66.34	
66.12/66.34	Lemma 91: multiplication(antidomain(X), domain(multiplication(codomain(Y), X))) = zero.
66.12/66.34	Proof:
66.12/66.34	  multiplication(antidomain(X), domain(multiplication(codomain(Y), X)))
66.12/66.34	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.34	  multiplication(one, multiplication(antidomain(X), domain(multiplication(codomain(Y), X))))
66.12/66.34	= { by lemma 35 }
66.12/66.34	  multiplication(antidomain(zero), multiplication(antidomain(X), domain(multiplication(codomain(Y), X))))
66.12/66.34	= { by axiom 21 (domain1) }
66.12/66.34	  multiplication(antidomain(multiplication(antidomain(X), X)), multiplication(antidomain(X), domain(multiplication(codomain(Y), X))))
66.12/66.34	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.34	  multiplication(antidomain(multiplication(antidomain(X), multiplication(one, X))), multiplication(antidomain(X), domain(multiplication(codomain(Y), X))))
66.12/66.34	= { by lemma 40 }
66.12/66.34	  multiplication(antidomain(multiplication(antidomain(X), multiplication(addition(one, codomain(Y)), X))), multiplication(antidomain(X), domain(multiplication(codomain(Y), X))))
66.12/66.34	= { by lemma 89 }
66.12/66.34	  multiplication(antidomain(multiplication(antidomain(X), multiplication(codomain(Y), X))), multiplication(antidomain(X), domain(multiplication(codomain(Y), X))))
66.12/66.34	= { by lemma 72 }
66.12/66.34	  zero
66.12/66.34	
66.12/66.34	Lemma 92: codomain(coantidomain(X)) = coantidomain(X).
66.12/66.34	Proof:
66.12/66.34	  codomain(coantidomain(X))
66.12/66.34	= { by axiom 20 (codomain4) }
66.12/66.34	  coantidomain(coantidomain(coantidomain(X)))
66.12/66.34	= { by axiom 20 (codomain4) }
66.12/66.34	  coantidomain(codomain(X))
66.12/66.34	= { by lemma 56 }
66.12/66.34	  coantidomain(X)
66.12/66.34	
66.12/66.34	Lemma 93: multiplication(domain(addition(X, Y)), X) = X.
66.12/66.34	Proof:
66.12/66.34	  multiplication(domain(addition(X, Y)), X)
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  multiplication(domain(addition(Y, X)), X)
66.12/66.34	= { by lemma 28 }
66.12/66.34	  addition(zero, multiplication(domain(addition(Y, X)), X))
66.12/66.34	= { by lemma 60 }
66.12/66.34	  addition(multiplication(antidomain(addition(X, Y)), X), multiplication(domain(addition(Y, X)), X))
66.12/66.34	= { by axiom 4 (left_distributivity) }
66.12/66.34	  multiplication(addition(antidomain(addition(X, Y)), domain(addition(Y, X))), X)
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  multiplication(addition(domain(addition(Y, X)), antidomain(addition(X, Y))), X)
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  multiplication(addition(domain(addition(Y, X)), antidomain(addition(Y, X))), X)
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  multiplication(addition(antidomain(addition(Y, X)), domain(addition(Y, X))), X)
66.12/66.34	= { by lemma 34 }
66.12/66.34	  multiplication(one, X)
66.12/66.34	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.34	  X
66.12/66.34	
66.12/66.34	Lemma 94: multiplication(X, antidomain(multiplication(coantidomain(X), Y))) = X.
66.12/66.34	Proof:
66.12/66.34	  multiplication(X, antidomain(multiplication(coantidomain(X), Y)))
66.12/66.34	= { by lemma 28 }
66.12/66.34	  addition(zero, multiplication(X, antidomain(multiplication(coantidomain(X), Y))))
66.12/66.34	= { by lemma 64 }
66.12/66.34	  addition(multiplication(X, domain(multiplication(coantidomain(X), Y))), multiplication(X, antidomain(multiplication(coantidomain(X), Y))))
66.12/66.34	= { by axiom 3 (right_distributivity) }
66.12/66.34	  multiplication(X, addition(domain(multiplication(coantidomain(X), Y)), antidomain(multiplication(coantidomain(X), Y))))
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  multiplication(X, addition(antidomain(multiplication(coantidomain(X), Y)), domain(multiplication(coantidomain(X), Y))))
66.12/66.34	= { by lemma 34 }
66.12/66.34	  multiplication(X, one)
66.12/66.34	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.34	  X
66.12/66.34	
66.12/66.34	Lemma 95: leq(antidomain(X), antidomain(multiplication(domain(X), Y))) = $$true.
66.12/66.34	Proof:
66.12/66.34	  leq(antidomain(X), antidomain(multiplication(domain(X), Y)))
66.12/66.34	= { by lemma 94 }
66.12/66.34	  leq(multiplication(antidomain(X), antidomain(multiplication(coantidomain(antidomain(X)), Y))), antidomain(multiplication(domain(X), Y)))
66.12/66.34	= { by lemma 88 }
66.12/66.34	  leq(multiplication(antidomain(X), antidomain(multiplication(coantidomain(antidomain(X)), Y))), antidomain(multiplication(coantidomain(antidomain(X)), Y)))
66.12/66.34	= { by lemma 78 }
66.12/66.34	  $$true
66.12/66.34	
66.12/66.34	Lemma 96: multiplication(antidomain(X), antidomain(multiplication(X, Y))) = antidomain(X).
66.12/66.34	Proof:
66.12/66.34	  multiplication(antidomain(X), antidomain(multiplication(X, Y)))
66.12/66.34	= { by lemma 28 }
66.12/66.34	  addition(zero, multiplication(antidomain(X), antidomain(multiplication(X, Y))))
66.12/66.34	= { by axiom 21 (domain1) }
66.12/66.34	  addition(multiplication(antidomain(multiplication(antidomain(X), domain(multiplication(X, Y)))), multiplication(antidomain(X), domain(multiplication(X, Y)))), multiplication(antidomain(X), antidomain(multiplication(X, Y))))
66.12/66.34	= { by lemma 65 }
66.12/66.34	  addition(multiplication(one, multiplication(antidomain(X), domain(multiplication(X, Y)))), multiplication(antidomain(X), antidomain(multiplication(X, Y))))
66.12/66.34	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.34	  addition(multiplication(antidomain(X), domain(multiplication(X, Y))), multiplication(antidomain(X), antidomain(multiplication(X, Y))))
66.12/66.34	= { by axiom 3 (right_distributivity) }
66.12/66.34	  multiplication(antidomain(X), addition(domain(multiplication(X, Y)), antidomain(multiplication(X, Y))))
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  multiplication(antidomain(X), addition(antidomain(multiplication(X, Y)), domain(multiplication(X, Y))))
66.12/66.34	= { by lemma 34 }
66.12/66.34	  multiplication(antidomain(X), one)
66.12/66.34	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.34	  antidomain(X)
66.12/66.34	
66.12/66.34	Lemma 97: addition(X, multiplication(antidomain(Y), addition(X, Z))) = addition(X, multiplication(antidomain(Y), Z)).
66.12/66.34	Proof:
66.12/66.34	  addition(X, multiplication(antidomain(Y), addition(X, Z)))
66.12/66.34	= { by axiom 5 (additive_commutativity) }
66.12/66.34	  addition(X, multiplication(antidomain(Y), addition(Z, X)))
66.12/66.34	= { by axiom 3 (right_distributivity) }
66.12/66.34	  addition(X, addition(multiplication(antidomain(Y), Z), multiplication(antidomain(Y), X)))
66.12/66.34	= { by lemma 79 }
66.12/66.34	  addition(multiplication(addition(one, antidomain(Y)), X), multiplication(antidomain(Y), Z))
66.12/66.34	= { by lemma 38 }
66.12/66.34	  addition(multiplication(one, X), multiplication(antidomain(Y), Z))
66.12/66.34	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.34	  addition(X, multiplication(antidomain(Y), Z))
66.12/66.34	
66.12/66.34	Lemma 98: multiplication(addition(X, Y), multiplication(Z, coantidomain(multiplication(X, Z)))) = multiplication(Y, multiplication(Z, coantidomain(multiplication(X, Z)))).
66.12/66.34	Proof:
66.12/66.34	  multiplication(addition(X, Y), multiplication(Z, coantidomain(multiplication(X, Z))))
66.12/66.34	= { by axiom 4 (left_distributivity) }
66.12/66.34	  addition(multiplication(X, multiplication(Z, coantidomain(multiplication(X, Z)))), multiplication(Y, multiplication(Z, coantidomain(multiplication(X, Z)))))
66.12/66.34	= { by axiom 11 (multiplicative_associativity) }
66.12/66.34	  addition(multiplication(multiplication(X, Z), coantidomain(multiplication(X, Z))), multiplication(Y, multiplication(Z, coantidomain(multiplication(X, Z)))))
66.12/66.34	= { by axiom 19 (codomain1) }
66.12/66.34	  addition(zero, multiplication(Y, multiplication(Z, coantidomain(multiplication(X, Z)))))
66.12/66.34	= { by lemma 28 }
66.12/66.35	  multiplication(Y, multiplication(Z, coantidomain(multiplication(X, Z))))
66.12/66.35	
66.12/66.35	Lemma 99: addition(multiplication(X, Y), multiplication(addition(one, X), Z)) = addition(Z, multiplication(X, addition(Y, Z))).
66.12/66.35	Proof:
66.12/66.35	  addition(multiplication(X, Y), multiplication(addition(one, X), Z))
66.12/66.35	= { by axiom 5 (additive_commutativity) }
66.12/66.35	  addition(multiplication(X, Y), multiplication(addition(X, one), Z))
66.12/66.35	= { by lemma 46 }
66.12/66.35	  addition(multiplication(X, Y), addition(Z, multiplication(X, Z)))
66.12/66.35	= { by lemma 45 }
66.12/66.35	  addition(multiplication(X, Z), addition(multiplication(X, Y), Z))
66.12/66.35	= { by lemma 45 }
66.12/66.35	  addition(Z, addition(multiplication(X, Z), multiplication(X, Y)))
66.12/66.35	= { by axiom 3 (right_distributivity) }
66.12/66.35	  addition(Z, multiplication(X, addition(Z, Y)))
66.12/66.35	= { by axiom 5 (additive_commutativity) }
66.12/66.35	  addition(Z, multiplication(X, addition(Y, Z)))
66.12/66.35	
66.12/66.35	Lemma 100: multiplication(codomain(X), domain(multiplication(codomain(X), Y))) = domain(multiplication(codomain(X), Y)).
66.12/66.35	Proof:
66.12/66.35	  multiplication(codomain(X), domain(multiplication(codomain(X), Y)))
66.12/66.35	= { by axiom 20 (codomain4) }
66.12/66.35	  multiplication(codomain(X), domain(multiplication(coantidomain(coantidomain(X)), Y)))
66.12/66.35	= { by lemma 28 }
66.12/66.35	  addition(zero, multiplication(codomain(X), domain(multiplication(coantidomain(coantidomain(X)), Y))))
66.12/66.35	= { by lemma 64 }
66.12/66.35	  addition(multiplication(coantidomain(X), domain(multiplication(coantidomain(coantidomain(X)), Y))), multiplication(codomain(X), domain(multiplication(coantidomain(coantidomain(X)), Y))))
66.12/66.35	= { by axiom 4 (left_distributivity) }
66.12/66.35	  multiplication(addition(coantidomain(X), codomain(X)), domain(multiplication(coantidomain(coantidomain(X)), Y)))
66.12/66.35	= { by lemma 31 }
66.12/66.35	  multiplication(one, domain(multiplication(coantidomain(coantidomain(X)), Y)))
66.12/66.35	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.35	  domain(multiplication(coantidomain(coantidomain(X)), Y))
66.12/66.35	= { by axiom 20 (codomain4) }
66.12/66.35	  domain(multiplication(codomain(X), Y))
66.12/66.35	
66.12/66.35	Lemma 101: addition(coantidomain(X), multiplication(antidomain(Y), codomain(X))) = addition(coantidomain(X), antidomain(Y)).
66.12/66.35	Proof:
66.12/66.35	  addition(coantidomain(X), multiplication(antidomain(Y), codomain(X)))
66.12/66.35	= { by lemma 97 }
66.12/66.35	  addition(coantidomain(X), multiplication(antidomain(Y), addition(coantidomain(X), codomain(X))))
66.12/66.35	= { by lemma 31 }
66.12/66.35	  addition(coantidomain(X), multiplication(antidomain(Y), one))
66.12/66.35	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.35	  addition(coantidomain(X), antidomain(Y))
66.12/66.35	
66.12/66.35	Lemma 102: addition(antidomain(X), multiplication(antidomain(Y), domain(X))) = addition(antidomain(X), antidomain(Y)).
66.12/66.35	Proof:
66.12/66.35	  addition(antidomain(X), multiplication(antidomain(Y), domain(X)))
66.12/66.35	= { by lemma 97 }
66.12/66.35	  addition(antidomain(X), multiplication(antidomain(Y), addition(antidomain(X), domain(X))))
66.12/66.35	= { by lemma 34 }
66.12/66.35	  addition(antidomain(X), multiplication(antidomain(Y), one))
66.12/66.35	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.35	  addition(antidomain(X), antidomain(Y))
66.12/66.35	
66.12/66.35	Lemma 103: antidomain(multiplication(X, domain(Y))) = antidomain(multiplication(X, Y)).
66.12/66.35	Proof:
66.12/66.35	  antidomain(multiplication(X, domain(Y)))
66.12/66.35	= { by axiom 23 (domain4) }
66.12/66.35	  antidomain(multiplication(X, antidomain(antidomain(Y))))
66.12/66.35	= { by axiom 18 (domain2) }
66.12/66.35	  addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
66.12/66.35	= { by axiom 23 (domain4) }
66.12/66.35	  addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
66.12/66.35	= { by lemma 102 }
66.12/66.35	  addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))
66.12/66.35	= { by lemma 54 }
66.12/66.35	  addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, multiplication(domain(Y), Y)))))
66.12/66.35	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.35	  addition(antidomain(multiplication(X, Y)), multiplication(one, multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, multiplication(domain(Y), Y))))))
66.12/66.35	= { by lemma 35 }
66.12/66.35	  addition(antidomain(multiplication(X, Y)), multiplication(antidomain(zero), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, multiplication(domain(Y), Y))))))
66.12/66.35	= { by lemma 43 }
66.12/66.35	  addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(multiplication(X, domain(Y))), multiplication(multiplication(X, domain(Y)), Y))), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, multiplication(domain(Y), Y))))))
66.12/66.35	= { by axiom 11 (multiplicative_associativity) }
66.12/66.35	  addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, multiplication(domain(Y), Y)))), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, multiplication(domain(Y), Y))))))
66.12/66.35	= { by lemma 72 }
66.12/66.35	  addition(antidomain(multiplication(X, Y)), zero)
66.12/66.35	= { by axiom 9 (additive_identity) }
66.12/66.35	  antidomain(multiplication(X, Y))
66.12/66.35	
66.12/66.35	Lemma 104: domain(multiplication(X, domain(Y))) = domain(multiplication(X, Y)).
66.12/66.35	Proof:
66.12/66.35	  domain(multiplication(X, domain(Y)))
66.12/66.35	= { by axiom 23 (domain4) }
66.12/66.35	  antidomain(antidomain(multiplication(X, domain(Y))))
66.12/66.35	= { by lemma 103 }
66.12/66.35	  antidomain(antidomain(multiplication(X, Y)))
66.12/66.35	= { by axiom 23 (domain4) }
66.12/66.35	  domain(multiplication(X, Y))
66.12/66.35	
66.12/66.35	Lemma 105: multiplication(domain(X), coantidomain(Y)) = multiplication(coantidomain(Y), domain(X)).
66.12/66.35	Proof:
66.12/66.35	  multiplication(domain(X), coantidomain(Y))
66.12/66.35	= { by lemma 61 }
66.12/66.35	  multiplication(domain(X), addition(coantidomain(Y), antidomain(X)))
66.12/66.35	= { by axiom 5 (additive_commutativity) }
66.12/66.35	  multiplication(domain(X), addition(antidomain(X), coantidomain(Y)))
66.12/66.35	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.35	  multiplication(domain(X), addition(antidomain(X), multiplication(coantidomain(Y), one)))
66.12/66.35	= { by lemma 34 }
66.12/66.35	  multiplication(domain(X), addition(antidomain(X), multiplication(coantidomain(Y), addition(antidomain(X), domain(X)))))
66.12/66.35	= { by axiom 5 (additive_commutativity) }
66.12/66.35	  multiplication(domain(X), addition(antidomain(X), multiplication(coantidomain(Y), addition(domain(X), antidomain(X)))))
66.12/66.35	= { by axiom 3 (right_distributivity) }
66.12/66.35	  multiplication(domain(X), addition(antidomain(X), addition(multiplication(coantidomain(Y), domain(X)), multiplication(coantidomain(Y), antidomain(X)))))
66.12/66.35	= { by lemma 79 }
66.12/66.35	  multiplication(domain(X), addition(multiplication(addition(one, coantidomain(Y)), antidomain(X)), multiplication(coantidomain(Y), domain(X))))
66.12/66.35	= { by lemma 37 }
66.12/66.35	  multiplication(domain(X), addition(multiplication(one, antidomain(X)), multiplication(coantidomain(Y), domain(X))))
66.12/66.35	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.35	  multiplication(domain(X), addition(antidomain(X), multiplication(coantidomain(Y), domain(X))))
66.12/66.35	= { by lemma 69 }
66.12/66.35	  multiplication(domain(X), multiplication(coantidomain(Y), domain(X)))
66.12/66.35	= { by axiom 11 (multiplicative_associativity) }
66.12/66.35	  multiplication(multiplication(domain(X), coantidomain(Y)), domain(X))
66.12/66.35	= { by lemma 74 }
66.12/66.35	  multiplication(addition(antidomain(X), multiplication(domain(X), coantidomain(Y))), domain(X))
66.12/66.35	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.35	  multiplication(addition(multiplication(antidomain(X), one), multiplication(domain(X), coantidomain(Y))), domain(X))
66.12/66.35	= { by lemma 37 }
66.12/66.35	  multiplication(addition(multiplication(antidomain(X), addition(one, coantidomain(Y))), multiplication(domain(X), coantidomain(Y))), domain(X))
66.12/66.35	= { by lemma 81 }
66.12/66.35	  multiplication(addition(antidomain(X), addition(multiplication(domain(X), coantidomain(Y)), multiplication(antidomain(X), coantidomain(Y)))), domain(X))
66.12/66.35	= { by axiom 4 (left_distributivity) }
66.12/66.35	  multiplication(addition(antidomain(X), multiplication(addition(domain(X), antidomain(X)), coantidomain(Y))), domain(X))
66.12/66.35	= { by axiom 5 (additive_commutativity) }
66.12/66.35	  multiplication(addition(antidomain(X), multiplication(addition(antidomain(X), domain(X)), coantidomain(Y))), domain(X))
66.12/66.35	= { by lemma 34 }
66.12/66.35	  multiplication(addition(antidomain(X), multiplication(one, coantidomain(Y))), domain(X))
66.12/66.35	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.35	  multiplication(addition(antidomain(X), coantidomain(Y)), domain(X))
66.12/66.35	= { by axiom 5 (additive_commutativity) }
66.12/66.35	  multiplication(addition(coantidomain(Y), antidomain(X)), domain(X))
66.12/66.35	= { by lemma 68 }
66.12/66.35	  multiplication(coantidomain(Y), domain(X))
66.12/66.35	
66.12/66.35	Lemma 107: multiplication(codomain(X), domain(Y)) = multiplication(domain(Y), codomain(X)).
66.12/66.35	Proof:
66.12/66.35	  multiplication(codomain(X), domain(Y))
66.12/66.35	= { by axiom 20 (codomain4) }
66.12/66.35	  multiplication(coantidomain(coantidomain(X)), domain(Y))
66.12/66.35	= { by lemma 105 }
66.12/66.35	  multiplication(domain(Y), coantidomain(coantidomain(X)))
66.12/66.35	= { by axiom 20 (codomain4) }
66.12/66.35	  multiplication(domain(Y), codomain(X))
66.12/66.35	
66.12/66.35	Lemma 107: multiplication(domain(Y), codomain(X)) = multiplication(codomain(X), domain(Y)).
66.12/66.35	Proof:
66.12/66.35	  multiplication(domain(Y), codomain(X))
66.12/66.35	= { by axiom 20 (codomain4) }
66.12/66.35	  multiplication(domain(Y), coantidomain(coantidomain(X)))
66.12/66.35	= { by lemma 105 }
66.12/66.35	  multiplication(coantidomain(coantidomain(X)), domain(Y))
66.12/66.35	= { by axiom 20 (codomain4) }
66.12/66.35	  multiplication(codomain(X), domain(Y))
66.12/66.35	
66.12/66.35	Lemma 108: multiplication(domain(X), multiplication(Y, coantidomain(multiplication(antidomain(X), Y)))) = multiplication(Y, coantidomain(multiplication(antidomain(X), Y))).
66.12/66.35	Proof:
66.12/66.35	  multiplication(domain(X), multiplication(Y, coantidomain(multiplication(antidomain(X), Y))))
66.12/66.35	= { by lemma 98 }
66.12/66.35	  multiplication(addition(antidomain(X), domain(X)), multiplication(Y, coantidomain(multiplication(antidomain(X), Y))))
66.12/66.35	= { by lemma 34 }
66.12/66.35	  multiplication(one, multiplication(Y, coantidomain(multiplication(antidomain(X), Y))))
66.12/66.35	= { by axiom 6 (multiplicative_left_identity) }
66.12/66.35	  multiplication(Y, coantidomain(multiplication(antidomain(X), Y)))
66.12/66.35	
66.12/66.35	Lemma 109: multiplication(domain(Y), multiplication(antidomain(X), Y)) = multiplication(antidomain(X), Y).
66.12/66.35	Proof:
66.12/66.35	  multiplication(domain(Y), multiplication(antidomain(X), Y))
66.12/66.35	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.35	  multiplication(domain(Y), multiplication(antidomain(X), multiplication(Y, one)))
66.12/66.35	= { by lemma 32 }
66.12/66.35	  multiplication(domain(Y), multiplication(antidomain(X), multiplication(Y, coantidomain(zero))))
66.12/66.35	= { by axiom 11 (multiplicative_associativity) }
66.12/66.35	  multiplication(domain(Y), multiplication(multiplication(antidomain(X), Y), coantidomain(zero)))
66.12/66.35	= { by lemma 90 }
66.12/66.35	  multiplication(domain(Y), multiplication(multiplication(antidomain(X), Y), coantidomain(multiplication(antidomain(Y), multiplication(antidomain(X), Y)))))
66.12/66.35	= { by lemma 108 }
66.12/66.36	  multiplication(multiplication(antidomain(X), Y), coantidomain(multiplication(antidomain(Y), multiplication(antidomain(X), Y))))
66.12/66.36	= { by axiom 11 (multiplicative_associativity) }
66.12/66.36	  multiplication(antidomain(X), multiplication(Y, coantidomain(multiplication(antidomain(Y), multiplication(antidomain(X), Y)))))
66.12/66.36	= { by lemma 90 }
66.12/66.36	  multiplication(antidomain(X), multiplication(Y, coantidomain(zero)))
66.12/66.36	= { by lemma 32 }
66.12/66.36	  multiplication(antidomain(X), multiplication(Y, one))
66.12/66.36	= { by axiom 7 (multiplicative_right_identity) }
66.12/66.36	  multiplication(antidomain(X), Y)
66.12/66.36	
66.12/66.36	Lemma 110: multiplication(X, multiplication(addition(Y, Z), coantidomain(multiplication(X, Z)))) = multiplication(X, multiplication(Y, coantidomain(multiplication(X, Z)))).
66.12/66.36	Proof:
66.12/66.36	  multiplication(X, multiplication(addition(Y, Z), coantidomain(multiplication(X, Z))))
66.12/66.36	= { by axiom 11 (multiplicative_associativity) }
66.12/66.36	  multiplication(multiplication(X, addition(Y, Z)), coantidomain(multiplication(X, Z)))
66.12/66.36	= { by axiom 3 (right_distributivity) }
66.12/66.36	  multiplication(addition(multiplication(X, Y), multiplication(X, Z)), coantidomain(multiplication(X, Z)))
66.12/66.36	= { by lemma 55 }
66.12/66.36	  multiplication(multiplication(X, Y), coantidomain(multiplication(X, Z)))
66.12/66.36	= { by axiom 11 (multiplicative_associativity) }
66.12/66.36	  multiplication(X, multiplication(Y, coantidomain(multiplication(X, Z))))
66.12/66.36	
66.12/66.36	Lemma 111: multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y)))) = zero.
66.12/66.36	Proof:
66.12/66.36	  multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))
66.12/66.36	= { by lemma 58 }
66.12/66.36	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, X))), multiplication(antidomain(X), domain(addition(X, Y))))
66.12/66.36	= { by axiom 5 (additive_commutativity) }
66.12/66.36	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))
66.12/66.36	= { by lemma 72 }
66.12/66.36	  zero
66.12/66.36	
66.12/66.36	Lemma 112: multiplication(codomain(X), multiplication(addition(coantidomain(X), Y), Z)) = multiplication(codomain(X), multiplication(Y, Z)).
66.12/66.36	Proof:
66.12/66.36	  multiplication(codomain(X), multiplication(addition(coantidomain(X), Y), Z))
66.12/66.36	= { by axiom 5 (additive_commutativity) }
66.12/66.36	  multiplication(codomain(X), multiplication(addition(Y, coantidomain(X)), Z))
66.12/66.36	= { by axiom 11 (multiplicative_associativity) }
66.12/66.36	  multiplication(multiplication(codomain(X), addition(Y, coantidomain(X))), Z)
66.12/66.36	= { by lemma 66 }
66.12/66.36	  multiplication(multiplication(codomain(X), Y), Z)
66.12/66.36	= { by axiom 11 (multiplicative_associativity) }
66.43/66.61	  multiplication(codomain(X), multiplication(Y, Z))
66.43/66.61	
66.43/66.61	Lemma 113: domain(multiplication(codomain(Y), X)) = multiplication(domain(X), codomain(Y)).
66.43/66.61	Proof:
66.43/66.61	  domain(multiplication(codomain(Y), X))
66.43/66.61	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), X)), one)
66.43/66.61	= { by lemma 32 }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), X)), coantidomain(zero))
66.43/66.61	= { by lemma 91 }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), X)), coantidomain(multiplication(antidomain(X), domain(multiplication(codomain(Y), X)))))
66.43/66.61	= { by lemma 108 }
66.43/66.61	  multiplication(domain(X), multiplication(domain(multiplication(codomain(Y), X)), coantidomain(multiplication(antidomain(X), domain(multiplication(codomain(Y), X))))))
66.43/66.61	= { by lemma 91 }
66.43/66.61	  multiplication(domain(X), multiplication(domain(multiplication(codomain(Y), X)), coantidomain(zero)))
66.43/66.61	= { by lemma 32 }
66.43/66.61	  multiplication(domain(X), multiplication(domain(multiplication(codomain(Y), X)), one))
66.43/66.61	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.61	  multiplication(domain(X), domain(multiplication(codomain(Y), X)))
66.43/66.61	= { by lemma 104 }
66.43/66.61	  multiplication(domain(X), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by lemma 74 }
66.43/66.61	  multiplication(addition(antidomain(multiplication(codomain(Y), domain(X))), domain(X)), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by axiom 6 (multiplicative_left_identity) }
66.43/66.61	  multiplication(addition(antidomain(multiplication(codomain(Y), domain(X))), multiplication(one, domain(X))), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by lemma 34 }
66.43/66.61	  multiplication(addition(antidomain(multiplication(codomain(Y), domain(X))), multiplication(addition(antidomain(multiplication(codomain(Y), domain(X))), domain(multiplication(codomain(Y), domain(X)))), domain(X))), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by axiom 5 (additive_commutativity) }
66.43/66.61	  multiplication(addition(antidomain(multiplication(codomain(Y), domain(X))), multiplication(addition(domain(multiplication(codomain(Y), domain(X))), antidomain(multiplication(codomain(Y), domain(X)))), domain(X))), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by axiom 4 (left_distributivity) }
66.43/66.61	  multiplication(addition(antidomain(multiplication(codomain(Y), domain(X))), addition(multiplication(domain(multiplication(codomain(Y), domain(X))), domain(X)), multiplication(antidomain(multiplication(codomain(Y), domain(X))), domain(X)))), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by lemma 81 }
66.43/66.61	  multiplication(addition(multiplication(antidomain(multiplication(codomain(Y), domain(X))), addition(one, domain(X))), multiplication(domain(multiplication(codomain(Y), domain(X))), domain(X))), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by lemma 41 }
66.43/66.61	  multiplication(addition(multiplication(antidomain(multiplication(codomain(Y), domain(X))), one), multiplication(domain(multiplication(codomain(Y), domain(X))), domain(X))), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.61	  multiplication(addition(antidomain(multiplication(codomain(Y), domain(X))), multiplication(domain(multiplication(codomain(Y), domain(X))), domain(X))), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by lemma 74 }
66.43/66.61	  multiplication(multiplication(domain(multiplication(codomain(Y), domain(X))), domain(X)), domain(multiplication(codomain(Y), domain(X))))
66.43/66.61	= { by axiom 11 (multiplicative_associativity) }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), multiplication(domain(X), domain(multiplication(codomain(Y), domain(X)))))
66.43/66.61	= { by lemma 69 }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), addition(antidomain(multiplication(codomain(Y), domain(X))), multiplication(domain(X), domain(multiplication(codomain(Y), domain(X))))))
66.43/66.61	= { by axiom 6 (multiplicative_left_identity) }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), addition(multiplication(one, antidomain(multiplication(codomain(Y), domain(X)))), multiplication(domain(X), domain(multiplication(codomain(Y), domain(X))))))
66.43/66.61	= { by lemma 41 }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), addition(multiplication(addition(one, domain(X)), antidomain(multiplication(codomain(Y), domain(X)))), multiplication(domain(X), domain(multiplication(codomain(Y), domain(X))))))
66.43/66.61	= { by lemma 79 }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), addition(antidomain(multiplication(codomain(Y), domain(X))), addition(multiplication(domain(X), domain(multiplication(codomain(Y), domain(X)))), multiplication(domain(X), antidomain(multiplication(codomain(Y), domain(X)))))))
66.43/66.61	= { by axiom 3 (right_distributivity) }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), addition(antidomain(multiplication(codomain(Y), domain(X))), multiplication(domain(X), addition(domain(multiplication(codomain(Y), domain(X))), antidomain(multiplication(codomain(Y), domain(X)))))))
66.43/66.61	= { by axiom 5 (additive_commutativity) }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), addition(antidomain(multiplication(codomain(Y), domain(X))), multiplication(domain(X), addition(antidomain(multiplication(codomain(Y), domain(X))), domain(multiplication(codomain(Y), domain(X)))))))
66.43/66.61	= { by lemma 34 }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), addition(antidomain(multiplication(codomain(Y), domain(X))), multiplication(domain(X), one)))
66.43/66.61	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), addition(antidomain(multiplication(codomain(Y), domain(X))), domain(X)))
66.43/66.61	= { by lemma 69 }
66.43/66.61	  multiplication(domain(multiplication(codomain(Y), domain(X))), domain(X))
66.43/66.61	= { by axiom 20 (codomain4) }
66.43/66.61	  multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X))
66.43/66.61	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.61	  multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), multiplication(domain(X), one))
66.43/66.61	= { by lemma 32 }
66.43/66.61	  multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), multiplication(domain(X), coantidomain(zero)))
66.43/66.61	= { by lemma 82 }
66.43/66.61	  multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), multiplication(domain(X), coantidomain(multiplication(coantidomain(Y), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X))))))
66.43/66.61	= { by axiom 11 (multiplicative_associativity) }
66.43/66.61	  multiplication(multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X)), coantidomain(multiplication(coantidomain(Y), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X)))))
66.43/66.61	= { by axiom 6 (multiplicative_left_identity) }
66.43/66.61	  multiplication(one, multiplication(multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X)), coantidomain(multiplication(coantidomain(Y), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X))))))
66.43/66.61	= { by lemma 31 }
66.43/66.61	  multiplication(addition(coantidomain(Y), codomain(Y)), multiplication(multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X)), coantidomain(multiplication(coantidomain(Y), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X))))))
66.43/66.61	= { by lemma 98 }
66.43/66.61	  multiplication(codomain(Y), multiplication(multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X)), coantidomain(multiplication(coantidomain(Y), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X))))))
66.43/66.61	= { by lemma 82 }
66.43/66.61	  multiplication(codomain(Y), multiplication(multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X)), coantidomain(zero)))
66.43/66.61	= { by axiom 11 (multiplicative_associativity) }
66.43/66.61	  multiplication(codomain(Y), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), multiplication(domain(X), coantidomain(zero))))
66.43/66.61	= { by lemma 32 }
66.43/66.61	  multiplication(codomain(Y), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), multiplication(domain(X), one)))
66.43/66.61	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.61	  multiplication(codomain(Y), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), domain(X)))
66.43/66.61	= { by lemma 112 }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), domain(X)))
66.43/66.61	= { by axiom 6 (multiplicative_left_identity) }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(one, domain(X))))
66.43/66.61	= { by lemma 40 }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(one, codomain(coantidomain(coantidomain(Y)))), domain(X))))
66.43/66.61	= { by axiom 5 (additive_commutativity) }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(codomain(coantidomain(coantidomain(Y))), one), domain(X))))
66.43/66.61	= { by lemma 34 }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(codomain(coantidomain(coantidomain(Y))), addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))))), domain(X))))
66.43/66.61	= { by axiom 12 (additive_associativity) }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(addition(codomain(coantidomain(coantidomain(Y))), antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))), domain(X))))
66.43/66.61	= { by axiom 5 (additive_commutativity) }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), codomain(coantidomain(coantidomain(Y)))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))), domain(X))))
66.43/66.61	= { by axiom 12 (additive_associativity) }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), addition(codomain(coantidomain(coantidomain(Y))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))))), domain(X))))
66.43/66.61	= { by lemma 85 }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), addition(domain(codomain(coantidomain(coantidomain(Y)))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))))), domain(X))))
66.43/66.61	= { by lemma 100 }
66.43/66.61	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), addition(domain(codomain(coantidomain(coantidomain(Y)))), multiplication(codomain(coantidomain(coantidomain(Y))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))))), domain(X))))
66.43/66.62	= { by lemma 54 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), addition(domain(codomain(coantidomain(coantidomain(Y)))), multiplication(multiplication(domain(codomain(coantidomain(coantidomain(Y)))), codomain(coantidomain(coantidomain(Y)))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))))), domain(X))))
66.43/66.62	= { by axiom 11 (multiplicative_associativity) }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), addition(domain(codomain(coantidomain(coantidomain(Y)))), multiplication(domain(codomain(coantidomain(coantidomain(Y)))), multiplication(codomain(coantidomain(coantidomain(Y))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))))))), domain(X))))
66.43/66.62	= { by lemma 47 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), multiplication(domain(codomain(coantidomain(coantidomain(Y)))), addition(multiplication(codomain(coantidomain(coantidomain(Y))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))), one))), domain(X))))
66.43/66.62	= { by lemma 85 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), multiplication(codomain(coantidomain(coantidomain(Y))), addition(multiplication(codomain(coantidomain(coantidomain(Y))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))), one))), domain(X))))
66.43/66.62	= { by axiom 5 (additive_commutativity) }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), multiplication(codomain(coantidomain(coantidomain(Y))), addition(one, multiplication(codomain(coantidomain(coantidomain(Y))), domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))))))), domain(X))))
66.43/66.62	= { by lemma 100 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), multiplication(codomain(coantidomain(coantidomain(Y))), addition(one, domain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))))), domain(X))))
66.43/66.62	= { by lemma 41 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), multiplication(codomain(coantidomain(coantidomain(Y))), one)), domain(X))))
66.43/66.62	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X))), codomain(coantidomain(coantidomain(Y)))), domain(X))))
66.43/66.62	= { by axiom 5 (additive_commutativity) }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(codomain(coantidomain(coantidomain(Y))), antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))), domain(X))))
66.43/66.62	= { by lemma 92 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(coantidomain(coantidomain(Y)), antidomain(multiplication(codomain(coantidomain(coantidomain(Y))), domain(X)))), domain(X))))
66.43/66.62	= { by lemma 92 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(coantidomain(coantidomain(Y)), antidomain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), domain(X))))
66.43/66.62	= { by lemma 86 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), multiplication(addition(coantidomain(coantidomain(Y)), coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))))), domain(X))))
66.43/66.62	= { by axiom 11 (multiplicative_associativity) }
66.43/66.62	  multiplication(codomain(Y), multiplication(multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), addition(coantidomain(coantidomain(Y)), coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))))), domain(X)))
66.43/66.62	= { by axiom 5 (additive_commutativity) }
66.43/66.62	  multiplication(codomain(Y), multiplication(multiplication(addition(coantidomain(Y), domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), addition(coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), coantidomain(coantidomain(Y)))), domain(X)))
66.43/66.62	= { by axiom 4 (left_distributivity) }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(multiplication(coantidomain(Y), addition(coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), coantidomain(coantidomain(Y)))), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), addition(coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), coantidomain(coantidomain(Y))))), domain(X)))
66.43/66.62	= { by lemma 48 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(multiplication(coantidomain(Y), coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))))), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), addition(coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), coantidomain(coantidomain(Y))))), domain(X)))
66.43/66.62	= { by lemma 52 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(multiplication(coantidomain(Y), coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))))), multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), coantidomain(coantidomain(Y)))), domain(X)))
66.43/66.62	= { by lemma 83 }
66.43/66.62	  multiplication(codomain(Y), addition(multiplication(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))), multiplication(coantidomain(coantidomain(Y)), domain(X))), multiplication(multiplication(coantidomain(Y), coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))))), domain(X))))
66.43/66.62	= { by lemma 54 }
66.43/66.62	  multiplication(codomain(Y), addition(multiplication(coantidomain(coantidomain(Y)), domain(X)), multiplication(multiplication(coantidomain(Y), coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))))), domain(X))))
66.43/66.62	= { by axiom 4 (left_distributivity) }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(coantidomain(Y)), multiplication(coantidomain(Y), coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X)))))), domain(X)))
66.43/66.62	= { by axiom 5 (additive_commutativity) }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(multiplication(coantidomain(Y), coantidomain(domain(multiplication(coantidomain(coantidomain(Y)), domain(X))))), coantidomain(coantidomain(Y))), domain(X)))
66.43/66.62	= { by lemma 86 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(multiplication(coantidomain(Y), antidomain(multiplication(coantidomain(coantidomain(Y)), domain(X)))), coantidomain(coantidomain(Y))), domain(X)))
66.43/66.62	= { by lemma 94 }
66.43/66.62	  multiplication(codomain(Y), multiplication(addition(coantidomain(Y), coantidomain(coantidomain(Y))), domain(X)))
66.43/66.62	= { by lemma 112 }
66.43/66.62	  multiplication(codomain(Y), multiplication(coantidomain(coantidomain(Y)), domain(X)))
66.43/66.62	= { by axiom 20 (codomain4) }
66.43/66.62	  multiplication(codomain(Y), multiplication(codomain(Y), domain(X)))
66.43/66.62	= { by axiom 11 (multiplicative_associativity) }
66.43/66.62	  multiplication(multiplication(codomain(Y), codomain(Y)), domain(X))
66.43/66.62	= { by lemma 48 }
66.43/66.62	  multiplication(multiplication(codomain(Y), addition(codomain(Y), coantidomain(codomain(Y)))), domain(X))
66.43/66.62	= { by lemma 44 }
66.43/66.62	  multiplication(multiplication(codomain(Y), one), domain(X))
66.43/66.62	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.62	  multiplication(codomain(Y), domain(X))
66.43/66.62	= { by lemma 107 }
66.43/66.64	  multiplication(domain(X), codomain(Y))
66.43/66.64	
66.43/66.64	Lemma 114: multiplication(codomain(X), domain(addition(Y, coantidomain(X)))) = multiplication(domain(Y), codomain(X)).
66.43/66.64	Proof:
66.43/66.64	  multiplication(codomain(X), domain(addition(Y, coantidomain(X))))
66.43/66.64	= { by axiom 5 (additive_commutativity) }
66.43/66.64	  multiplication(codomain(X), domain(addition(coantidomain(X), Y)))
66.43/66.64	= { by lemma 107 }
66.43/66.64	  multiplication(domain(addition(coantidomain(X), Y)), codomain(X))
66.43/66.64	= { by lemma 56 }
66.43/66.64	  multiplication(domain(addition(coantidomain(codomain(X)), Y)), codomain(X))
66.43/66.64	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.64	  multiplication(domain(addition(coantidomain(multiplication(codomain(X), one)), Y)), codomain(X))
66.43/66.64	= { by axiom 15 (additive_idempotence) }
66.43/66.64	  multiplication(domain(addition(coantidomain(multiplication(codomain(X), addition(one, one))), Y)), codomain(X))
66.43/66.64	= { by lemma 44 }
66.43/66.64	  multiplication(domain(addition(coantidomain(multiplication(codomain(X), addition(one, addition(codomain(X), coantidomain(codomain(X)))))), Y)), codomain(X))
66.43/66.64	= { by axiom 5 (additive_commutativity) }
66.43/66.64	  multiplication(domain(addition(coantidomain(multiplication(codomain(X), addition(one, addition(coantidomain(codomain(X)), codomain(X))))), Y)), codomain(X))
66.43/66.64	= { by axiom 12 (additive_associativity) }
66.43/66.64	  multiplication(domain(addition(coantidomain(multiplication(codomain(X), addition(addition(one, coantidomain(codomain(X))), codomain(X)))), Y)), codomain(X))
66.43/66.64	= { by axiom 3 (right_distributivity) }
66.43/66.64	  multiplication(domain(addition(coantidomain(addition(multiplication(codomain(X), addition(one, coantidomain(codomain(X)))), multiplication(codomain(X), codomain(X)))), Y)), codomain(X))
66.43/66.64	= { by lemma 48 }
66.43/66.64	  multiplication(domain(addition(coantidomain(addition(multiplication(codomain(X), one), multiplication(codomain(X), codomain(X)))), Y)), codomain(X))
66.43/66.64	= { by axiom 3 (right_distributivity) }
66.43/66.64	  multiplication(domain(addition(coantidomain(multiplication(codomain(X), addition(one, codomain(X)))), Y)), codomain(X))
66.43/66.64	= { by axiom 5 (additive_commutativity) }
66.43/66.64	  multiplication(domain(addition(coantidomain(multiplication(codomain(X), addition(codomain(X), one))), Y)), codomain(X))
66.43/66.64	= { by lemma 113 }
66.43/66.64	  domain(multiplication(codomain(X), addition(coantidomain(multiplication(codomain(X), addition(codomain(X), one))), Y)))
66.43/66.64	= { by lemma 100 }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(coantidomain(multiplication(codomain(X), addition(codomain(X), one))), Y))))
66.43/66.64	= { by lemma 52 }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(coantidomain(codomain(X)), addition(coantidomain(multiplication(codomain(X), addition(codomain(X), one))), Y)))))
66.43/66.64	= { by lemma 45 }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, addition(coantidomain(codomain(X)), coantidomain(multiplication(codomain(X), addition(codomain(X), one))))))))
66.43/66.64	= { by axiom 5 (additive_commutativity) }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, addition(coantidomain(multiplication(codomain(X), addition(codomain(X), one))), coantidomain(codomain(X)))))))
66.43/66.64	= { by lemma 47 }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, addition(coantidomain(addition(codomain(X), multiplication(codomain(X), codomain(X)))), coantidomain(codomain(X)))))))
66.43/66.64	= { by lemma 46 }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, addition(coantidomain(multiplication(addition(codomain(X), one), codomain(X))), coantidomain(codomain(X)))))))
66.43/66.64	= { by axiom 6 (multiplicative_left_identity) }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, addition(coantidomain(multiplication(addition(codomain(X), one), codomain(X))), coantidomain(multiplication(one, codomain(X))))))))
66.43/66.64	= { by lemma 57 }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, addition(coantidomain(multiplication(addition(codomain(X), one), codomain(X))), coantidomain(multiplication(codomain(addition(codomain(X), one)), codomain(X))))))))
66.43/66.64	= { by axiom 20 (codomain4) }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, addition(coantidomain(multiplication(addition(codomain(X), one), codomain(X))), coantidomain(multiplication(coantidomain(coantidomain(addition(codomain(X), one))), codomain(X))))))))
66.43/66.64	= { by axiom 16 (codomain2) }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, coantidomain(multiplication(coantidomain(coantidomain(addition(codomain(X), one))), codomain(X)))))))
66.43/66.64	= { by axiom 20 (codomain4) }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, coantidomain(multiplication(codomain(addition(codomain(X), one)), codomain(X)))))))
66.43/66.64	= { by lemma 57 }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, coantidomain(multiplication(one, codomain(X)))))))
66.43/66.64	= { by axiom 6 (multiplicative_left_identity) }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), addition(Y, coantidomain(codomain(X))))))
66.43/66.64	= { by lemma 48 }
66.43/66.64	  multiplication(codomain(X), domain(multiplication(codomain(X), Y)))
66.43/66.64	= { by lemma 100 }
66.43/66.64	  domain(multiplication(codomain(X), Y))
66.43/66.64	= { by lemma 113 }
66.43/66.66	  multiplication(domain(Y), codomain(X))
66.43/66.66	
66.43/66.66	Lemma 115: antidomain(multiplication(antidomain(Y), X)) = addition(antidomain(X), domain(Y)).
66.43/66.66	Proof:
66.43/66.66	  antidomain(multiplication(antidomain(Y), X))
66.43/66.66	= { by lemma 87 }
66.43/66.66	  antidomain(multiplication(codomain(antidomain(Y)), X))
66.43/66.66	= { by axiom 2 (order_1) }
66.43/66.66	  $$fresh($$true, $$true, coantidomain(antidomain(Y)), antidomain(multiplication(codomain(antidomain(Y)), X)))
66.43/66.66	= { by lemma 77 }
66.43/66.66	  $$fresh(leq(multiplication(coantidomain(antidomain(Y)), antidomain(multiplication(coantidomain(coantidomain(antidomain(Y))), X))), antidomain(multiplication(coantidomain(coantidomain(antidomain(Y))), X))), $$true, coantidomain(antidomain(Y)), antidomain(multiplication(codomain(antidomain(Y)), X)))
66.43/66.66	= { by lemma 94 }
66.43/66.66	  $$fresh(leq(coantidomain(antidomain(Y)), antidomain(multiplication(coantidomain(coantidomain(antidomain(Y))), X))), $$true, coantidomain(antidomain(Y)), antidomain(multiplication(codomain(antidomain(Y)), X)))
66.43/66.66	= { by axiom 20 (codomain4) }
66.43/66.66	  $$fresh(leq(coantidomain(antidomain(Y)), antidomain(multiplication(codomain(antidomain(Y)), X))), $$true, coantidomain(antidomain(Y)), antidomain(multiplication(codomain(antidomain(Y)), X)))
66.43/66.66	= { by axiom 13 (order_1) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), antidomain(multiplication(codomain(antidomain(Y)), X)))
66.43/66.66	= { by lemma 66 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), antidomain(multiplication(codomain(antidomain(Y)), addition(X, coantidomain(antidomain(Y))))))
66.43/66.66	= { by lemma 103 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), antidomain(multiplication(codomain(antidomain(Y)), domain(addition(X, coantidomain(antidomain(Y)))))))
66.43/66.66	= { by axiom 9 (additive_identity) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), addition(antidomain(multiplication(codomain(antidomain(Y)), domain(addition(X, coantidomain(antidomain(Y)))))), zero))
66.43/66.66	= { by lemma 72 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), addition(antidomain(multiplication(codomain(antidomain(Y)), domain(addition(X, coantidomain(antidomain(Y)))))), multiplication(antidomain(multiplication(codomain(antidomain(Y)), domain(addition(X, coantidomain(antidomain(Y)))))), multiplication(codomain(antidomain(Y)), domain(domain(addition(X, coantidomain(antidomain(Y)))))))))
66.43/66.66	= { by lemma 47 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(codomain(antidomain(Y)), domain(addition(X, coantidomain(antidomain(Y)))))), addition(multiplication(codomain(antidomain(Y)), domain(domain(addition(X, coantidomain(antidomain(Y)))))), one)))
66.43/66.66	= { by lemma 114 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(multiplication(codomain(antidomain(Y)), domain(domain(addition(X, coantidomain(antidomain(Y)))))), one)))
66.43/66.66	= { by axiom 5 (additive_commutativity) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(codomain(antidomain(Y)), domain(domain(addition(X, coantidomain(antidomain(Y)))))))))
66.43/66.66	= { by axiom 23 (domain4) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(codomain(antidomain(Y)), antidomain(antidomain(domain(addition(X, coantidomain(antidomain(Y))))))))))
66.43/66.66	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(codomain(antidomain(Y)), multiplication(antidomain(antidomain(domain(addition(X, coantidomain(antidomain(Y)))))), one)))))
66.43/66.66	= { by lemma 34 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(codomain(antidomain(Y)), multiplication(antidomain(antidomain(domain(addition(X, coantidomain(antidomain(Y)))))), addition(antidomain(antidomain(addition(X, coantidomain(antidomain(Y))))), domain(antidomain(addition(X, coantidomain(antidomain(Y)))))))))))
66.43/66.66	= { by axiom 23 (domain4) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(codomain(antidomain(Y)), multiplication(antidomain(antidomain(domain(addition(X, coantidomain(antidomain(Y)))))), addition(domain(addition(X, coantidomain(antidomain(Y)))), domain(antidomain(addition(X, coantidomain(antidomain(Y)))))))))))
66.43/66.66	= { by lemma 30 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(codomain(antidomain(Y)), multiplication(antidomain(antidomain(domain(addition(X, coantidomain(antidomain(Y)))))), addition(domain(addition(X, coantidomain(antidomain(Y)))), antidomain(domain(addition(X, coantidomain(antidomain(Y)))))))))))
66.43/66.66	= { by lemma 58 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(codomain(antidomain(Y)), multiplication(antidomain(antidomain(domain(addition(X, coantidomain(antidomain(Y)))))), domain(addition(X, coantidomain(antidomain(Y)))))))))
66.43/66.66	= { by axiom 23 (domain4) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(codomain(antidomain(Y)), multiplication(domain(domain(addition(X, coantidomain(antidomain(Y))))), domain(addition(X, coantidomain(antidomain(Y)))))))))
66.43/66.66	= { by lemma 54 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(codomain(antidomain(Y)), domain(addition(X, coantidomain(antidomain(Y))))))))
66.43/66.66	= { by lemma 114 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), addition(one, multiplication(domain(X), codomain(antidomain(Y))))))
66.43/66.66	= { by lemma 58 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), one))
66.43/66.66	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), antidomain(multiplication(domain(X), codomain(antidomain(Y)))))
66.43/66.66	= { by lemma 101 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), codomain(antidomain(Y))))
66.43/66.66	= { by axiom 2 (order_1) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication($$fresh($$true, $$true, antidomain(X), antidomain(multiplication(domain(X), codomain(antidomain(Y))))), codomain(antidomain(Y))))
66.43/66.66	= { by lemma 95 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication($$fresh(leq(antidomain(X), antidomain(multiplication(domain(X), codomain(antidomain(Y))))), $$true, antidomain(X), antidomain(multiplication(domain(X), codomain(antidomain(Y))))), codomain(antidomain(Y))))
66.43/66.66	= { by axiom 13 (order_1) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(addition(antidomain(X), antidomain(multiplication(domain(X), codomain(antidomain(Y))))), codomain(antidomain(Y))))
66.43/66.66	= { by lemma 102 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(addition(antidomain(X), multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), domain(X))), codomain(antidomain(Y))))
66.43/66.66	= { by lemma 83 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), addition(multiplication(antidomain(multiplication(domain(X), codomain(antidomain(Y)))), multiplication(domain(X), codomain(antidomain(Y)))), multiplication(antidomain(X), codomain(antidomain(Y)))))
66.43/66.66	= { by axiom 21 (domain1) }
66.43/66.66	  addition(coantidomain(antidomain(Y)), addition(zero, multiplication(antidomain(X), codomain(antidomain(Y)))))
66.43/66.66	= { by lemma 28 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), multiplication(antidomain(X), codomain(antidomain(Y))))
66.43/66.66	= { by lemma 101 }
66.43/66.66	  addition(coantidomain(antidomain(Y)), antidomain(X))
66.43/66.66	= { by lemma 88 }
66.43/66.66	  addition(domain(Y), antidomain(X))
66.43/66.66	= { by axiom 5 (additive_commutativity) }
66.43/66.68	  addition(antidomain(X), domain(Y))
66.43/66.68	
66.43/66.68	Lemma 116: multiplication(domain(X), multiplication(antidomain(addition(X, Y)), Z)) = zero.
66.43/66.68	Proof:
66.43/66.68	  multiplication(domain(X), multiplication(antidomain(addition(X, Y)), Z))
66.43/66.68	= { by lemma 28 }
66.43/66.68	  multiplication(addition(zero, domain(X)), multiplication(antidomain(addition(X, Y)), Z))
66.43/66.68	= { by lemma 60 }
66.43/66.68	  multiplication(addition(multiplication(antidomain(addition(one, Y)), one), domain(X)), multiplication(antidomain(addition(X, Y)), Z))
66.43/66.68	= { by axiom 7 (multiplicative_right_identity) }
66.43/66.68	  multiplication(addition(antidomain(addition(one, Y)), domain(X)), multiplication(antidomain(addition(X, Y)), Z))
66.43/66.68	= { by axiom 5 (additive_commutativity) }
66.43/66.68	  multiplication(addition(antidomain(addition(Y, one)), domain(X)), multiplication(antidomain(addition(X, Y)), Z))
66.43/66.68	= { by lemma 115 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(antidomain(addition(X, Y)), Z))
66.43/66.68	= { by axiom 11 (multiplicative_associativity) }
66.43/66.68	  multiplication(multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), antidomain(addition(X, Y))), Z)
66.43/66.68	= { by lemma 50 }
66.43/66.68	  multiplication(multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), addition(multiplication(antidomain(X), addition(Y, one)), antidomain(addition(X, Y)))), Z)
66.43/66.68	= { by axiom 11 (multiplicative_associativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(addition(multiplication(antidomain(X), addition(Y, one)), antidomain(addition(X, Y))), Z))
66.43/66.68	= { by axiom 5 (additive_commutativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(addition(antidomain(addition(X, Y)), multiplication(antidomain(X), addition(Y, one))), Z))
66.43/66.68	= { by axiom 5 (additive_commutativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(addition(antidomain(addition(Y, X)), multiplication(antidomain(X), addition(Y, one))), Z))
66.43/66.68	= { by axiom 5 (additive_commutativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(addition(antidomain(addition(Y, X)), multiplication(antidomain(X), addition(one, Y))), Z))
66.43/66.68	= { by axiom 5 (additive_commutativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(addition(multiplication(antidomain(X), addition(one, Y)), antidomain(addition(Y, X))), Z))
66.43/66.68	= { by lemma 81 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(addition(antidomain(X), addition(antidomain(addition(Y, X)), multiplication(antidomain(X), Y))), Z))
66.43/66.68	= { by lemma 58 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(addition(antidomain(X), addition(antidomain(addition(Y, X)), multiplication(antidomain(X), addition(Y, X)))), Z))
66.43/66.68	= { by axiom 12 (additive_associativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(addition(addition(antidomain(X), antidomain(addition(Y, X))), multiplication(antidomain(X), addition(Y, X))), Z))
66.43/66.68	= { by lemma 49 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(addition(addition(antidomain(X), antidomain(addition(Y, X))), multiplication(addition(antidomain(X), antidomain(addition(Y, X))), addition(Y, X))), Z))
66.43/66.68	= { by lemma 47 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication(addition(antidomain(X), antidomain(addition(Y, X))), addition(addition(Y, X), one)), Z))
66.43/66.68	= { by axiom 5 (additive_commutativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication(addition(antidomain(X), antidomain(addition(X, Y))), addition(addition(Y, X), one)), Z))
66.43/66.68	= { by axiom 5 (additive_commutativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication(addition(antidomain(addition(X, Y)), antidomain(X)), addition(addition(Y, X), one)), Z))
66.43/66.68	= { by axiom 13 (order_1) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication($$fresh(leq(antidomain(addition(X, Y)), antidomain(X)), $$true, antidomain(addition(X, Y)), antidomain(X)), addition(addition(Y, X), one)), Z))
66.43/66.68	= { by lemma 93 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication($$fresh(leq(antidomain(addition(X, Y)), antidomain(multiplication(domain(addition(X, Y)), X))), $$true, antidomain(addition(X, Y)), antidomain(X)), addition(addition(Y, X), one)), Z))
66.43/66.68	= { by lemma 95 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication($$fresh($$true, $$true, antidomain(addition(X, Y)), antidomain(X)), addition(addition(Y, X), one)), Z))
66.43/66.68	= { by axiom 2 (order_1) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication(antidomain(X), addition(addition(Y, X), one)), Z))
66.43/66.68	= { by axiom 12 (additive_associativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication(antidomain(X), addition(Y, addition(X, one))), Z))
66.43/66.68	= { by lemma 45 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication(antidomain(X), addition(X, addition(one, Y))), Z))
66.43/66.68	= { by lemma 50 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication(antidomain(X), addition(one, Y)), Z))
66.43/66.68	= { by axiom 5 (additive_commutativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), addition(Y, one))), multiplication(multiplication(antidomain(X), addition(Y, one)), Z))
66.43/66.68	= { by lemma 43 }
66.43/66.68	  zero
66.43/66.68	
66.43/66.68	Lemma 117: multiplication(antidomain(X), domain(addition(X, multiplication(domain(multiplication(X, Y)), Z)))) = zero.
66.43/66.68	Proof:
66.43/66.68	  multiplication(antidomain(X), domain(addition(X, multiplication(domain(multiplication(X, Y)), Z))))
66.43/66.68	= { by axiom 6 (multiplicative_left_identity) }
66.43/66.68	  multiplication(one, multiplication(antidomain(X), domain(addition(X, multiplication(domain(multiplication(X, Y)), Z)))))
66.43/66.68	= { by lemma 35 }
66.43/66.68	  multiplication(antidomain(zero), multiplication(antidomain(X), domain(addition(X, multiplication(domain(multiplication(X, Y)), Z)))))
66.43/66.68	= { by lemma 43 }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(multiplication(antidomain(X), domain(multiplication(X, Y)))), multiplication(multiplication(antidomain(X), domain(multiplication(X, Y))), Z))), multiplication(antidomain(X), domain(addition(X, multiplication(domain(multiplication(X, Y)), Z)))))
66.43/66.68	= { by lemma 65 }
66.43/66.68	  multiplication(antidomain(multiplication(one, multiplication(multiplication(antidomain(X), domain(multiplication(X, Y))), Z))), multiplication(antidomain(X), domain(addition(X, multiplication(domain(multiplication(X, Y)), Z)))))
66.43/66.68	= { by axiom 6 (multiplicative_left_identity) }
66.43/66.68	  multiplication(antidomain(multiplication(multiplication(antidomain(X), domain(multiplication(X, Y))), Z)), multiplication(antidomain(X), domain(addition(X, multiplication(domain(multiplication(X, Y)), Z)))))
66.43/66.68	= { by axiom 11 (multiplicative_associativity) }
66.43/66.68	  multiplication(antidomain(multiplication(antidomain(X), multiplication(domain(multiplication(X, Y)), Z))), multiplication(antidomain(X), domain(addition(X, multiplication(domain(multiplication(X, Y)), Z)))))
66.43/66.68	= { by lemma 111 }
69.01/69.24	  zero
69.01/69.24	
69.01/69.24	Lemma 118: addition(domain(X), domain(Y)) = domain(addition(X, Y)).
69.01/69.24	Proof:
69.01/69.24	  addition(domain(X), domain(Y))
69.01/69.24	= { by axiom 5 (additive_commutativity) }
69.01/69.24	  addition(domain(Y), domain(X))
69.01/69.24	= { by axiom 23 (domain4) }
69.01/69.24	  addition(antidomain(antidomain(Y)), domain(X))
69.01/69.24	= { by lemma 115 }
69.01/69.24	  antidomain(multiplication(antidomain(X), antidomain(Y)))
69.01/69.24	= { by lemma 109 }
69.01/69.24	  antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y))))
69.01/69.24	= { by axiom 6 (multiplicative_left_identity) }
69.01/69.24	  multiplication(one, antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))))
69.01/69.24	= { by lemma 38 }
69.01/69.24	  multiplication(addition(one, antidomain(domain(antidomain(Y)))), antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))))
69.01/69.24	= { by lemma 96 }
69.01/69.24	  multiplication(addition(one, multiplication(antidomain(domain(antidomain(Y))), antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))))), antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))))
69.01/69.24	= { by lemma 87 }
69.01/69.24	  multiplication(addition(one, multiplication(antidomain(domain(antidomain(Y))), antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))))), codomain(antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y))))))
69.01/69.24	= { by lemma 70 }
69.01/69.24	  addition(codomain(antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y))))), multiplication(antidomain(domain(antidomain(Y))), multiplication(antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))), codomain(antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y))))))))
69.01/69.24	= { by lemma 87 }
69.01/69.24	  addition(antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))), multiplication(antidomain(domain(antidomain(Y))), multiplication(antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))), codomain(antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y))))))))
69.01/69.24	= { by lemma 53 }
69.01/69.24	  addition(antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))), multiplication(antidomain(domain(antidomain(Y))), antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y))))))
69.01/69.24	= { by lemma 96 }
69.01/69.24	  addition(antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))), antidomain(domain(antidomain(Y))))
69.01/69.24	= { by axiom 5 (additive_commutativity) }
69.01/69.24	  addition(antidomain(domain(antidomain(Y))), antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))))
69.01/69.24	= { by lemma 59 }
69.01/69.24	  addition(antidomain(antidomain(Y)), antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))))
69.01/69.24	= { by axiom 23 (domain4) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(domain(antidomain(Y)), multiplication(antidomain(X), antidomain(Y)))))
69.01/69.24	= { by lemma 109 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(Y))))
69.01/69.24	= { by lemma 67 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), addition(antidomain(Y), domain(X)))))
69.01/69.24	= { by lemma 115 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(antidomain(X), Y)))))
69.01/69.24	= { by axiom 6 (multiplicative_left_identity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(one, multiplication(antidomain(X), Y))))))
69.01/69.24	= { by axiom 9 (additive_identity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, zero), multiplication(antidomain(X), Y))))))
69.01/69.24	= { by lemma 58 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, zero), multiplication(antidomain(X), addition(Y, X)))))))
69.01/69.24	= { by axiom 11 (multiplicative_associativity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(multiplication(addition(one, zero), antidomain(X)), addition(Y, X))))))
69.01/69.24	= { by lemma 103 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(multiplication(addition(one, zero), antidomain(X)), domain(addition(Y, X)))))))
69.01/69.24	= { by axiom 11 (multiplicative_associativity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, zero), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 42 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y)))), multiplication(coantidomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y)))))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 11 (multiplicative_associativity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(multiplication(antidomain(X), domain(addition(X, Y))), multiplication(coantidomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 115 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(addition(antidomain(Y), domain(X)), multiplication(multiplication(antidomain(X), domain(addition(X, Y))), multiplication(coantidomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 5 (additive_commutativity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(addition(domain(X), antidomain(Y)), multiplication(multiplication(antidomain(X), domain(addition(X, Y))), multiplication(coantidomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 110 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(addition(domain(X), antidomain(Y)), multiplication(multiplication(antidomain(X), domain(addition(X, Y))), multiplication(addition(coantidomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))), codomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y)))))), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 31 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(addition(domain(X), antidomain(Y)), multiplication(multiplication(antidomain(X), domain(addition(X, Y))), multiplication(one, coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y))))))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 6 (multiplicative_left_identity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(addition(domain(X), antidomain(Y)), multiplication(multiplication(antidomain(X), domain(addition(X, Y))), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(addition(X, Y)))))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 111 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(addition(domain(X), antidomain(Y)), multiplication(multiplication(antidomain(X), domain(addition(X, Y))), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero)))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 11 (multiplicative_associativity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(addition(domain(X), antidomain(Y)), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 4 (left_distributivity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, addition(multiplication(domain(X), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero)))))), multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero)))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 11 (multiplicative_associativity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, addition(multiplication(multiplication(domain(X), antidomain(X)), multiplication(domain(addition(X, Y)), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero))))), multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero)))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 33 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, addition(multiplication(zero, multiplication(domain(addition(X, Y)), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero))))), multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero)))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 8 (left_annihilation) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, addition(zero, multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero)))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 28 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(multiplication(antidomain(X), domain(addition(X, Y))), codomain(zero))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 11 (multiplicative_associativity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(antidomain(X), multiplication(domain(addition(X, Y)), codomain(zero)))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 20 (codomain4) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(coantidomain(zero))))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 32 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(one)))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 26 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(antidomain(X), multiplication(domain(addition(X, Y)), zero))))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 10 (right_annihilation) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(multiplication(antidomain(X), zero)))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 10 (right_annihilation) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), coantidomain(zero))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by lemma 32 }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), multiplication(domain(addition(X, Y)), one)))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 7 (multiplicative_right_identity) }
69.01/69.24	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), domain(addition(X, Y))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.24	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(addition(one, multiplication(antidomain(Y), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(X), domain(addition(Y, X))))))))
69.01/69.25	= { by lemma 70 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))))))))
69.01/69.25	= { by lemma 109 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))))))
69.01/69.25	= { by axiom 7 (multiplicative_right_identity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), one))))
69.01/69.25	= { by lemma 32 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)))))
69.01/69.25	= { by axiom 7 (multiplicative_right_identity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), multiplication(coantidomain(zero), one)))))
69.01/69.25	= { by lemma 32 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), multiplication(coantidomain(zero), coantidomain(zero))))))
69.01/69.25	= { by lemma 116 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), multiplication(coantidomain(zero), coantidomain(multiplication(domain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)))))))))
69.01/69.25	= { by axiom 11 (multiplicative_associativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)), coantidomain(multiplication(domain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero))))))))
69.01/69.25	= { by axiom 23 (domain4) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)), coantidomain(multiplication(antidomain(antidomain(multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero))))))))
69.01/69.25	= { by lemma 108 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(domain(antidomain(multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)), coantidomain(multiplication(antidomain(antidomain(multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)))))))))
69.01/69.25	= { by axiom 23 (domain4) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(antidomain(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))))), multiplication(multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)), coantidomain(multiplication(antidomain(antidomain(multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)))))))))
69.01/69.25	= { by axiom 23 (domain4) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(domain(multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)), coantidomain(multiplication(antidomain(antidomain(multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)))))))))
69.01/69.25	= { by lemma 59 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)), coantidomain(multiplication(antidomain(antidomain(multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)))))))))
69.01/69.25	= { by axiom 23 (domain4) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)), coantidomain(multiplication(domain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)))))))))
69.01/69.25	= { by lemma 116 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero)), coantidomain(zero))))))
69.01/69.25	= { by axiom 11 (multiplicative_associativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), multiplication(coantidomain(zero), coantidomain(zero)))))))
69.01/69.25	= { by lemma 32 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), multiplication(coantidomain(zero), one))))))
69.01/69.25	= { by axiom 7 (multiplicative_right_identity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(zero))))))
69.01/69.25	= { by lemma 117 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), coantidomain(multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), domain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))))))))))
69.01/69.25	= { by lemma 110 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(addition(antidomain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))), domain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))))))), coantidomain(multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), domain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))))))))))
69.01/69.25	= { by lemma 34 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), multiplication(one, coantidomain(multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), domain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X)))))))))))))))
69.01/69.25	= { by axiom 6 (multiplicative_left_identity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), coantidomain(multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), domain(addition(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(domain(multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))), multiplication(antidomain(Y), multiplication(multiplication(antidomain(X), domain(addition(Y, X))), multiplication(antidomain(X), domain(addition(Y, X))))))))))))))
69.01/69.25	= { by lemma 117 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), coantidomain(zero)))))
69.01/69.25	= { by lemma 32 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), multiplication(antidomain(multiplication(antidomain(X), domain(addition(Y, X)))), one))))
69.01/69.25	= { by axiom 7 (multiplicative_right_identity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(multiplication(antidomain(X), domain(addition(Y, X)))))))
69.01/69.25	= { by lemma 115 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), addition(antidomain(domain(addition(Y, X))), domain(X)))))
69.01/69.25	= { by lemma 59 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), addition(antidomain(addition(Y, X)), domain(X)))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), addition(domain(X), antidomain(addition(Y, X))))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), addition(domain(X), antidomain(addition(X, Y))))))
69.01/69.25	= { by lemma 73 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(addition(X, Y)))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(X), antidomain(addition(Y, X)))))
69.01/69.25	= { by lemma 62 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(antidomain(X), domain(addition(Y, X))), antidomain(addition(Y, X)))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(domain(addition(Y, X)), antidomain(X)), antidomain(addition(Y, X)))))
69.01/69.25	= { by axiom 6 (multiplicative_left_identity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(domain(addition(Y, X)), multiplication(one, antidomain(X))), antidomain(addition(Y, X)))))
69.01/69.25	= { by lemma 34 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(domain(addition(Y, X)), multiplication(addition(antidomain(addition(Y, X)), domain(addition(Y, X))), antidomain(X))), antidomain(addition(Y, X)))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(domain(addition(Y, X)), multiplication(addition(domain(addition(Y, X)), antidomain(addition(Y, X))), antidomain(X))), antidomain(addition(Y, X)))))
69.01/69.25	= { by axiom 4 (left_distributivity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(domain(addition(Y, X)), addition(multiplication(domain(addition(Y, X)), antidomain(X)), multiplication(antidomain(addition(Y, X)), antidomain(X)))), antidomain(addition(Y, X)))))
69.01/69.25	= { by lemma 45 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(multiplication(domain(addition(Y, X)), antidomain(X)), addition(multiplication(antidomain(addition(Y, X)), antidomain(X)), domain(addition(Y, X)))), antidomain(addition(Y, X)))))
69.01/69.25	= { by lemma 45 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(multiplication(antidomain(addition(Y, X)), antidomain(X)), addition(domain(addition(Y, X)), multiplication(domain(addition(Y, X)), antidomain(X)))), antidomain(addition(Y, X)))))
69.01/69.25	= { by lemma 47 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(multiplication(antidomain(addition(Y, X)), antidomain(X)), multiplication(domain(addition(Y, X)), addition(antidomain(X), one))), antidomain(addition(Y, X)))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(multiplication(antidomain(addition(Y, X)), antidomain(X)), multiplication(domain(addition(Y, X)), addition(one, antidomain(X)))), antidomain(addition(Y, X)))))
69.01/69.25	= { by lemma 38 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(multiplication(antidomain(addition(Y, X)), antidomain(X)), multiplication(domain(addition(Y, X)), one)), antidomain(addition(Y, X)))))
69.01/69.25	= { by axiom 7 (multiplicative_right_identity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(addition(multiplication(antidomain(addition(Y, X)), antidomain(X)), domain(addition(Y, X))), antidomain(addition(Y, X)))))
69.01/69.25	= { by lemma 62 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(multiplication(antidomain(addition(Y, X)), antidomain(X)), antidomain(addition(Y, X)))))
69.01/69.25	= { by axiom 11 (multiplicative_associativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), multiplication(antidomain(X), antidomain(addition(Y, X))))))
69.01/69.25	= { by lemma 73 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), addition(domain(addition(Y, X)), multiplication(antidomain(X), antidomain(addition(Y, X)))))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), addition(multiplication(antidomain(X), antidomain(addition(Y, X))), domain(addition(Y, X))))))
69.01/69.25	= { by axiom 6 (multiplicative_left_identity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), addition(multiplication(antidomain(X), antidomain(addition(Y, X))), multiplication(one, domain(addition(Y, X)))))))
69.01/69.25	= { by lemma 38 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), addition(multiplication(antidomain(X), antidomain(addition(Y, X))), multiplication(addition(one, antidomain(X)), domain(addition(Y, X)))))))
69.01/69.25	= { by lemma 99 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), addition(domain(addition(Y, X)), multiplication(antidomain(X), addition(antidomain(addition(Y, X)), domain(addition(Y, X))))))))
69.01/69.25	= { by lemma 34 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), addition(domain(addition(Y, X)), multiplication(antidomain(X), one)))))
69.01/69.25	= { by axiom 7 (multiplicative_right_identity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), addition(domain(addition(Y, X)), antidomain(X)))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), addition(antidomain(X), domain(addition(Y, X))))))
69.01/69.25	= { by lemma 67 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(addition(Y, X)), antidomain(X))))
69.01/69.25	= { by lemma 59 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(domain(addition(Y, X))), antidomain(X))))
69.01/69.25	= { by lemma 93 }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(domain(addition(Y, X))), antidomain(multiplication(domain(addition(X, Y)), X)))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(multiplication(antidomain(domain(addition(Y, X))), antidomain(multiplication(domain(addition(Y, X)), X)))))
69.01/69.25	= { by lemma 96 }
69.01/69.25	  addition(domain(Y), antidomain(antidomain(domain(addition(Y, X)))))
69.01/69.25	= { by lemma 59 }
69.01/69.25	  addition(domain(Y), antidomain(antidomain(addition(Y, X))))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(Y), antidomain(antidomain(addition(X, Y))))
69.01/69.25	= { by axiom 23 (domain4) }
69.01/69.25	  addition(domain(Y), domain(addition(X, Y)))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(addition(X, Y)), domain(Y))
69.01/69.25	= { by axiom 7 (multiplicative_right_identity) }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(domain(Y), one))
69.01/69.25	= { by lemma 34 }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(domain(Y), addition(antidomain(addition(X, Y)), domain(addition(X, Y)))))
69.01/69.25	= { by lemma 99 }
69.01/69.25	  addition(multiplication(domain(Y), antidomain(addition(X, Y))), multiplication(addition(one, domain(Y)), domain(addition(X, Y))))
69.01/69.25	= { by lemma 41 }
69.01/69.25	  addition(multiplication(domain(Y), antidomain(addition(X, Y))), multiplication(one, domain(addition(X, Y))))
69.01/69.25	= { by axiom 6 (multiplicative_left_identity) }
69.01/69.25	  addition(multiplication(domain(Y), antidomain(addition(X, Y))), domain(addition(X, Y)))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(domain(Y), antidomain(addition(X, Y))))
69.01/69.25	= { by lemma 86 }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(domain(Y), coantidomain(domain(addition(X, Y)))))
69.01/69.25	= { by lemma 105 }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(coantidomain(domain(addition(X, Y))), domain(Y)))
69.01/69.25	= { by lemma 86 }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(antidomain(addition(X, Y)), domain(Y)))
69.01/69.25	= { by axiom 5 (additive_commutativity) }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(antidomain(addition(Y, X)), domain(Y)))
69.01/69.25	= { by axiom 6 (multiplicative_left_identity) }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(one, multiplication(antidomain(addition(Y, X)), domain(Y))))
69.01/69.25	= { by lemma 35 }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(antidomain(zero), multiplication(antidomain(addition(Y, X)), domain(Y))))
69.01/69.25	= { by lemma 60 }
69.01/69.25	  addition(domain(addition(X, Y)), multiplication(antidomain(multiplication(antidomain(addition(Y, X)), Y)), multiplication(antidomain(addition(Y, X)), domain(Y))))
69.01/69.25	= { by lemma 72 }
69.01/69.25	  addition(domain(addition(X, Y)), zero)
69.01/69.25	= { by axiom 9 (additive_identity) }
69.01/69.25	  domain(addition(X, Y))
69.01/69.25	
69.01/69.25	Goal 1 (goals): addition(domain(multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(multiplication(sK3_goals_X0, domain(sK1_goals_X2)))) = domain(multiplication(sK3_goals_X0, addition(domain(sK2_goals_X1), domain(sK1_goals_X2)))).
69.01/69.25	Proof:
69.01/69.25	  addition(domain(multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(multiplication(sK3_goals_X0, domain(sK1_goals_X2))))
69.01/69.25	= { by lemma 104 }
69.01/69.25	  addition(domain(multiplication(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(sK3_goals_X0, domain(sK1_goals_X2))))
69.01/69.25	= { by lemma 104 }
69.01/69.25	  addition(domain(multiplication(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(sK3_goals_X0, sK1_goals_X2)))
69.01/69.25	= { by lemma 118 }
69.01/69.25	  domain(addition(multiplication(sK3_goals_X0, sK2_goals_X1), multiplication(sK3_goals_X0, sK1_goals_X2)))
69.01/69.25	= { by axiom 3 (right_distributivity) }
69.01/69.25	  domain(multiplication(sK3_goals_X0, addition(sK2_goals_X1, sK1_goals_X2)))
69.01/69.25	= { by lemma 104 }
69.01/69.25	  domain(multiplication(sK3_goals_X0, domain(addition(sK2_goals_X1, sK1_goals_X2))))
69.01/69.25	= { by lemma 118 }
69.01/69.25	  domain(multiplication(sK3_goals_X0, addition(domain(sK2_goals_X1), domain(sK1_goals_X2))))
69.01/69.25	% SZS output end Proof
69.01/69.25	
69.01/69.25	RESULT: Theorem (the conjecture is true).
69.01/69.26	EOF